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Anatomically based human hand modeling and simulation
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Anatomically based human hand modeling and simulation
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Content
Anatomically Based Human Hand Modeling and Simulation
by
Bohan Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(Computer Science)
December 2021
Copyright 2021 Bohan Wang
I dedicate this thesis to my family for love and encouragement, my PhD advisor Prof.
Jernej Barbic for grateful support, inspiration, and guidance, my lab mates for help
and discussions, and committee members for reading and improving the thesis.
ii
ii
vi
vii
xvi
Table of Contents
Dedication
List of Tables
List of Figures
Abstract
Chapter 1: Introduction 1
1.1 Hand modeling and simulation using stabilized magnetic resonance imaging . . . . 1
1.2 Modeling of personalized anatomy using plastic strains . . . . . . . . . . . . . . . 6
Chapter 2: Related Work 10
2.1 Anatomically based simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Animating the human hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Acquiring medical images of human hand . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Segmentation techniques for medical images of human hands . . . . . . . . . . . . 13
2.5 Medical image registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Non-rigid iterative closest point method . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Geometric shape modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.9 Anatomy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 3: Acquiring MRI Scans in Many Poses 18
Chapter 4: Hand Simulation using Hand Skeleton and Soft Tissue 23
4.1 Segmenting MRI data into bone meshes . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Skeleton kinematic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Generation of the tet mesh outer surface . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Resolving discrepancies due to a small pose change of the subject in the mold prior
to MRI scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Generation of the tet mesh inner boundary . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Constraining the tet mesh to the bones . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.8 Embedding of surface triangle meshes into tet mesh . . . . . . . . . . . . . . . . . 39
4.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
iii
4.10 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter 5: Extracting Anatomy using Plastic Strains 45
5.1 Attachments and medical image constraints . . . . . . . . . . . . . . . . . . . . . 47
5.2 Plastic deformation gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Shape deformation of attached objects . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Solving the optimization problem for attached objects . . . . . . . . . . . . . . . . 53
5.5 Singular lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.6 Unattached objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.7 Gradient and Hessian of Polar(F) . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.8 Comparison to standard shape deformation methods . . . . . . . . . . . . . . . . . 67
5.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.9.1 Hand muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.9.1.1 Marking the muscles in MRI scans . . . . . . . . . . . . . . . . 72
5.9.1.2 Attachments to bones . . . . . . . . . . . . . . . . . . . . . . . 72
5.9.1.3 Direct attempt using segmentation: . . . . . . . . . . . . . . . . 72
5.9.1.4 Removing interpenetrations of muscles . . . . . . . . . . . . . . 73
5.9.2 Hip bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.9.3 Liver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.9.4 Hip muscle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.9.5 Comparison to variational methods . . . . . . . . . . . . . . . . . . . . . 75
5.9.6 Comparison to iterative closest point methods . . . . . . . . . . . . . . . . 78
5.9.7 Comparison to ShapeOP . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.9.8 Comparison to incremental plasticity . . . . . . . . . . . . . . . . . . . . 81
5.10 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter 6: Comprehensive Modeling of the Human Hand 85
6.1 Hand muscle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.1 Muscle pose space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.2 Muscle plastic strain extraction . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.3 Initial guess of muscle tet mesh in non-neutral example poses . . . . . . . 93
6.1.4 Generating muscle volumetric plastic strains in example poses . . . . . . . 95
6.1.5 Muscle simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Hand tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.1 Tendon simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.2 Tendon preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2.1 Tendon extraction . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2.2 Tendon registration . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2.3 Tendon sliding locations . . . . . . . . . . . . . . . . . . . . . . 106
6.3 Hand fascia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4 Hand fat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter 7: Conclusion and Future Work 121
iv
References 126
Appendices 135
A Plastic strain Laplacian and its nullspace . . . . . . . . . . . . . . . . . . . . . . . 136
B First and second derivatives of elastic energy with respect to plastic strain . . . . . 137
C Proof of singular lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
D Proof of nullspace lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
E Second derivative of polar decomposition . . . . . . . . . . . . . . . . . . . . . . 140
F Formulas for A
k
,b
k
,c
k
(Equation 5.6) . . . . . . . . . . . . . . . . . . . . . . . . 141
G Meeting constraints by scaling the elastic stiffness . . . . . . . . . . . . . . . . . . 142
v
List of Tables
4.1 Bone rig errors for each joint. We list them separately for rotations (R) and
translations (x). Here, FM
R
and FM
x
are the absolute errors under our Full Model
(i.e., quadratic fitting of
ˆ
R and ˆ x), while PM
R
and PM
x
are the errors under a
Partial Mode, in which
ˆ
R= I and ˆ x= 0. The error is computed as the average
difference between the output of our bone rig and the ground truth (segmented
MRI data), across all 12 poses (male subject). It can be seen that the fitting of
ˆ
R
and ˆ x decreases the error. The first and second halves of the table present 1-DOF
and 2-DOF joints, respectively. Observe that the pinky finger has the largest error;
this is because it is the smallest. Consequently, it is resolved with fewer voxels in
the MRI scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1 Solving a single linear system of equations with H, using the conjugate gradients
(CG) and our method. The naive direct solver failed in all cases. Note that H is a
dense 6m× 6m matrix. The column t
prep
gives a common pre-processing time for
both CG and our method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 The statistics for our examples: #vtx = number of vertices; #ele = number of
tetrahedra inM ; #iter = number of ICP iterations; “time” = total computation time
to compute the output shape; “attached” indicates whether the object is attached;
e
init
= error between our template mesh and the ICP constraints; e
final
= error be-
tween our result and the ICP markers. The first and second reported error numbers
are the average and maximum errors, respectively. In the hand example, there are
17 groups of muscles; “min”, “med” and “max” refer to the smallest, representa-
tive median, and largest muscle groups; “max-m” is the example with the largest
number of ICP constraints. Observe that our markers are sparse; the ratio between
the number of markers and the number of vertices is 0.3%–7.3% in our examples. 70
6.1 Comparisons between simulated and ground truth skins. Model m
1
simulates
each organ separately (as proposed in this Chapter), whereas model m
2
simulates
all soft tissues using a single mesh (as proposed in this Chapter 4). Column “%”
denotes the ratio between the value in m
1
and the value in m
2
. The values in bold
face are cases where our model is slightly worse than m
2
. The first five rows are
the example poses used for our simulation, whereas the last six rows are the unseen
poses. All distances are represented in millimeter units. . . . . . . . . . . . . . . . 116
vi
List of Figures
1.1 Opposition of the thumb: Our method produces an accurate data-driven skele-
ton mesh kinematic “rig.” Here, we used our rig to reliably reproduce the well-
known opposition of the thumb to all other four fingers. The solid soft tissue was
computed using an FEM simulation attached to the articulated bone meshes. The
fingers do not merely touch but also orient to be co-planar at the contact location,
subject to biomechanical limits. The right-most four images show representative
FEM frames of the opposition between the thumb and the pinky finger. . . . . . . 2
1.2 Left: Kurihara’s CT scan [71]. Reprinted with permission from Eurographics.
Right: Our MRI scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Second-order methods produce spiky outputs as the tet mesh is refined. Here,
we illustrate the output of an FEM static solver under a single hard point constraint
(seen in (a)), that is, we minimize the FEM elastic energy under the depicted hard
point constraint. Bunny is fixed at the bottom. Poissons’s ratio is 0.49. As we
increase the tet mesh resolution in (b)–(e), the spike becomes progressively nar-
rower, which is undesirable. Changing the elastic stiffness (Young’s modulus) of
the bunny does not help (see Appendix G). Converting the constraint into a soft
spring constraint also does not help (f); now, the constraint is not satisfied. . . . . 8
3.1 12 MRI-scanned poses (1 subject; male; late 20s). Four rows are the cloned
plastic hand (the ”ground truth” shapes), our FEM result, skeleton mesh obtained
from the MRI scan, and the MRI scan. The last four plastic poses have a plastic
stand (seen on the left of each image) to easily stand up on a table. FEM closely
matches the plastic hand in all poses, despite not actually using plastic shapes
anywhere in our system (except the neutral shape: column 5 in the top set). The
solid black arrows indicate the logical sequence of steps; the FEM and cloned
hands are shown adjacent to each other for easier comparison. The image can be
zoomed in on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Stabilized MRI scanning: (a) The mold with a plastic hand before cutting. (b1,2,3)
Mold has been cut into two parts. (c) Hand secured into the mold. (d) MRI scan-
ning with the mold. Clinical MRI scanner is manufactured by General Electric.
Magnetic field strength of 3T, and resolution of 0 .5 x 0.5 x 0.5mm
3
. . . . . . . . . 20
vii
3.3 We “cloned” this hand onto a plastic hand. (a) Mixing the AljaSafe. (b) The
hand in a bucket filled with liquid AljaSafe. (c) Hand removed, leaving a hand-
shaped hole. (d, e) Casting a liquid plastic into solidified AljaSafe. (f) Removed
the AljaSafe to obtain the plastic hand. (g) Photo of plastic hand. (h) Rendered
3D-scanned mesh of plastic hand. Scanner: Artec Spider. Note the high-resolution
detail in g, h (This can be zoomed in on in PDF). . . . . . . . . . . . . . . . . . . 20
3.4 Our method outperforms prior hand bone segmentation methods. It is chal-
lenging to segment the “metacarpal II” bone (red rectangle) for two reasons: (1)
the MRI signal (depicted in A) on the bone is not of uniform intensity, and (2)
the neighboring bone “metacarpal III” (yellow rectangle; only head of the bone is
visible in this 2D view) has very similar signal intensity to “metacarpal II.” Our
method (depicted in B) successfully segmented this challenging case. The prior
method [99] uses a region-based active contour method and produces a suboptimal
result: a significant part of II’s bone head is missing (C). . . . . . . . . . . . . . . 22
4.1 Our segmentation interface. The result of the first step of our 3D Laplacian-
based segmentation. The segmented bone is red. Yellow is the local segmentation
region selected by the user for this bone. . . . . . . . . . . . . . . . . . . . . . . 24
4.2 The bones and joints of a human hand. Our joint hierarchy is indicated with a
superimposition in blue, with the number of joint DOFs indicated. . . . . . . . . . 27
4.3 Transformation (x
i
,R
i
) is expressed in the parent’s coordinate system. It trans-
forms the neutral child bone into pose i. . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 The rotation axes and rotation centers of our joints. The bone meshes are
shown in a transparent style. The axes are shown as red lines. The centers are
shown as black dots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Fitting the residual ˆ x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Simulation tet mesh. Top row: outer and inner tet surface mesh. InT
outer
, note
the greater resolution at the folds. Bottom: three cutaway views of the FEM mesh. 34
4.7 Tet mesh creation. Top: meshT
inner
; removing tets at the joints; tet mesh con-
strained vertices. Bottom-left: the two-step remeshing process at the folds pro-
duces good triangles and is necessary: under a single-step process, bad triangles
appear. Bottom-right: illustration of sweeping in 2D. . . . . . . . . . . . . . . . . 36
4.8 Young’s modulus of various human musculoskeletal tissues. Note the unit
kPa= 1000N/m
2
. From [36]. Reprinted with permission from the AAAS. . . . . . 37
4.9 Spatially varying mesh resolution and materials. Fold formation is improved by
increasing the tet mesh resolution under the finger joints (via T
outer
; Sections 4.3
and 4.4), and by making the material in the same region 6x softer (Section 4.7).
Our method produces a quality horizontal fold across the entire length of the palm.
Other methods only produce a partial fold, or no fold at all. . . . . . . . . . . . . 38
viii
4.10 Anatomical bone rig stabilizes FEM soft tissue simulations. Left: non-anatomical
rig, created by pivoting bones around the center of the parent bone’s head. Bones
pinch elastic material, causing artefacts. Right: our rig produces stable soft tissue
simulations. Both results use FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.11 Comparison of FEM simulation to skinning. Skinning weights were computed
using Maya’s geodesic voxel method. Other Maya skinning weight methods yield
even worse results. Both methods use our bone rig. . . . . . . . . . . . . . . . . . 40
4.12 The opposition of the thumb for our female subject (late 40s). The bone and
FEM geometry of three animation frames. . . . . . . . . . . . . . . . . . . . . . . 41
4.13 Comparison of our FEM results to photographs, on two “in-between” poses
(i.e., not a part of the acquired 12 pose-dataset). The FEM results are rendered
using our high-resolution mesh (11M triangles) with subsurface scattering; indi-
vidual pores are visible as geometric features. The image can be zoomed in on.
Generally, our FEM method produces a good qualitative visual match to the pho-
tographs: the finger and palm appear “organic”, and the two main palmar lines
are in the correct location and are produced by FEM as geometric mesh features
(folds), as opposed to only a texture map illusion. This figure also demonstrates
the limitations of our method. We do not model the wrinkling of the skin, which
is absent in the FEM result, but visible in photographs. We also do not simulate
muscle activation and, hence, the soft tissue at the root of the thumb is flatter in
FEM vs. in photographs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.14 Our bone rig also improves skinning. Left: skinning with a non-anatomical
bone rig, created manually by placing the joints at the center of the parent bone’s
bonehead. Right: skinning with our anatomical rig. No FEM simulation was used
in this example, only skinning. It can be seen that skinning works better when it
uses our anatomical bone rig (see the dent at the joint). Both results use Maya’s
“geodesic voxel” skinning weights. . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 We optimized the shape of this hand muscle (Adductor Pollicis) to match the
MRI scan. (a) Template shape [26]; (b) representative MRI slice [134]; we rigidly
aligned the template mesh onto the markers in the MRI scan, producing the white
contour that is obviously incorrect (deeply penetrates the bone and extends out of
the volume of the MRI-scanned hand); (c) our output shape, optimized to the MRI
scan; the white contour now matches the scan; (d, e) large anisotropic spatially
varying strains accommodated by our method (d: maximum principal stretch; e:
ratio between maximum and minimum principal stretch). . . . . . . . . . . . . . . 46
5.2 Our shape deformation setup. Our optimization discovers plastic strains F
p
and
vertex positions x so that the model is in elastic equilibrium under the attachments
while meeting the medical image landmark and closest point constraints as closely
as possible. The presence of attachments is optional; our work address both at-
tached and unattached objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
ix
5.3 Comparison to augmented deformation transfer. The beam’s attachments (red)
cause the beam to bend, whereas the ICP markers (blue) cause it to stretch 2x in
one of the two transverse directions. Our method can easily recover such a shape
deformation, whereas deformation transfer [118] cannot, even if augmented with
an elastic energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Plastic and elastic deformation gradient for a single tet. . . . . . . . . . . . . . 49
5.5 Seventeen muscles of the human hand extracted from MRI. Observe that the
template hand is larger than the scanned hand. The pose is also different. Our
method solves this using bone attachments. . . . . . . . . . . . . . . . . . . . . . 49
5.6 Comparison to variational shape modeling. In a variational method [21], the
wiggles increase if one imposes a stricter constraint satisfaction. First row: un-
der a small number of landmarks, variational methods with k = 2,3 produce a
smooth and reasonable result, albeit somewhat smoothening the rest shape. Mid-
dle row: under more landmarks, it becomes more difficult for variational methods
to meet the landmarks while avoiding the wiggles. Bottom row: variational meth-
ods produce wavy results. Our method meets the landmarks and produces fewer
wiggles. This is because the plastic deformation field can arbitrarily rotate and
non-uniformly scale to adapt to the inputs; the elastic energy then finally irons out
the kinks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.7 Elastic energy methods only work with dense markers. In this figure, we com-
pare to a state-of-the art medical imaging technique [91], whereby the output shape
is calculated by minimizing an elastic energy of a template shape, subject to dense
medical image markers. With dense markers, elastic energy methods work well
(left). As the constraints sparsify, elastic energy produces artefacts (middle). Our
plasticity method (right) produces a good shape, even with sparse markers. . . . . 55
5.8 Convergence plots. The iterations are presented on the X-axis, and the optimiza-
tion energy is marked on the Y-axis. The initial optimization energies are normal-
ized to 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.9 Illustration of the Singular Lemma. . . . . . . . . . . . . . . . . . . . . . . . . 61
5.10 Unattached optimization of a hip bone shape to a CT scan. The scanned bone
is smaller and has a substantially different shape from the template. . . . . . . . . 62
5.11 Unattached optimization of a liver shape to a CT scan. Our method success-
fully captures the large shape variation between the template and the scan. This
figure also demonstrates that our method makes it possible to transfer the rendering
textures and UV coordinates from the template onto the output. . . . . . . . . . . . 62
5.12 Attached optimization of a hip muscle (gluteus medius) to an MRI scan. Our
method successfully captures the large shape variation between the template and
the scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
x
5.13 Standard volume-based shape deformation methods result in wiggly and spiky
artefacts. The displayed hand Palmar interossei muscle has a tendon on one side
and no tendon on the other; both ends are attached to a bone (light blue). The
acronyms are defined in Section 5.8. The MRI landmark constraints are shown in
dark blue. The shape deformation between the template muscle and the shape in
the MRI consists of large and spatially varying stretches. Our method successfully
models this deformation. We note that spikes cannot be avoided by using, say,
a spherical region for the constraints as opposed to a point; the non-smoothness
merely moves to the spherical region boundary. . . . . . . . . . . . . . . . . . . . 66
5.14 Comparison between surface energy and volumetric energy. In both exam-
ples (bunny and hand), we performed non-rigid iterative closest point alignment
between a template triangle mesh and a collection of target ICP markers (13 for
bunny and 456 for the hand). For the bunny, we manually placed the markers to
greatly enlarge the bunny’s left ear. For the hand, we placed the markers on a
pre-existing target hand surface mesh that has different geometric proportions and
mesh topology as the template. The template is a man and target is a woman. We
then solved the ICP problem using the surface-based ARAP energy, volume-based
ARAP energy, and our volumetric plastic strains. Our method produces smooth
artefact-free outputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.15 Output ICP error histograms. Each medical image marker contributes one entry
to the histogram. The hand muscles histogram is a combined histogram for all the
17 hand muscles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.16 Minimum tetrahedral dihedral angles before and after our optimization. It
can be seen that the output angles are still relatively large. As expected, the output
angles are somewhat worse than the initial ones, as the object has undergone a
plastic deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.17 Our extracted organs match the medical image. The intersection of the output
mesh with the medical image slices is shown in green. . . . . . . . . . . . . . . . 71
5.18 Interpenetration removal. The yellow lines are muscle cross-sections in this
representative MRI slice of the hand. It is clear that our interpenetration removal
method successfully removes penetrations without modifying the interpenetration-
free sections of the boundaries of muscles. . . . . . . . . . . . . . . . . . . . . . 73
5.19 Quantitative evaluation on ground truth. We first performed a dynamic FEM
simulation with plasticity [57], whereby the back of the dragon is fixed (red), and
the head was pulled to the left with uniform force. This produced our ground truth
plastic deformation gradients. We then selected 488 sparse triangle mesh vertices
as landmarks and ICP markers, and ran our method to compute F
p
and shape x. It
is evident that F
p
and x closely match the ground truth. . . . . . . . . . . . . . . . 76
xi
5.20 Comparison to [54]. Top row: hip bone. Bottom row: liver. Red points are the
markers from the CT scan. We used the publicly available implementation of [54]
to compute the normals, followed by screened Poisson surface reconstruction us-
ing the points and computed normals [67]. We used this combination because it
produced better results than running [54] directly. Our method produces shapes
that more closely match the ground truth data. . . . . . . . . . . . . . . . . . . . 77
5.21 Comparison to surface-based methods. We compare to the “PRIMO” method [22]
because this method is a good representative choice. It also has fewest artefacts in
Figure 10 of the shape deformation survey [20]. Our method produces a clearly
superior result in the challenging scenario where the ICP markers were placed
to anisotropically stretch the beam in one transverse direction. Our method also
passes the standard shape deformation benchmark [20]. . . . . . . . . . . . . . . 77
5.22 Comparison to variational methods. A 1D string of length 1 is attached on
both ends. The left attachment is fixed. The right attachment is moved to coor-
dinate 2. Thus, the spring stretches longitudinally while attempting to obey the
two landmarks at t = 1/4 and t = 1/2. Y-axis gives the deformed 1D position of
the corresponding material point on the X-axis. Big and small dots denote the
attachments and landmarks, respectively. For each method, we plot the result un-
der a few representative parameters. The blue, red, and green curves represent the
different strengths of the landmarks. With variational methods, one has to either
give up meeting the landmarks, or the curve becomes wiggly. Similar behavior
can also be observed in 3D variational results (Figure 5.6). Our method produces a
deformation profile whereby the slope (total 1D deformation gradient) on all three
subintervals [0,1/4],[1/4,1/2],[1/2,1] approximately matches the slope implied
by the attachments and landmarks (1, 5, 1, respectively). Note that the shown
output curves are onlyC
r− 1
and not inC
r
at the two juncture points 1/4,1/2;
however, their r-th derivative is integrable and they are the optimal Cauchy limit
of a score-decreasing sequence of curves inC
r
. . . . . . . . . . . . . . . . . . . . 78
5.23 Comparison to methods that penalize the difference of neighboring affine
transformations [F b]. These methods produce a result similar to the variational
method and suffer from wiggle artefects. . . . . . . . . . . . . . . . . . . . . . . . 79
5.24 Comparison to E
bend
. We deformed the surface using the ShapeOp library. Here,
α is the strength of the marker constraints (attachments and ICP markers) relative
to the bending energy. In the beam example, our method is presented in green
wireframe and E
bend
is shown as solid. Low values of α (such as, for example,
α = 10
4
) cannot satisfy the constrains; therefore, we useα = 10
7
andα = 10
8
to
meet the constraints in the beam and muscle examples, respectively. It can be seen
that the E
bend
method suffers from volume loss in the beam example. In addition, it
fails to grow the beam (the middle is collapsed). Wiggle artefacts can be observed
in the E
bend
method on the muscle. . . . . . . . . . . . . . . . . . . . . . . . . . 80
xii
5.25 Comparison to [42]. We compare our method to two different deformation ener-
gies used in [42]. Equation 1 penalizes the volume change relative to the previous
ICP iteration. Equation 5 penalizes E
bend
relative to the previous ICP iteration (“in-
cremental plasticity”). On the beam example, our method is depicted via a green
wireframe, whereas the compared method is depicted as a solid. It can be seen that
the “incremental plasticity” method suffers from similar artefacts as other prior
methods (but to a lesser degree). However, “incremental plasticity” introduces its
own problems, namely volume loss and waviness (observable during bending the
beam) and sharp creases (that occur while growing the muscle). To eliminate any
effects of incorrect correspondence from the experiment, both examples use land-
mark constraints (depicted as blue spheres); the results are even more favorable to
our method when using ICP constraints. . . . . . . . . . . . . . . . . . . . . . . . 82
5.26 Comparison to E
bend
on the shape benchmark of [20]. . . . . . . . . . . . . . . 83
6.1 MRI slices in the coronal plane and transversal plane. Here, we show two MRI
slices of human hand. It is evident that there are several different tissues inside the
hand represented by different intensities in the MRI image. . . . . . . . . . . . . . 86
6.2 The overview of our human hand rig. We model tendons (dark purple), fat (flesh
color), muscles (light pink), bones (light yellow), and ligaments (green) around
joint regions. To improve the simulation quality, we also model two additional
fascia layers: bone fascia (light blue) and muscle fascia (light pruple). The fat
tissue is directly attached to fascia layers and ligaments. . . . . . . . . . . . . . . 86
6.3 Muscle attachments. The muscle attachment vertices are depicted as red dots. To
show the attachments clearly, the muscles are visualized using dark wireframes in
the left two figures. Further, to present the relative locations to the bones, we show
the bones in dark wireframes in the right two figures. . . . . . . . . . . . . . . . . 87
6.4 Overview of muscle simulation. A muscle is attached to more than one bone (or
tendons). The muscle surface is embedded into a tetrahedral mesh. The muscle
contraction is controlled by the plastic strain of each tetrahedron. . . . . . . . . . . 89
6.5 The muscle and joint ligament surface meshes. The muscles are presented in
red and the joint ligaments are presented in green. . . . . . . . . . . . . . . . . . . 92
6.6 Muscle preprocessing results. In step 2, we remesh the muscle surface mesh. As
demonstrated on the left, remeshing improves the triangle quality. On the right of
the figure, the target attachment locations are depicted as red dots, and the green
dots are constraints to preserve the shape of the muscle. The target attachment
locations are obtained by rigidly transforming the positions of the attachment ver-
tices at the neutral pose to the target example pose. By applying step 3, we make
the surface vertices better match the attachment locations. . . . . . . . . . . . . . . 95
xiii
6.7 Tendon simulation model. The left of the figure presents our rod simulation
model. We break the rod into linked rigid segments. Each segment is represented
by the positions of two end vertices and a normal showing the orientation. The
normal (orange) is perpendicular to the segment. The tendon in our system is sim-
ulated using such a rod. The rod goes through a few locations (red circles) that are
skinned to the closest bones. One end of the tendon is attached to the bone with a
fixed constraint. In addition, we apply a constant force (purple) at the end of the
tendon to stretch it as much as possible without invalidating the constraints. . . . . 101
6.8 Tendon simulation. Here, we show the rods representing tendons during simula-
tion. The red lines are the rigid segments and their normals. The blue dots are the
sliding locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.9 Tendon stages. In stage 1, we simulate the tendons based on the sliding con-
straints that are only rigidly transforming with the closest bones. This gives a rel-
atively correct position of the tendon. However, it does not match the MRI shape
(blue wireframe). To make it match as close as possible, we first create an initial
guess for our non-rigid registration (stage 2). As shown in (2), the resulting mesh
matches the MRI shape better. In stage 3, we do non-rigid registration to match
the MRI shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.10 Fascia meshes. (a) Bone meshes, (b) corresponding bone fascia mesh, (c) muscle
meshes, and (d) muscle fascia mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.11 Bone fascia. (a) The constraints applied to bone fascia simulation. Vertices in
green are attached to the closest bones using fixed constraints. Vertices in red are
in contact with the bone meshes. (b, c) Bone and bone fascia meshes in the fist pose.108
6.12 Muscle fascia constraints. Here, we show the constraints applied during mus-
cle fascia simulation. Vertices in green are attached to the muscles using fixed
constraints. Vertices in red are sliding against the muscle surface meshes. . . . . . 109
6.13 Fascia animation. Top: a few frames of an animation sequence of bones and
muscles. Bottom: the fascia simulation results. . . . . . . . . . . . . . . . . . . . 109
6.14 The exterior and interior surface of the fat tissue. (a) The exterior surface of
the fat tissue is the skin. (b) The interior surface of the fat tissue is composed of
the bone fascia, muscle fascia, and joint ligaments. . . . . . . . . . . . . . . . . . 110
6.15 The constraints for simulating the fat. The fat is attached to bone fascia, muscle
fascia, ligaments, and tendons. Green dots are the fixed constraints to the bone
fascia. Yellow dots are the contact constraints to the ligaments. Pink dots are
the sliding constraints against the muscle fascia, while the red dots are contact
constraints against the muscle fascia. In addition, the purple dots are the constraints
to mimic the rigidity of the nails. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.16 Hand poses used for joint ligaments, muscles, and tendons. . . . . . . . . . . . 113
xiv
6.17 Extracted muscle meshes in example poses. . . . . . . . . . . . . . . . . . . . . 114
6.18 Comparisons between muscle simulation results and the ground truth meshes
in example poses. We simulated hand muscles using our model to reach a few
example poses that we believe are the most extreme poses among all six example
poses. Simulation results (green wireframe) closely overlap with the ground truth
meshes (red wireframe). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.19 Comparisons between tendon simulation results and the ground truth meshes
in example poses. We simulated hand tendons using our model to reach a few
example poses (same as in the muscle experiment). Simulation results (green wire-
frame) closely overlap with the ground truth meshes (red wireframe). . . . . . . . . 115
6.20 Visual comparisons to Wang et al. [134]. The palm area bulges more under our
simulation model, which is closer to the behavior in the real world. . . . . . . . . . 117
6.21 Visual comparisons of joint regions to Wang et al. [134] in example poses. The
ground truth mesh is depicted in red wireframe, whereas the simulation results are
shown in green wireframe. The palm and joint areas bulge more correctly in our
simulation model compared to the single-layer soft tissue model. . . . . . . . . . . 118
6.22 Visual comparisons to MRI slices in example poses. We compare our simulated
skin and muscle surfaces with the MRI images in example poses 1, 2 and 12. Pose
1 is the neutral pose. Pose 2 (fist) and 12 (thumb moving to the most opposite
location) are extreme example poses. We intersect our surface meshes with the
MRI slices. The intersections between skin and the slices are presented in green.
Further, the intersections between muscles and the slices are depicted in yellow.
We observe a very close match to the MRI data for both skin and muscles. . . . . . 119
6.23 Visual comparisons to MRI slices in non-example poses. We compare our sim-
ulated skin and muscle surfaces with the MRI images in non-example poses (poses
6 and 10). The intersections between skin and the slices are shown in green, and
the intersections between muscles and the slices are depicted in yellow. We can
observe a reasonably close match to the MRI data for both skin and the muscles. . . 120
7.1 The mismatches between simulation results and the MRI. The silhouettes of
muscles are marked in white. Mismatches are highlighted by red rectangles. We
show two typical types of mismatches that are not handled by our model. . . . . . 123
D.1 Illustration of the nullspace proof. The original forces F
i
sum to zero. We have
G
i
= RF
i
; note that R is the same for all tets. Therefore, the rotated forces G
i
also sum to zero. Hence, there is no change in the internal elastic force under a
rotation—that is, infinitesimal rotations are in the nullspace of K. . . . . . . . . . 140
xv
Abstract
Human hands and their modeling and animation are of paramount importance in many applications,
such as computer games, films, ergonomic design, tracking, and medical treatment. Unfortunately,
complex hand biomechanics today is not modeled, measured, or resolved in any qualitative manner.
While there are several computational models of hand kinematics, few have attempted to model
internal structures, let alone produce anatomically precise models that match real-world data.
This thesis demonstrates how to acquire complete human hand anatomy in multiple poses us-
ing magnetic resonance imaging (MRI), how to extract each different organ—such as bones and
muscles from MRI, and how to build an accurate animation-ready “rig” for the entire hand. Ac-
quiring human hand anatomy in multiple poses was previously difficult because MRI scans must
take a long duration for high-precision results (over 10 minutes) and because humans cannot hold
their hands perfectly still in non-trivial and badly supported poses. Thus, we invent a manufactur-
ing process whereby lifecasting materials commonly employed in the film special effects industry
are used to generate hand molds that are personalized to the subject and to each pose. These
molds are both ergonomic and encasing, and they also stabilize the hand during scanning. We also
demonstrate how to efficiently segment the MRI scans into individual bone meshes in all poses,
and how to correspond each bone’s mesh to the same mesh connectivity across all poses. Next,
we interpolate and extrapolate the MRI-acquired bone meshes to the entire range of motion of the
hand, thereby producing an accurate data-driven animation-ready rig for bone meshes. We also
demonstrate how to acquire not just bone geometry (using MRI) in each pose, but also a matching
highly accurate surface geometry (using optical scanners) in each pose, modeling skin pores and
wrinkles. We also give a soft tissue Finite Element Method simulation “rig”, consisting of novel
xvi
tet meshing for stability at the joints, spatially varying geometric and material detail, and quality
constraints to the acquired skeleton kinematic rig.
To further improve the hand model in both qualitative and quantitative aspects, we also sep-
arately extract and model other organs such as muscles and tendons. Organs like muscles are
often difficult to extract from MRI, because the boundaries between muscles and other organs
are often blurry. To extract them, we propose a method for modeling solid objects undergoing
large spatially varying and/or anisotropic strains given sparse observations (blurry boundary). Our
novel shape deformation method uses plastic strains and the Finite Element Method to successfully
model shapes undergoing large and/or anisotropic strains, specified by sparse point constraints on
the boundary of the object. We extensively compare our method to standard second-order shape
deformation methods, variational methods, and surface-based methods and demonstrate that our
method avoids the spikiness, wiggliness, and other artefacts of previous methods. Further, we
demonstrate how to perform such shape deformation both for attached and unattached (“free fly-
ing”) objects, using a novel method to solve linear systems with singular matrices with a known
nullspace. While our method is applicable to general large-strain shape deformation modeling, we
use it to create personalized 3D triangle and volumetric meshes of human organs, based on MRI or
CT scans. Given a medically accurate anatomy template of a generic individual, we optimize the
geometry of the organ to match the MRI or CT scan of a specific individual. Using our method,
we successfully extract all hand muscles in six example poses.
Finally, we developed a comprehensive system for hand animation by modeling bones, mus-
cles, tendons, joint ligaments, and fat separately. We gave a pipeline that pre-processes MRI and
optical scans into a simulation-ready hand rig. We demonstrated that our model is accurate. Our
model does not only match the ground truth skin, with average errors less than 1mm in all exam-
ple poses, but it also produces matched interior structures in all example poses. Furthermore, our
model produces realistic hand motions. It qualitatively reproduces important features seen in the
photographs of the subject’s hand, such as similar overall organic shape and fold formation.
xvii
Chapter 1
Introduction
1.1 Hand modeling and simulation using stabilized magnetic
resonance imaging
Hands and their modeling and animation are of paramount importance in numerous applications.
In computer games and film, clothing may occlude the body of the characters, but the hands are
often exposed and important for the story. In engineering, hand anatomical models can be used
to design tools and equipment. In computer vision, anatomically based modeling can improve
the tracking of hands, because it provides better statistical priors on hand shapes. In healthcare,
accurate hand shapes and motions can be used to design better finger and partial hand prosthetics,
as well as better tools for surgeons. As is well-known and argued by medical authorities in the
field [62], the dexterity of a hand stems from a thumb opposing four fingers (Figure 1.1), thereby
making the hand perfectly suited for precise grasping, lifting, climbing and daily manipulations of
objects. To improve realism, virtual hands should be modeled similarly to biological hands, and
this requires building precise anatomical and kinematic models of real human hands.
Unfortunately, complex hand biomechanics is currently not modeled, measured, or resolved in
any qualitative manner. While there are several computational models of hand kinematics, few have
attempted to model internal structures, let alone produce anatomically precise models that match
real-world data. Existing detailed human hand anatomy books like [62] focus on applications in
1
Figure 1.1: Opposition of the thumb: Our method produces an accurate data-driven skeleton
mesh kinematic “rig.” Here, we used our rig to reliably reproduce the well-known opposition of
the thumb to all other four fingers. The solid soft tissue was computed using an FEM simulation
attached to the articulated bone meshes. The fingers do not merely touch but also orient to be co-
planar at the contact location, subject to biomechanical limits. The right-most four images show
representative FEM frames of the opposition between the thumb and the pinky finger.
medicine, where the goal is to treat and repair hand injuries, not analyze the functioning of healthy
hands.
Figure 1.2: Left: Kuri-
hara’s CT scan [71].
Reprinted with permis-
sion from Eurographics.
Right: Our MRI scan.
It is challenging to acquire the motion of the internal human hand
anatomy. It must be performed in vivo (on a live person), as the hand
motion is greatly affected by the tendons, fat tissue, and active muscles.
It cannot be reliably performed using exterior scanning techniques (mo-
tion capture) due to tissue deformations and sliding. We give a method
to acquire high-resolution geometry of the bones of the human hand
(collectively referred to as the “skeleton”) in multiple hand poses, using
MRI scanning. We scan two subjects (1 male and 1 female) in 12 poses.
Our poses were selected to reasonably sample the range of motion of
each important hand joint. Note that in medicine, it is common to use
cadavers for human anatomy research. We exclude such studies because
our aim is to study the kinematics of healthy hands, as activated by a live
person.
In modern medicine, there are two imaging techniques that are potentially suitable to acquire
3D hand anatomy. In Computed Tomography (CT), a 3D image is generated from a large number
of two-dimensional X-ray images taken around a fixed axis of rotation. In computer animation,
2
hand bone skeletons have been acquired using CT scans in the pioneering work of Kurihara et.
al. [71]. Similarly, Marai et al. have also used CT scans to acquire the carpal bone motion of the
human hand [81]. Unfortunately, a CT machine emits ionizing radiation, which can cause cancer in
high radiation doses. Building a quality skeleton animation model requires acquiring many poses
for each individual, leading to unacceptably high radiation doses. Therefore, we decided to exclude
CT from our work. Magnetic Resonance Imaging (MRI) does not use any ionizing radiation nor
pose other healthy risks for the vast majority of people. Compared to Kurihara’s CT work, which
produced CT scans for one individual in five poses, we produced MRI scans for two individuals in
twelve poses without any health risks. We performed our MRI scans in a hospital, using clinical
3T scanners manufactured by General Electric. We obtained human subjects research approval
for these experiments from our institution’s review board (IRB). An additional advantage of MRI
over CT is that CT only generates sharp images of bones, whereas MRI reproduces all major hand
organs, including bones, muscles, and fat. For example, in the CT images in Kurihara’s work [71],
one can barely observe any significant hand components other than the bones and the surface
(Figure 1.2). Although we do not utilize muscles and fat in this work, their availability bodes well
for further research in this area.
Although MRI scanners can provide great anatomical detail, there is a practical challenge that
prior work has not addressed: the hand must be kept perfectly still in the scanner for 10-15 minutes.
Long scanning times are necessary to decrease the signal-to-noise ratio, which decreases with the
square root of the scanning time [115]. If the hand is not still, the scanning image quickly loses
resolution and becomes useless. If the goal is to scan just one pose, this is doable (not easily
comfortable, but doable) for most people—lie down on the scanner bed, firmly press the hand
against the bed, and hold it still during the scan. However, general hand poses cannot be scanned
in this manner because they position the palm and fingers in a certain general configuration—for
example, bending the fingers, closing the hand into a fist, partially closing the hand, etc. There is
no means to physically support the hand inside the scanner. Without support, it is impossible to
3
sharply scan such hand poses, as a human simply cannot keep an unsupported hand pose still for
more than a few seconds.
Our solution is to generate molds that hold the hand in a known and fixed pose. We generate
our molds by placing a hand in a chosen pose into a liquid lifecasting solution that solidifies into
an elastic rubber-like solid and then retrieve the hand. Thereafter, we fill the hole with liquid
plastic, generating a plastic replica of the hand in each pose. This replica captures extremely high
detail on the skin’s surface. We then scan the plastic replica using an optical 3D scanner (Artec
Spider [8]), and repeat the lifecasting process to generate a rubber mold (negative image of plastic
hand). We then cut the mold into two parts, place the hand into the mold, cover it with the upper
part, and place the hand into the MRI scanner. To the best of our knowledge (including that of our
co-author Dr. Matcuk who is a medical doctor/radiologist and active researcher in radiology), we
are the first to provide a solution for a stable long-time ( >10 minutes) MRI scanning of the hand
in multiple poses. Another key benefit of our approach is that we can obtain, for each pose, not
just the anatomy in that pose, but also a high-resolution surface skin detail in that pose, which can
be used for high-resolution hand rendering.
Next, we demonstrate how to segment the MRI scans into bone meshes for each pose. We
show how to register the bone meshes to have equal number of vertices and triangle connectivity.
Although there is substantial literature on body MRI data segmentation, the literature on hand MRI
segmentation is very sparse. We give new methods that improve the state-of-the-art (Section 4.1).
We then demonstrate how to build a data-driven skeleton kinematic model (a skeleton animation
“rig”). This rig is able to articulate the skeleton mesh into an arbitrary pose, using interpolation
and extrapolation of the acquired MRI meshes. Our rig captures complex real-world rotations and
translations of bones relative to their parent bone.
To model the motion of the human hand, we first provide a new Finite Element Method (FEM)
simulation method that treats the entire hand as a single soft tissue (“soft tissue rig”). Naive tet
meshing produces unstable tet meshes that explode due to pinching of the elastic material at the
joints, or over-constraining at the joints. We show how to stably constrain the single soft tissue
4
tetrahedral mesh to the skeleton mesh animated using our bone rig. We give a method to increase
the spatial mesh resolution and adjust elastic material properties in selected regions, which can
reliably and automatically simulate folds and creases, both on the palm and under fingers. Further,
we show experimentally that our bone rig greatly improves the stability of our soft tissue rig,
compared to naive bone rigs. We also demonstrate that our single soft tissue rig outperforms
skinning with standard Maya skinning weights.
As is well known, different organs of the hand have very different mechanical properties. For
example, a muscle tissue is much stiffer than a fat tissue; and is an active tissue. A hand tendon
behaves like an inextensible rod. In addition, adjacent tissues usually move relatively to each other,
such as muscles sliding against each other. These effects cannot be captured by merely modeling
the entire hand as a single soft tissue. Therefore, to further improve the accuracy of our hand
simulation model and capture key features of the human hand, it is important to treat different
organs of the human hand separately. While different hand organs have been widely studied in
many research fields independently, to the best of our knowledge, nobody has attempted to model
the entire hand as a complete biomechanical volumetric system. Moreover, few methods have
attempted to match medical data (namely MRI in our case) in both qualitative and quantitative
manner. First, without our proposed capturing method, it is difficult to capture the hand of a
specific live subject in different poses. As a result, existing methods use anatomy either obtained
from a single medical image or constructed based on medical literature. Second, due to the lack of
medical data, existing volumetric simulation models are not designed for matching medical scans
such as MRI.
With the MRI scans captured using our method, we are able to explore and design such a vol-
umetric model whereby the organs are modeled as separate but coupled objects, and that matches
both internal structure and external appearance of the human hand in example poses. Our work
models bones, muscles, joint ligaments, tendons, and fat. For each organ group, we give a complete
pipeline that takes medical and/or optical scans as inputs and produces a simulation-ready “rig”.
We use a comprehensive layered simulation system that begins with the kinematic bones, and is
5
followed by the bone fascia layer, tendon simulation, muscle simulation and muscle fascia simula-
tion. Finally, we simulate the fat tissue, which gives us the external shape of the hand. We employ
a layered simulation, as opposed to a coupled simulation of all hand organs, for two reasons: (1)
even layered simulation is at the limit of what we can achieve computationally today with state of
the art algorithms—coupled simulation is computationally infeasible, and (2) layered simulation is
better suited for our goal of the organs matching the MRI scans in example poses. Namely, a fully
coupled hand has too many diverse tissues and too complex interdependencies, which make it in-
tractable to simultaneously match the MRI scans of the different hand organs in multiple poses. We
demonstrate that we can closely match the ground truth medical images in each example pose: all
the organs are volumetrically matching their shapes in the MRI scans, with average errors less than
1mm in all example poses for skin, as compared to an optical scan. Furthermore, our volumetric
simulation model produces realistic volumetric deformations of the organs across the entire range
of the hand’s motion (ROM). Our method nonlinearly “interpolates” the organs in the MRI scans
across the entire ROM of the hand. We are not aware of any other work that has demonstrated a
volumetric simulation method for the entire hand’s musculoskeletal system that matches medical
image ground truth data, and that stably interpolates to the entire hand’s ROM. Furthermore, our
model qualitatively reproduces important features seen in the photographs of the subject’s hand,
such as similar overall organic shape and formation of bulges due to activated muscles.
1.2 Modeling of personalized anatomy using plastic strains
In this thesis, we demonstrate how to model anatomically realistic personalized three-dimensional
shapes of human organs, based on medical images of a real person, such as MRI or CT. Such
modeling is crucial for personalized medicine. For example, after scanning the patient with an
MRI or CT scanner, doctors can use the resulting 3D meshes to perform pre-operative surgery
planning. Such models are also a starting point for anatomically based human simulation for
applications in computer graphics, animation, and virtual reality. Constructing volumetric meshes
6
that match an organ in a medical image can also help with building volumetric correspondences
between multiple MRI or CT scans of the same person [96]—for example, for medical education
purposes.
Although the types, number, and function of organs in the human body are generally the same
for any human, the shape of each individual organ varies greatly from person to person due to
natural genetic variation across the human population. The shape variation is substantial: any
two individuals’ organsΩ
1
⊂ R
3
andΩ
2
⊂ R
3
generally vary by a non-trivial shape deformation
function Φ :Ω
1
→Ω
2
that often consists of large and spatially varying anisotropic strains (see
Figure 5.1). A “large anisotropic strain” implies that the singular values of the 3x3 gradient matrix
ofΦ are both different from each other and substantially different from 1.0, that is, the material lo-
cally stretches (or compresses) by large amounts; and this amount is different in different directions
and varies spatially across the model.
We tackle the problem of how to model such large shape variations by using volumetric 3D
medical imaging (such as MRI or CT scan), and a new shape deformation method capable of mod-
eling large spatially varying anisotropic strains. We note that the boundary between the different
organs in medical images is often blurry. For example, in an MRI of a human hand, the muscles of-
ten “blend” into each other and into fat without clear boundaries; a CT scan has even less contrast.
Therefore, we manually select as many reliable points (”markers”) as possible on the boundary of
the organ in the medical image; some with correspondence (“landmark constraints”) to the same
anatomical landmark in the template organ, and some without (“ICP constraints”). Given a tem-
plate volumetric mesh of an organ of a generic individual, a medical image of the same organ of
a new individual, and a set of landmark and Iterative Closest Point (ICP) constraints, we identify
how to deform the template mesh to match the medical image.
Our first attempt to solve this problem was to use standard shape deformation methods com-
monly used in computer graphics, such as as-rigid-as-possible energy (ARAP) [113], bounded
biharmonic weights (BBW) [59], biharmonic weights with linear precision (LBW) [137], and a fi-
nite element method static solver (FEM) [11]. As presented in Section 5.8, none of these methods
7
Figure 1.3: Second-order methods produce spiky outputs as the tet mesh is refined. Here, we
illustrate the output of an FEM static solver under a single hard point constraint (seen in (a)), that
is, we minimize the FEM elastic energy under the depicted hard point constraint. Bunny is fixed
at the bottom. Poissons’s ratio is 0.49. As we increase the tet mesh resolution in (b)–(e), the spike
becomes progressively narrower, which is undesirable. Changing the elastic stiffness (Young’s
modulus) of the bunny does not help (see Appendix G). Converting the constraint into a soft spring
constraint also does not help (f); now, the constraint is not satisfied.
was able to capture the large strains observed in medical images. Namely, these standard methods
either cannot model point constraints when the shape undergoes large spatially varying strains, or
introduce excessive curvature. For example, in the limit where the tetrahedral mesh is refined to
finer and finer tets, the FEM static solver produces spikes (Figure 1.3, Appendix G). We explore
the limitations of other methods in Section 5.8.
We provide a new shape deformation method that uses plastic strains and the Finite Element
Method to successfully model shapes undergoing large and/or anisotropic strains, controlled by
8
the sparse landmark and ICP point constraints on the boundary of the object. In order to do so,
we formulate a nonlinear optimization problem for the unknown plastic deformation gradients of
the template shapeΩ
1
, such that under these gradients, the shapeΩ
1
transforms into a shape that
matches the medical image landmark and ICP constraints. The ICP constraints are handled by
appropriately incorporating the ICP algorithm into our method. We note that in solid mechanics,
plastic deformation gradients are a natural tool to model large volumetric shape variations. We
are, however, unaware of any prior work that has used plastic deformation gradients and the Finite
Element Method to model large-strain shape deformation.
In order to make our method work, we needed to overcome several numerical obstacles. The
large-strain shape optimization problem is highly nonlinear and cannot be reliably solved with off-
the-shelf optimizers such as the interior point method [9]. Furthermore, a naive solution requires
solving large dense linear systems of equations. We demonstrate how to adapt the Gauss-Newton
optimization method to robustly and efficiently solve our shape deformation problem, and how to
numerically avoid dense linear systems. In order to optimize our shapes, it is essential to derive
analytical gradients of internal elastic forces and the tangent stiffness matrix with respect to the
plastic strain, which will be useful for further work on using plasticity for optimization and design
of 3D objects. In addition, we address objects that are attached to other objects, such as a hand
muscle attached to one or more bones as well as unattached objects. An example of an unattached
object is a liver, where the attachments to the surrounding tissue certainly exist, but are not easy to
define. It is practically easier to just model the liver as an unattached object. In order to address
unattached objects, we present a novel numerical method to solve linear systems with singular
matrices with a known nullspace. Such linear systems are commonly encountered in applications
in geometric shape modeling and nonlinear elastic simulation. Our examples include human hand
muscles, a liver, a hip bone, and a hip abductor muscle (“gluteus medius”), all of which undergo
substantial and non-trivial shape change between the template and the medical image.
9
Chapter 2
Related Work
2.1 Anatomically based simulation
Anatomically based simulation of the human body has been explored in multiple publications. For
example, researchers simulated human facial muscles [109], the entire upper human body [73],
volumetric muscles for motion control [74] and hand bones and soft tissue [134]. Anatomically
based simulation is also popular in film industry [121]. Existing papers largely simulate generic
humans because it is not easy to create accurate anatomy personalized to each specific person.
Our method can provide such an input anatomy, based on a medical image of any specific new
individual.
2.2 Animating the human hand
Hands are biomechanically complex and it is therefore natural to model their shapes using physi-
cally based simulation; such as, for example, simulating a solid mesh constrained to the underlying
skeleton [27, 68, 78]. Lee et al. comprehensively modeled the upper human body using anatom-
ically based simulation [73]. However, they excluded hands from simulation and modeled them
kinematically. The skin and tendons of the human hand have been simulated to enhance the vi-
sual appearance or control of hand articulation [76, 117, 100]. McAdams et al. [82] and Smith
et al. [110] used FEM to simulate cartoon hands. For human hands, Kry et al. [70] and Garre et
10
al. [41] modeled hand deformations using a layer of FEM soft tissue around articulated bones. We
give a hand FEM simulation model that is more elaborate than previous approaches. We model
spatially varying geometric detail and material properties, automatically remove elastic material in
between the heads of bones, and determine attachments to bones at the joints to avoid pinching. If
such features are omitted, it is in our experience impossible to stably simulate hands that aspire to
look like real hands. Our approach can, for example, model the fold formation on the entire hand.
To a first-degree approximation, hands can be animated using skinning techniques [60]. Skin-
ning can be improved using several techniques, such as dual quaternions [63], or implicit meth-
ods [127]. Pose-Space Deformation (PSD) is a method that combines skeleton subspace defor-
mation [79] with artist-corrected pose shapes [75]. It is widely used in industry due to its speed,
simplicity and the ability to incorporate real-world scans and arbitrary artist corrections.Kurihara
et. al. presented a variant of PSD suitable for hand animation [71], and Rhee et al. demonstrated
how to efficiently implement it on a GPU [97].
Recently, Romero et al. [98] gave a surface scanning approach to animate human hands to-
gether with the rest of the body. Their model is purely data-driven and does not model interior
anatomy. Different from our approach, their goal is to generate a statistical model to parameterize
the variations of human hand shapes due to personalization and pose. In contract to data-driven
methods [71, 98] and model-based methods [70, 41], we demonstrate how to simultaneously ac-
quire both a kinematic model of the internal anatomy and high-quality surface scans in matching
poses. We are not aware of any prior work that has achieved this for hands.
Our hands are driven by standard and familiar skeleton joint angle animations. Such anima-
tions can be created manually by artists, or acquired from real performances, for example, using
gloves [136, 32, 50], or computer vision techniques [72, 135, 92]. In computer animation, there
are many methods that compute kinematic hand skeleton models based on surface tracking meth-
ods, such as optical or magnetic motion capture; see [140] for a good survey. Unlike surface-
based techniques, our imaging-based method can generate internal anatomy, which is required for
anatomically based simulation. Because of their small motion and combined effect on the surface
11
of the palm, surface-based techniques have difficulties tracking the bones in the palm. Our MRI
method works equally well for all bones. We were able to extract the rigid motion of palmar bones
without any difficulty.
2.3 Acquiring medical images of human hand
Common medical imaging techniques that are in principle applicable to in-vivo measurements
include X-rays, Computed Tomography (CT), Positron Emission Tomography (PET), Magnetic
Resonance Imaging (MRI), and Ultrasound (for a good review, see [38, 40]). PET requires in-
jecting the subjects with positron-emitting radionuclides, equivalent to 8-years of natural back-
ground radiation [95]. Although ultrasound is a safe imaging techniques and has been studied for
decades [40, 111], it is not suitable for hands. This is because ultrasound signals are blocked by
bones, and are thus unable to generate complex three-dimensional geometry due to sonic occlu-
sions. MRI provides good contrast for all anatomical structures [33, 138], and can capture soft
tissues such as muscles and fat. Commercial companies have used MRI to build complete hu-
man body 3D anatomy [147]. A few publications analyzed the hand bone geometry using MRI
scans [88, 99, 128, 115]. However, because the MRI scanning times need to be long to decrease
the signal to noise ratio (10-15 minutes in our work), prior work has not addressed an impor-
tant limitation. Namely, if the hand is not held perfectly still during the scan, the MRI image is
blurred (useless). Of course, humans cannot hold the hand still in arbitrary poses for 10 minutes.
Researchers attempted to overcome this limitation by employing clay support, and by asking the
subject to hold objects during the scan. Using such an approach, Miyata et al. [88] succeeded in
MRI-scanning and modeling 2 finger joints on 4 fingers, in 4 poses. In our work, we scanned the
complete hand in 12 poses. We use ergonomic molds which, due to complete hand encasement,
provide superior support to using clay or relying on grasping. To combat hand shake, Stillfriend’s
work [115] used a balancedSSFP MRI scanning sequence to shorten the scanning time to 2-3 min-
utes. In medicine, this sequence is usually employed to scan moving objects (e.g., a beating heart).
12
While this reduces hand fatigue, it also decreases tissue detail and sacrifices the overall image qual-
ity. Furthermore, even if such procedures are used, the specific MRI-scanned poses in prior work
do not correspond to any optical scan of the surface geometry. Note that the quality of any hand
skin surface extracted from a MRI scan is far inferior to what can be achieved via surface optical
scanning. In our work, we use the PD Cube MRI sequence, which is a common MRI sequence to
image stationary objects. That said, the “fast MRI” (balancedSSFP) approaches are orthogonal to
our mold stabilization approach and could be combined if so desired.
2.4 Segmentation techniques for medical images of human hands
Most medical imaging segmentation techniques focus on the general body (and many on the hip
area), and not on hands (see [84] for a good review). Although segmentation algorithms with
prior knowledge have succeeded in automatic musculoskeletal segmentation of the hip [47, 107,
44, 106, 16], they have not been demonstrated to work on hands. Note that hands have a much
smaller and more complicated structure. Compared to the hip, a human hand (without the wrist)
has 27 bones and 19 flexible joints, which dramatically increases the segmentation complexity.
The most common approach (aimed at general human bodies, but also applicable to hands) in
practice is to mostly perform a manual segmentation, with the help of a few basic computer vision
techniques such as thresholding, filtering and water-shedding [128, 115]. Such capabilities are
often integrated into commercial medical image analysis software, such as Amira [7], VTK [130]
and ITK-SNAP [58]. We tried Amira, but found it too inefficient for hands and not automated
enough for our application. Recently, with the rapid development of GPUs, deep neural network
techniques (DNN) became a popular and powerful technique for image segmentation, including
MRI segmentation [66, 34]. However, these techniques are still in their infancy, e.g., [66] does not
solve the inhomogeneous bone problem. DNN approaches are currently inhibited due to a lack of
hand medical imaging datasets and approaches to stably build such datasets.
13
2.5 Medical image registration
Deformable models are widely used in medical image analysis [83]. Extracting quality anatomy
geometry from medical images is difficult. For example, Sifakis and Fedkiw [109] reported that it
took them “six months” (including implementing the tools) to extract the facial muscles from the
visible human dataset [126], and even with the tools implemented it would still take “two weeks.”
With our tools, we are able to extract all the 17 muscles of the human hand in 1 day (including
computer and user-interaction time). Bones generally have good contrast against the surrounding
tissue and can be segmented using active contour methods [120] or Laplacian-based segmenta-
tion [48, 134]. For bones, it is therefore generally possible to obtain a “dense” set of boundary
points in the medical image. Gilles et al. [46] used this to deform template skeleton models to
match a subject-specific MRI scan and posture. They used the ARAP energy and deformed sur-
face meshes. In contrast, we give a method that is suitable for soft tissues where the image contrast
is often low (our hand muscles and liver examples) and that accommodates volumetric meshes
and large volumetric scaling variations between the template and the subject. If one assumes that
the template mesh comes with a registered MRI scan (or if one manually creates a template mesh
that matches a MRI scan), musculoskeletal reshaping becomes more defined because one can now
use the pair of MRI images, namely the template and target, to aid with reshaping the template
mesh [45, 105, 43]. The examples in these papers demonstrate non-trivial musculoskeletal reshap-
ing involving translation and spatially varying large rotations with a limited amount of volumetric
stretching (Figure 14 in [43]). This is consistent with their choice of the similarity metric between
the template and output shapes: their reshaping energy tries to keep the distance of the output mesh
to the medial axis the same as the distance in the template [43], which biases the output against
volume growth. For bones, a similar idea was also presented in [108, 104], where they did not use
a medial-axis term to establish similarity to a source mesh, but instead relied on a PCA prior on the
shapes of bones, based on a database of 29 hip and femur bone shapes. Our work does not require
any pre-existing database of shapes. Because our method uses plasticity, it can accommodate large
and spatially varying volumetric stretching between the template and the subject. We do not need
14
a medical image for the template mesh. We only assume that the template mesh is plausible. Of
course, the template mesh itself might have been derived from or inspired by some MRI or CT
scan, but there is no requirement that it matches any such scan.
2.6 Non-rigid iterative closest point method
Non-rigid ICP methods are widely used for deforming the template mesh to a given target surface
mesh (or a dense point cloud) [5, 6, 77]. To apply ICP to medical imaging, one can first delineate
the organs in medical images using segmentation techniques, and then deform a template mesh
to the segmented surface. In prior work, this has been done by segmenting dense organ bound-
aries [91, 42]. Organ boundaries in medical images are often not clear, and therefore segmenting
dense boundaries is a laborious process, and even then, the boundaries are often unreliable. Our
work only requires identifying a sparse set of boundary points in the medical image. Combining
such sparse inputs with ICP methods has not been investigated in prior work. One could apply
existing ICP methods to a sparsely sampled boundaries, but this produces suboptimal results under
large strains and rotations (Figures 5.7, 5.23, 5.25). It is important to emphasize that the source of
errors in previous methods is not the miss-correspondence of sparse markers: suboptimal results
are produced even if given perfect correspondence (Figure 5.25).
2.7 Geometric shape modeling
Geometric shape modeling is an important topic in computer graphics research; e.g., see the Botsch
and Sorkine [20] survey and the SIGGRAPH course notes by Alexa et al. [3]. Popular methods
include variational methods [21], Laplacian surface editing [112], as-rigid-as-possible (ARAP)
deformation [56, 113], coupled prisms [22] and partition-of-unity methods such as bounded bi-
harmonic weights (BBW) [59] and biharmonic weights with linear precision [137]; we provide
a comparison in Section 5.8 and in several other Figures in the thesis. Our method reconstructs
the surface shape from a set of un-oriented point observations; this goal is similar to variational
15
implicit surface methods [123, 54]; we give a comparison in Section 5.9. Point clouds can also
be used to optimize rest shapes [125] and material properties of 3D solids [133]. Such a method
cannot be applied to our problem because it assumes a 4D dense point cloud input; whereas we
assume 3D sparse point inputs as commonly encountered in medical imaging. Point constraint
artefacts of second-order methods can be addressed using spatial averaging [17, 64]; however this
requires specifying the averaging functions (often by hand) and, by the nature of averaging, causes
the constraints to be met only approximately. Our method can meet the constraints very closely
(under 0.5 mm error in our examples), i.e., in the precision range of the medical scanners.
2.8 Plasticity
Elastoplastic simulations are widely used in computer animation. O’Brien et al. [93] and Muller
and Gross [89] used an additive plasticity formulation, whereas Irving et al. [57] presented a multi-
plicative formulation and argued that it is better for handling large plasticity; we adopt multiplica-
tive formulation in our work. The multiplicative model was used in many subsequent publications
to simulate plasticity, e.g., [15, 116, 29]. Because plasticity models shapes that undergo permanent
and large deformation, it is in principle a natural choice also for geometric shape modeling. How-
ever, such an application is not straightforward: an incorrect choice of the optimization energy will
produce degenerate outputs, elastoplastic simulations in equilibrium lead to linear systems with
singular matrices, optimization requires second-order derivatives for fast convergence, and easily
produces large linear systems with dense matrices. We present a solution to these obstacles. To
the best of our knowledge, we are the first to present such a comprehensive approach for using
plasticity for geometric shape modeling with large and anisotropic strains.
2.9 Anatomy transfer
Recently, great progress has been made on anatomically based simulations of humans. Anatomy
transfer has been pioneered by Dicko and colleagues [35]. Anatomical muscle growth and shrink-
age have been demonstrated in the “computational bodybuilding” work [102]. Kadlecek et al. [61]
16
demonstrated how to transfer simulation-ready anatomy to a novel human, and Ichim et al. [55]
gave a state-of-the-art pipeline for anatomical simulation of human faces. Anatomy transfer and
a modeling method such as ours are complementary because the former can interpolate known
anatomies to new subjects, whereas the latter provides a means to create the anatomies in the first
place. Namely, anatomy transfer requires a quality anatomy template to serve as its source, which
brings up the question of how one obtains such a template. Human anatomy is both extremely
complex for each specific subject, and exhibits large variability in geometry across the population.
Accurate templates can therefore only be created by matching them to medical images. Even if one
creates such a template, new templates will always be needed to model the anatomical variability
across the entire population; and this requires an anatomy modeling method such as ours.
17
Chapter 3
Acquiring MRI Scans in Many Poses
We now describe our procedure to acquire bone geometry and detailed external hand surface ge-
ometry and color texture in multiple hand poses, spanning the typical range of motion of the human
hand (Figure 3.1). Our skeleton geometry is acquired using MRI, and the external geometry is ac-
quired using laser scanner metrology (for shape) and photogrammetry (for color texture). For each
bone, we generate a mesh for each pose, with equal mesh connectivity. We then use the skeleton
geometry to run FEM simulations to generate high-quality unscanned “in-between” poses, produc-
ing registered high-resolution surface geometry that can be articulated to any continuous pose of
interest.
We stabilize the human hand inside the MRI scanners in known prescribed poses, by manu-
facturing rigid molds. Our technology “silver bullet” is the observation that a human hand can
be physically cloned into a shape made of a variety of materials (plastic, silicon, etc.) using life-
casting materials (Figure 3.3). Inspired by the film special and visual effects industry, we observe
that it is possible to generate extremely precise replicas of human hands, using commonly avail-
able rubber-like materials such as the AljaSafe
TM
material from Smooth-On, Inc [4]. We note that
lifecasting has been used in the visual effects industry to model human faces; we are not aware of
prior usage to model hands in multiple poses. Lifecasting has been used for fabricating an accurate
neutral shape of a human hand in robotics [69, 103], to perform hand control, such as for grasping.
We perform lifecasting using AljaSafe [4], an alginate skin-safe material naturally occurring
in the cells of brown algae. The resulting hand replicas capture extremely detailed surface hand
18
Figure 3.1: 12 MRI-scanned poses (1 subject; male; late 20s). Four rows are the cloned plastic
hand (the ”ground truth” shapes), our FEM result, skeleton mesh obtained from the MRI scan, and
the MRI scan. The last four plastic poses have a plastic stand (seen on the left of each image) to
easily stand up on a table. FEM closely matches the plastic hand in all poses, despite not actually
using plastic shapes anywhere in our system (except the neutral shape: column 5 in the top set).
The solid black arrows indicate the logical sequence of steps; the FEM and cloned hands are shown
adjacent to each other for easier comparison. The image can be zoomed in on.
features, down to individual pores and tiny lines on the hand surface (Figure 3.3, g). In order
to make the replicas, we prepare a liquid AljaSafe solution (mixture of powder and water) in
a properly hand-sized bucket, and then position the hand into it, in a chosen pose. The solution
19
Figure 3.2: Stabilized MRI scanning: (a) The mold with a plastic hand before cutting. (b1,2,3)
Mold has been cut into two parts. (c) Hand secured into the mold. (d) MRI scanning with the
mold. Clinical MRI scanner is manufactured by General Electric. Magnetic field strength of 3T,
and resolution of 0.5 x 0.5 x 0.5mm
3
.
Figure 3.3: We “cloned” this hand onto a plastic hand. (a) Mixing the AljaSafe. (b) The hand in
a bucket filled with liquid AljaSafe. (c) Hand removed, leaving a hand-shaped hole. (d, e) Casting
a liquid plastic into solidified AljaSafe. (f) Removed the AljaSafe to obtain the plastic hand. (g)
Photo of plastic hand. (h) Rendered 3D-scanned mesh of plastic hand. Scanner: Artec Spider.
Note the high-resolution detail in g, h (This can be zoomed in on in PDF).
solidifies in approximately 8-10 minutes into a rubbery solid. The shape of the hand and the micro-
detail on the surface get imprinted into the AljaSafe surface. We note that AljaSafe gently encases
the hand. It does not change volume when it solidifies, so there is minimal hand compression; only
small water-like hydrostatic pressure due to gravity. Because AljaSafe is sturdy, but sufficiently
20
flexible, it is easily possible to withdraw the hand, without damaging the imprinted surface detail.
This process produces a bucket of solidified AljaSafe rubber with a hole in the shape of the hand
pose. We then fill this hole with liquid plastic (Smooth-Cast
TM
300Q), which solidifies in a few
minutes. We then remove the AljaSafe rubber. The result is a high-quality plastic replica of the
hand in the specific pose (Figure 3.3, g, h). This manufacturing process is also shown in our video.
We note that the subject just has to approximately eyeball each pose during lifecasting. The created
mold and the plastic hand are automatically consistent.
We then scan the plastic hand using a precise laser scanner, such as Artec Spider [8]. We
scan the plastic hand, as opposed to the real hand directly, because the plastic hand is perfectly
still, and therefore such a scan can achieve high precision (Figure 3.3, h), on the order of 0.05
- 0.1mm with Artec Spider, producing a mesh with 11M triangles. For comparison, the number
of hand triangles in the dataset on the MANO website [98] is 100K. The Artec Spider precision
already is an order of magnitude better than the precision of the skin geometry acquired by our
MRI scan (which is 0.5 - 1mm). We are not aware of any prior computer graphics work to use
such a chemistry-based solution to “clone” hands. We observe that computer graphics and vision
acquisition research typically focuses on optical surface scanning technologies, e.g., reconstructing
shapes from multiple camera views. Chemistry-based approaches offer clear advantages in their
ability to capture high-precision surface detail on human body parts with minimal costs: we only
spent about $600 worth of chemical materials to make our replicas for 1 subject.
By repeating the above process for every pose, we obtain a high-quality plastic hand replica in
each pose, as well as a high-precision surface geometry scan of each shape. Next, we use these
plastic hands to stabilize the real hand during the MRI scan, by generating a mold into which the
subject inserts the hand prior to MRI scanning (Figure 3.2). The mold keeps the hand still in a
fixed, known, optically scanned pose. The presence of the mold does not corrupt the MRI signal
in any way. We tested this in practice and discussed the issue with MRI technicians. Our mold
generates almost no signal outside of the hand, and zero signal inside the hand. We generate the
mold by repeating the AljaSafe casting process. We prepare the liquid AljaSafe solution, and then
21
place the plastic hand into it. After AljaSafe solidifies, we cut the resulting mold in two parts along
the hand’s frontal (i.e., coronal) plane, using a precision knife (Figure 3.2, a). For MRI scanning,
we place the lower part of the mold onto the scanner bed. The subject inserts her/his hand into
the mold, and then the hand is covered and secured by the top part of the mold, and additionally
secured and fastened with tape (Figure 3.2, c). Because the top mold piece is largely supported by
the bottom piece, and because our molds are precise hand negative images, there is only a minimal
gravity and twisting during MRI scanning. Note that if one does not need the plastic hand, one can
already use the first AljaSafe cast for the MRI mold. However, molds obtained from plastic hands
can reorient the hand, for MRI ergonomics. The mold has to be sturdy to keep the hand still, but
still somewhat flexible, so that the subject can “nudge” the hand position slightly before the scan,
for ergonomics during the scan. This will inevitably result in a small mismatch between the plastic
hand pose that was used to make the mold, and the actual scanned MRI pose. We address this in
Section 4.4.
Figure 3.4: Our method outperforms prior hand bone segmentation methods. It is challenging
to segment the “metacarpal II” bone (red rectangle) for two reasons: (1) the MRI signal (depicted in
A) on the bone is not of uniform intensity, and (2) the neighboring bone “metacarpal III” (yellow
rectangle; only head of the bone is visible in this 2D view) has very similar signal intensity to
“metacarpal II.” Our method (depicted in B) successfully segmented this challenging case. The
prior method [99] uses a region-based active contour method and produces a suboptimal result: a
significant part of II’s bone head is missing (C).
22
Chapter 4
Hand Simulation using Hand Skeleton and Soft Tissue
4.1 Segmenting MRI data into bone meshes
The MRI scan data consist of a regular volumetric grid of scalar values proportional to tissue mass
density, and must be segmented into individual bone meshes. Although this task is well-understood
for general full-body MRI scans, the literature on segmenting hand MRI scans is very sparse. We
found only a single research paper (in any field) that discusses (in any substantial detail) how
to segment bones from hand MRI images [99], and even this publication only showed a single
example. We implemented their method and found it challenging to use: (1) there are more than
four parameters to tune, and (2) the result cannot be incrementally enhanced by users. Moreover, in
many cases it did not work well (for example, see Figure 3.4), so we abandoned it. We give a new
technique to perform such a segmentation. In general, common issues in MRI images include (a)
inhomogeneous bone tissue intensity: cortical bone (i.e, outer bone layer) is darker than cancellous
bone (inner layer), and (b) unclear and fuzzy bone boundaries in articulation areas, as described
in [106].
We give a method that segments the image with as few parameters as possible and is as auto-
matic as possible. We use a variant of Laplacian-based segmentation [47], but show how to extend
it to 3D in a scalable manner. We also give an intuitive interface specifically for segmentation of
hands. Our method has only one parameter, and users are able to incrementally improve the result
by delineating additional bone voxels which the segmentation classified incorrectly. The algorithm
23
Figure 4.1: Our segmentation interface. The result of the first step of our 3D Laplacian-based
segmentation. The segmented bone is red. Yellow is the local segmentation region selected by the
user for this bone.
in [47] is unacceptably slow when solving the Laplace equation in 3D on the entire volume. Our
solution is to segment each bone separately and combine the results together. When merging seg-
mentations into a global labeled volume, we resolve any labeling conflicts by treating conflicted
voxels as unlabeled voxels again, and resolve the Laplacian problem again in a local region. Using
our method, we were able to segment a 512x512x512 volume of the cancellous bones of the entire
hand in about 3 hours (5-10 minutes per bone). For cortical bones, there is no existing work on
segmentation without using training data (which is unavailable for hands) [106]. The Laplacian
method does not work here because the cortical bone is dark in the MRI images and too similar to
tendons. We segment it by the classical computer vision watershed method [85] in 3D. We deem
the connected component closest to the cancellous bone as the cortical segmentation. Finally, we
merge each cortical and cancellous pair into a single bone using dilation and/or random walk [47].
24
After the medical image segmentation, it is not difficult to build the implicit function for each bone
from the labeled images. With such implicit functions, we can easily generate the explicit 3D
geometry using an isosurface meshing algorithm [19].
We now describe how we register each bone’s pose-varying meshes to the same number of
vertices and triangle connectivity, and how we compute a rigid transformation between the bone
in the neutral pose and all other poses. For the neutral pose, we perform complete segmentation
of the bone as described in the previous paragraph. This gives us a mesh of the cancellous part
of the bone, and a complete bone mesh combining both the cancellous and cortical parts of the
bone, in the neutral pose. For all other poses, we only perform cancellous segmentation. This
saves a lot of time and is all that is needed. Unlike the cancellous bone which is white and has
good contrast, the cortical bone is dark and intensive to segment in terms of manual work; and its
segmentation in multiple poses is redundant. We bypass the need for segmenting cortical bones
in non-neutral poses by executing a rigid Iterative Closest Point (ICP) algorithm [18] to align the
neutral cancellous bone mesh onto the cancellous mesh of each pose, extracting the translation
x∈R
3
and rotation matrix R for each pose. We then transform the entire neutral bone mesh using
the rigid transformation (x,R). This produces the output of our segmentation: a mesh for each
bone in each pose, with the same connectivity across all poses. We smooth the bone meshes using
2 levels of the Laplacian smoothing in MeshLab [30]. Our final skeleton has 155,450 triangles
total for 23 bones. Using these methods, we successfully segmented 12 hand poses of our two
subjects (Figure 3.1). Figure 4.1 shows a screenshot of our segmentation user interface. Next, we
demonstrate how we can animate this skeleton mesh, by building a “skeleton rig.”
4.2 Skeleton kinematic model
In computer animation, the motion of the human hand is typically modeled using a hierarchical
joint structure; and we adopt this approach also in our work. Controlling motion by prescribing
joint angles using commonly available inputs (mocap, IK, keyframing, etc.) is commonly done in
25
practice because it is easier (and admittedly less principled) than optimizing muscle actuators [73].
Given the vector of joint angles θ, a typical pipeline in computer animation industry today is
not to compute any bone geometry, but instead to bypass bones and apply a skinning algorithm
that computes the positions of the skin vertices, based on θ and some suitably defined skinning
weights, optionally with sculpted pose-varying corrections. Such modeling is imprecise because
the fat layer and the skin are sliding relative to the bones, and because real bones undergo general
6-DOF rigid body motion relative to one another. The rigid transformations between bones depend
nonlinearly onθ, and are dictated by the anatomical constraints between the bones and/or muscles,
such as ligaments and tendons, and muscle activations. Accurate anatomical modeling requires
computing the bone geometry (mesh vertex positions) in arbitrary hand poses. We therefore depart
from previous work and give a new kinematic model (a “bone rig”) that computes the positions
of all vertices in all bone meshes, based on θ. The input to our rig is θ, and the output is the
anatomically based rigid transformation (translation and rotation) of each bone. We obtain our rig
by fitting it to our segmented bone MRI meshes in the captured poses (Section 3). Our model can
both interpolate and extrapolate the MRI-acquired poses. In later sections, we will use our bone
rig to compute FEM simulations of the soft tissue of the hand.
We note that Kurihara et. al. [71] optimized bone rotation centers to data, but did not optimize
joint rotation axes. Instead, they specified the relative rotation of each joint via three Euler angles
corresponding to unoptimized axes that in general do not match biomechanical axes. For example,
the finger DIP and PIP joints rotate mostly around one axis. With unoptimized axes, none of
the elemental Euler rotations correspond to the main biomechanical axis of rotation. Hence, the
animator has to either accept inaccurate motion, or animate all three Euler angles simultaneously.
In contrast, we optimize rotation axes to the MRI-acquired bone motion. We give a rig where 1-
DOF and 2-DOF joints can be animated by specifying only 1 and 2 angles, respectively; namely the
angles of rotations around our optimized joint axes (Figure 4.4). Our rig drives the biomechanical
joint axes directly, and as such requires fewer parameters to animate.
26
Figure 4.2: The bones and joints of a human hand. Our joint hierarchy is indicated with a
superimposition in blue, with the number of joint DOFs indicated.
Hand kinematics Modeling the hand joints and bone motion is a complex problem. The human
hand (see Figure 4.2) has 27 bones: 5 distal phalanges, 4 intermediate phalanges, 5 proximal pha-
langes, 5 metacarpals and 8 carpals, most of which are movable. The hand has 9 inter-phalangeal
joints (DIPs and PIPs), 5 metacarpo-phalangeal joints (MP) and 5 carpo-metacarpal joints (CMC).
The thumb is the only finger capable of opposing the other four fingers; therefore, it has a larger
range of motion and greater complexity. The CMC joint of the thumb is usually called Trapezo-
Metacarpal joint (TM). Researchers also study the inter-carpal joints, i.e., scapho-trapezial joint
(ST) [62]. The employed models vary largely according to the specific setup and application. The
number of modeled joints can range from 5 to 36 [143]. The number of DOFs of a single joint can
range from 1 to 3. The motion of carpal bones is small and on (or slightly over) the boundary of
our MRI scanning volume for our male subject; as commonly done [115], we do not model this
motion. The IP and CMC joint are usually modeled to have one rotation axis (or zero for CMC
27
Table 4.1: Bone rig errors for each joint. We list them separately for rotations (R) and translations
(x). Here, FM
R
and FM
x
are the absolute errors under our Full Model (i.e., quadratic fitting of
ˆ
R
and ˆ x), while PM
R
and PM
x
are the errors under a Partial Mode, in which
ˆ
R= I and ˆ x= 0. The
error is computed as the average difference between the output of our bone rig and the ground truth
(segmented MRI data), across all 12 poses (male subject). It can be seen that the fitting of
ˆ
R and
ˆ x decreases the error. The first and second halves of the table present 1-DOF and 2-DOF joints,
respectively. Observe that the pinky finger has the largest error; this is because it is the smallest.
Consequently, it is resolved with fewer voxels in the MRI scan.
Joint FM
R
[deg] PM
R
[deg] FM
x
[mm] PM
x
[mm]
DIP I 2.40 2.98 0.52 0.67
DIP II 2.10 3.25 0.31 0.40
DIP III 1.93 2.87 0.22 0.36
DIP IV 2.41 2.78 0.30 0.33
DIP V 2.84 3.30 0.35 0.73
PIP II 1.90 2.08 0.32 0.36
PIP III 1.58 2.19 0.27 0.50
PIP IV 1.32 2.81 0.34 0.46
PIP V 2.77 3.44 0.26 0.31
CMC IV 2.06 2.07 0.69 0.70
CMC V 2.74 2.79 1.07 1.15
MP I 0.59 1.03 0.44 0.80
MP II 1.58 1.82 0.55 0.76
MP III 1.76 1.87 0.37 0.65
MP IV 0.87 2.57 0.24 0.51
MP V 1.67 1.81 0.46 0.79
CMC I 0.79 1.94 0.90 2.26
Average 1.84 2.45 0.45 0.69
II and III), and MP and TM joints to have two rotation axes [62, 49]. We adopt these conventions
in our work. There are 17 joints in total in our human hand model, of which 11 are single-axis
and 6 are two-axis, for a total of 23 degrees of freedom (Figure 4.2). We assign no degrees of
freedom to CMC II and III because it is well-known that these joints almost do no move [10]; in
our dataset, maximum motion was under 2 degrees. The configuration of the entire joint hierarchy
is then specified via the vector θ of 23 joint angles. These angles can be driven using any suitable
computer animation method, such as keyframe animation, motion capture or even motion control.
28
Figure 4.3: Trans-
formation (x
i
,R
i
) is ex-
pressed in the parent’s
coordinate system. It
transforms the neutral
child bone into pose i.
Our bone rig We first describe our approach for 1-DOF joints. Con-
sider one specific parent-child bone pair. During hand articulation, both
the parent and child undergo rigid body motion. In this section, we con-
ceptually undo the parent’s transformation so that the parent is stationary
in the world, and the child undergoes some rigid body motion relative
to parent (Figure 4.3). All motions of the child bone in this section are
to be understood in this way. Denote the angle of the 1-DOF rotation of
the child bone by the scalarφ. Our MRI scan and its segmentation give
us the rigid transformation (x
i
, p
i
) of the child bone in each pose i, for
i= 1,...,N, where N is the number of acquired poses. Here, x
i
∈R
3
is the translation vector and p
i
∈R
4
is the unit quaternion. In pose i,
the child bone undergoes rigid transformation X7→ R
i
X+ x
i
, where R
i
is the rotation matrix cor-
responding to p
i
, and X ∈R
3
is an arbitrary point on the child bone. Our task is to discover a
unit rotation axis a∈R
3
, and anglesφ
i
∈R, such that p
i
is as close as possible to a rotation byφ
i
around axis a. This leads to a constrained optimization problem for a and anglesφ
i
:
max
a,φ
1
,...,φ
N
s.t. a
T
a=1
N
∑
i=1
dot
p
i
, cos(
φ
i
2
)+ sin(
φ
i
2
)a
. (4.1)
Here dot(p,q) denotes the dot product of quaternions p and q. We robustly solve this problem using
the IPOPT constrained nonlinear optimizer [131] with ADOL-C automatic differentiation [132].
After a and φ
i
are known, we construct our rig as follows. Given an arbitrary angle φ, we
model the rigid transformation of the child bone relative to its parent as
X7→
ˆ
R(φ)R(φ)
X− C
+C+ ˆ x(φ), (4.2)
where R(φ) is the rotation around axis a by angleφ, and
ˆ
R and ˆ x are the residual rotation and trans-
lation, respectively (to be determined). Here, C∈R
3
is a center of rotation, which we determine
using least squares so that ˆ x(φ) is minimized, assuming
ˆ
R= I,
29
Figure 4.4: The rotation axes and rotation centers of our joints. The bone meshes are shown in
a transparent style. The axes are shown as red lines. The centers are shown as black dots.
min
C
N
∑
i=1
||x
i
+(R
i
− I)C||
2
. (4.3)
Figure 4.5: Fitting the
residual ˆ x.
Finally, we compute the residual translations ˆ x(φ
i
)= x
i
+(R
i
− I)C
and residual rotations
ˆ
R(φ
i
)= R
i
R(φ
i
)
T
at each sampled pose i. The only
remaining task is to interpolate / extrapolate these residuals to arbitrary
φ. We achieve this by representing
ˆ
R using Euler angles (note that
ˆ
R
is a small rotation), and then interpolating / extrapolating each degree of
freedom of translations and rotations by fitting a 1D quadratic function to
samples(φ
i
, ˆ x
i
(φ
i
)) and(φ
i
,
ˆ
R
i
(φ
i
)), respectively. We initially employed
natural cubic splines, and then also polyharmonic splines; however, this resulted in a lot of spu-
rious wiggles in the rig; hence, we settled for quadratic functions which gave smooth residuals.
We extrapolate the functions using the tangents at each end of the samples. We experimentally
observed that the residuals ˆ x
i
are small, on the order or smaller than the accuracy of our MRI scan
30
(0.5 mm) (Figure 4.5), and the wiggles in ˆ x are therefore mostly just experimental noise. Our con-
clusion is that bones in real hands, to a large extent, do rotate around parent bones around a fixed
center C. However, the deviation away from the “fixed-rotation-center” model is still substantial.
Table 4.1 analyzes the accuracy of our bone rig. It demonstrates that using our quadratically in-
terpolated offsets
ˆ
R and ˆ x improves the accuracy of the rig. Even if one accepts fixed centers, one
still needs our method to compute the centers and the rotation axes; otherwise suboptimal shapes
occur (Figures 4.10, 4.14).
Our approach for two-dimensional joints is similar, except that we now have to find two or-
thogonal unit rotation axes a and b and a pair of joint anglesφ
i
andψ
i
. Our kinematic model is that
we first perform a rotation around a byφ
i
, followed by a rotation around the rotated axis b byψ
i
.
Therefore, we maximize
max
a,b,φ
1
,...,φ
N
,θ
1
,...,θ
N
s.t. a
T
a=1,b
T
b=1,a
T
b=0
N
∑
i=1
dot
p
i
, r
i
q
i
, where (4.4)
q
i
= cos(
φ
i
2
)+ sin(
φ
i
2
)a, r
i
= cos(
θ
i
2
)+ sin(
θ
i
2
)b
′
i
, (4.5)
and b
′
i
is obtained by rotating vector b using q
i
. As in the 1D case, this optimization is greatly
facilitated by the ADOL-C automatic differentiation library. The computation of the entire rig
took 1 minute total for all poses and the entire skeleton. The rotation axes are shown in Figure 4.4.
With our bone rig, we can simulate the human hand by considering the rest of the human hand
as a single soft tissue. Therefore, Our simulation model consists of bone geometry (Section 3),
surrounded by an elastic soft tissue modeled as a tetrahedral mesh. The bone geometry is animated
using our skeleton rig (Section 4.2). We now describe how we generate a stable tetrahedral mesh
(Figure 4.6), assign its material properties, and how we constrain the tetrahedral mesh to the bones
for FEM simulation.
31
4.3 Generation of the tet mesh outer surface
We start with the high-resolution (11M triangles) mesh of the hand in the neutral pose, obtained us-
ing optical scanning (Artec Spider) of the plastic neutral hand. Because the fingers are sufficiently
spread in this pose, this mesh had no webbing or occlusion issues, and only tiny imperfections
which we easily cleaned in Maya using smoothing. We then simplify this mesh to a smaller mesh
(0.5M triangles) using MeshLab; this is done to make the rest of the pipeline more tractable. The
average edge length in this mesh is 0.5mm, however the edge lengths are not uniform and some tri-
angles are of bad quality. We then manually (using painting in Maya) separate the triangles in two
groups: those on creases (where higher deformation precision is needed; both at the finger joints
and on the palm), and those elsewhere (“non-crease”). We then constrain the crease triangles, and
remesh the non-crease triangles using CGAL’s isotropic remeshing tool, with a 2.5mm target edge
length. Next, we constrain the non-crease triangles and remesh the crease triangles using the same
tool, at 0.5mm. We note that such a 2-stage remeshing process is necessary; otherwise, artefacts
appear (Figure 4.7, bottom-left). This procedure gives us a good isotropic manifold triangle mesh
ˆ
T
outer
(31,986 triangles) of the hand’s neutral pose surface, with smaller edge lengths in the crease
areas. We need to correct this mesh, as described next.
4.4 Resolving discrepancies due to a small pose change of the
subject in the mold prior to MRI scanning
Note that
ˆ
T
outer
encloses the surface of the plastic hand, which is the negative image of the MRI
mold of the neutral pose. However, when the hand is placed into the rubber-like mold for MRI
scanning, for ergonomics reasons, the hand slightly changes its pose as the subject settles it into
the mold prior to scanning. Also, the rubber-like mold somewhat gently squeezes the hand, which
slightly alters its volume. We resolve these discrepancies by constructing the hand surface geom-
etry (the skin) from the MRI scan. This is feasible for the neutral shape because the fingers are
32
well-separated. Note that due to limited MRI scanning resolution, this mesh has significantly lower
quality than
ˆ
T
outer
, and cannot serve directly as an input mesh for tetrahedral meshing. Instead, we
align
ˆ
T
outer
onto the MRI-scanned surface mesh, using a nonlinear Iterative Closest Point (ICP)
tool Wrap3 [142]. Here, we set the parameters in Wrap3 so that the wrap is nearly rigid and only
adjusts the pose, preserving the local surface detail in
ˆ
T
outer
. This is a slight, but important adjust-
ment, to compensate for subject’s ergonomic settling. The result is a triangle meshT
outer
of the
surface of the neutral hand that has good quality and local surface detail, does not suffer from a
loss of volume, and is positioned consistently with the MRI-scanned skeleton (see Figure 4.6, top).
This mesh will serve as the outer mesh for our simulation tet mesh.
4.5 Generation of the tet mesh inner boundary
We now describe how we generate a triangle meshT
inner
which will serve as the inner boundary
for our tet mesh. This mesh has to conform to the bones; however, it should not mesh the space
in between the bones at the joints; otherwise, those tets will be immediately pinched as the joints
articulate, leading to tet inversions and simulation instabilities. We start with the skeleton rig mesh
B (155,450 triangles; animated in Section 4.2), and use CGAL’s isotropic remeshing function (at
2.5mm) to produce a meshB
coarse
with 10,408 triangles for the entire skeleton. This mesh can-
not serve asT
inner
because the mesh simplification and remeshing sometimes introduce collisions
between the bone meshes. To remove collisions, we use the TetWild algorithm [52] to create a
tetrahedral mesh of the union of the volumes enclosed by bones inB
coarse
. The surface of this tet
mesh is our inner meshT
inner
. It is a good-quality isotropic mesh (10,682 triangles) whose surface
closely matchesB
coarse
, except thatT
inner
joins (“welds”) bone meshes in the collision areas. Note
that the vertices ofT
inner
are different to those ofB
coarse
. Here, we considered an alternative: com-
pute mesh boolean union of the bone meshes inB
coarse
, and then use CGAL’s isotropic remeshing
tool; however, this destroyed some geometric detail and introduced new collisions, so we aban-
doned this approach.
33
Figure 4.6: Simulation tet mesh. Top row: outer and inner tet surface mesh. InT
outer
, note the
greater resolution at the folds. Bottom: three cutaway views of the FEM mesh.
Next, we execute constrained Delaunay tetrahedralization with refinement (we use TetGen [51])
on inputT
outer
∪T
inner
. We set TetGen’s parameters so thatT
outer
andT
inner
are not refined, but
the interior tetrahedra are refined. The running time of TetGen was under 1 second. The result-
ing tet mesh is not our final tet mesh because we need to remove tetrahedra located at the joints
between two bones. For each bone, we manually select the triangles ofB
coarse
on the proximal
epiphysis (the head closer to the parent bone), roughly corresponding to the region of articular car-
tilage (smooth white tissue where bone forms a joint to its parent bone); call this setS
child
. Next,
34
we execute a procedure that we call “sweeping”: for each vertex of a triangle fromS
child
, find the
closest triangle t on the parent bone. If the distance is less than a threshold (we use 2.5mm), and
the normals are approximately opposite (dot product is smaller than -0.7), we add t to the swept
setS
parent
. We then form the convex hull (using CGAL’s convex hull 3) ofS
child
∪S
parent
. We
then remove from the tet mesh every tet whose center is inside the convex hull (see Figure 4.7,
top-middle). This gives us our simulation tet mesh (101,941 tets, 29,583 vertices). Note that this
mesh has an external surface and an internal surface (i.e., there are internal voids corresponding to
the bones).
4.6 Constraining the tet mesh to the bones
Our tet mesh is attached to the bones using soft constraints. We constrain a subset of the vertices
of the internal tet mesh surface. Constrained vertices are determined by sweeping (in the sense of
Section 4.5) the child bone against the parent bone, across the entire range of motion of the joint.
Note that hand joints may have 1-DOF or 2-DOFs. Our sweep resolution is 1 degree for each
joint degree of freedom, both for 1-DOF and 2-DOF joints. For each value of the joint degrees of
freedom, we computeS
parent
. We form the unionU ofS
parent
across the entire range of motion
(Figure 4.7, bottom-right). The sweeping takes 10 minutes total for the entire skeleton. We then
traverse all tet mesh vertices that are on the internal surface. If the closest triangle onB
coarse
is
further than 0.01mm, or is inU, we do not constrain this vertex, otherwise we constrain it to the
closest location onB
coarse
, using a spring of stiffness 10 N/mm. This avoids constrainingU, and
the region on the tet mesh adjacent to the removed tets in between the bones. The constraint springs
are integrated using implicit integration by computing their force gradients, for simulation stability.
We compute the motion of the coarse bone meshB
coarse
by skinning it to the output meshB of our
bone rig; each vertex ofB
coarse
is skinned to the closest bone inB. Because nails are rigid, we need
to also constrain nail vertices. We do this manually, by painting the nail triangles in Maya. The nail
vertices are constrained to follow the transformation of the distal phalanx finger bones. In addition
35
Figure 4.7: Tet mesh creation. Top: meshT
inner
; removing tets at the joints; tet mesh constrained
vertices. Bottom-left: the two-step remeshing process at the folds produces good triangles and is
necessary: under a single-step process, bad triangles appear. Bottom-right: illustration of sweeping
in 2D.
to the constraints, we also apply collision response between the bones and the soft tissue, and soft
tissue self-collision response. For the former, we use the technique presented in [13], whereas
for the latter, we detect pairs of non-neighboring overlapping triangles and push them apart using
penalty springs. In one of our examples (connect-all-poses), we observed small spurious wrinkles
36
Figure 4.8: Young’s modulus of various human musculoskeletal tissues. Note the unit kPa=
1000N/m
2
. From [36]. Reprinted with permission from the AAAS.
on the skin at the dorsal (i.e., back) side of the PIP joint, when the PIP joint is bent to its extreme.
We attribute this to the fact that the skin is pre-folded in that region in the neutral configuration,
and “unfolds” as PIP articulates; pre-folding is not modeled by our system. We remedy this issue
by post-processing the small local region using the “Delta Mush” deformer [80].
4.7 Material properties
For elastic material properties, we use the Saint-Venant Kirchhoff material. We add a volume
preservation term which prevents large compressions and successfully guards against collapse.
Specifically, we model volume preservation by adding a term λ(J− 1)
2
/2 to the elastic strain en-
ergy density function. Here,λ is the first Lam ´ e parameter, and J is the local volume growth factor
(product of the three principal stretches [57]). The combination of StVK and volume preservation
produced organic hand shapes that visually match photographs of the subject’s hand (Figure 4.13).
We model our tissue as a nearly incompressible isotropic hyperelastic solid, and therefore use
a Poisson’s ratio of 0.495. Although biological tissues are not completely incompressible, in-
compressibility and isotropy is appropriate for many biological materials [139], because human
tissue largely consists of an incompressible liquid (water). Our choice of Young’s modulus is
motivated by experimental measurements available in literature, such as Figure 4.8 reproduced
from [36], which the authors assembled from various experimental studies. As can be seen, fat
(2500-4000N/m
2
) is softer than muscles (9000-15000N/m
2
). Because we do not model fat and
37
Figure 4.9: Spatially varying mesh resolution and materials. Fold formation is improved by
increasing the tet mesh resolution under the finger joints (via T
outer
; Sections 4.3 and 4.4), and by
making the material in the same region 6x softer (Section 4.7). Our method produces a quality
horizontal fold across the entire length of the palm. Other methods only produce a partial fold, or
no fold at all.
muscles as separate objects, we set our Young’s modulus to a single value of 12000N/m
2
. In lo-
cations where there are visible hand lines, such as on the palm and at finger joins, we decrease
Young’s modulus to 2000N/m
2
, and Poisson’s ratio to 0.45, which helps with crease formation.
Because there are no muscles on the dorsal side of the hand, we decrease Young’s modulus in
that region to 6000N/m
2
, including on the back side of the fingers. For mass density, we choose
the density of water (1000kg/m
3
). This is because the reported densities of muscles and fat are
1060kg/m
3
and 920kg/m
3
, respectively [39].
38
Figure 4.10: Anatomical bone rig stabilizes FEM soft tissue simulations. Left: non-anatomical
rig, created by pivoting bones around the center of the parent bone’s head. Bones pinch elastic
material, causing artefacts. Right: our rig produces stable soft tissue simulations. Both results use
FEM.
4.8 Embedding of surface triangle meshes into tet mesh
We deform any of our surface triangle meshes by embedding it into the tet simulation mesh. Some-
times, some vertices of these surface triangle meshes in the neutral pose may be slightly outside of
the tet mesh due to coarsening or discretization errors; in which case we simply deform the vertex
using the closest tetrahedron. There are two surface meshes that we deform in this way and that
appear in our rendering results: the high-resolution mesh obtained by scanning the neutral plas-
tic hand using Artec Spider (11M triangles), and a low-resolution version with 15,232 triangles,
obtained by simplifying the high-resolution mesh using MeshLab.
4.9 Results
We used a 3 Tesla GE MRI scanner with a PD CUBE (3D fast spin echo) sequence with the
following parameters: NEX (number of excitations) 1; FA (flip angle) 90 degrees; TR (time to
repetition) 1500 ms; TE (time to echo) 25.93 ms; slice thickness 1 mm; slice spacing 0.5 mm;
matrix 256 x 256 pixels; and FOV (field of view) 28.3 x 25.6 cm. We scanned, processed and
simulated two subjects: a male (age: late 20s; Figure 3.1), and a female (age: late 40s; Figure 4.12).
We tested our bone rig and FEM simulation on 4 motion sequences. Two sequences were
recorded using a LeapMotion system [72], one was created using Inverse Kinematics (IK) to per-
form the opposition of the thumb to the other fingers, and one connects all of our scanned poses
into one continuous sequence, also using IK. We implemented a simple inverse kinematics engine,
39
Figure 4.11: Comparison of FEM simulation to skinning. Skinning weights were computed
using Maya’s geodesic voxel method. Other Maya skinning weight methods yield even worse
results. Both methods use our bone rig.
using the ADOL-C symbolic differentiation library. In our IK, we can constrain both positions and
orientations of the end effectors; and we did so for the fingers subject to biological limits. The
running time of our FEM simulations is approximately 1 sec / timestep, on an Intel i7 6950X PC
(manufactured in 2016, 1 processors x 10 cores at 3.00 GHz), 128GB of RAM. We are performing
60 timesteps per frame, at 25 FPS; and therefore, the computational cost is 1 minute per frame,
or 25 minutes per second of motion. We do not attempt to simulate accurate tissue dynamics,
only static shapes under the animated bone meshes. Hence mass, damping and timestep properties
only matter for simulation stability. The four sequences (recorded motions 1, 2; thumb opposition,
connect-all-poses) are 13, 16, 10 and 16 seconds long, and took 5.4, 6.7, 4.2 and 6.7 hours to
compute, respectively.
Our pipeline produces a quality un-colored surface mesh consistent with internal anatomy. In
order to add a color texture, we took photos of the subject’s hand from approximately 60 different
angles using a DSLR camera (Canon EOS 7). The photos were taken rapidly while the subject
held the hand firmly against a table. The subject then reversed the hand and another set of 60
photos was taken. We then constructed two separate textured surfaces from these photographs
40
rest shape frame 196 frame 435
Figure 4.12: The opposition of the thumb for our female subject (late 40s). The bone and FEM
geometry of three animation frames.
using the Photoscan software [2]. We then separately wrapped each of these two surfaces onto
our un-colored low-resolution surface mesh, using Wrap3. At the seams, the surfaces and colors
overlap because each of the two half-scans covers slightly more than one half of the hand. We
adjusted and smoothed the seams using Photoshop and Maya. We then transferred the texture onto
the high-resolution mesh, using Maya. This produced a good quality color texture map for both
low-resolution and high-resolution surface meshes, which we then use for rendering (Figure 4.14).
In this project, we discovered that it is actually not easy to obtain a complete hand static model
with a high-resolution geometry and texture. Our Photoscan approach benefited from the fact that
41
Figure 4.13: Comparison of our FEM results to photographs, on two “in-between” poses (i.e.,
not a part of the acquired 12 pose-dataset). The FEM results are rendered using our high-resolution
mesh (11M triangles) with subsurface scattering; individual pores are visible as geometric features.
The image can be zoomed in on. Generally, our FEM method produces a good qualitative visual
match to the photographs: the finger and palm appear “organic”, and the two main palmar lines are
in the correct location and are produced by FEM as geometric mesh features (folds), as opposed
to only a texture map illusion. This figure also demonstrates the limitations of our method. We do
not model the wrinkling of the skin, which is absent in the FEM result, but visible in photographs.
We also do not simulate muscle activation and, hence, the soft tissue at the root of the thumb is
flatter in FEM vs. in photographs.
we already obtained high-precision geometry by scanning the (perfectly still) plastic hand using
Artec Spider, and only needed to add color texture. Before using Photoscan, we consulted a lo-
cal professional photogrammetry studio who attempted to create both geometry and color texture.
They failed to produce a good scan, despite taking over 60 simultaneous photographs of the sub-
ject’s hand with high-quality cameras from multiple angles. Because a live hand is moving, Artec
Spider also failed to simultaneously create geometry and color texture, despite repeated scanning
attempts.
We compare our anatomy-driven bone rig to a non-anatomical bone rig created in Maya, by
placing the joint pivots at the end of each bone, and then running FEM soft tissue simulation.
Our rig clearly outperforms the non-anatomical rig, where bone collisions artificially squeeze the
elastic material, causing instabilities in FEM simulations (Figure 4.10). We also compare our FEM
method to skinning (Figure 4.11). Both FEM and skinning used our bone rig. The comparison
42
Figure 4.14: Our bone rig also improves skinning. Left: skinning with a non-anatomical bone
rig, created manually by placing the joints at the center of the parent bone’s bonehead. Right:
skinning with our anatomical rig. No FEM simulation was used in this example, only skinning. It
can be seen that skinning works better when it uses our anatomical bone rig (see the dent at the
joint). Both results use Maya’s “geodesic voxel” skinning weights.
shows that FEM produces higher quality shapes. Figure 4.9 demonstrates that our spatially varying
materials enable the generation of quality folds near the finger joints. In Figure 4.13, we compare
our FEM result to photographs of the hand of the same subject. Figure 4.14 demonstrates that our
bone rig also improves the quality of skinning (i.e., without using FEM). Table 4.1 analyzed the
accuracy of our bone rig.
4.10 Discussion and conclusion
We demonstrated how to acquire highly accurate human skeleton geometry in multiple poses using
MRI. We achieved this using a novel molding procedure to stabilize hands during MRI scanning.
We registered all poses to the same bones mesh connectivity, and built a skeleton mesh kinematic
model (“skeleton rig”) to interpolate or extrapolate the acquired pose to the entire hand range of
motion. We demonstrated that our skeleton rig can be used to drive soft-tissue FEM simulation
to produce anatomically plausible organic hand shapes that qualitatively match photographs. Our
accurate hand model can potentially benefit virtual hands in games / film / VR, robotic hands,
grasping, and medical education, such as visualizations of internal hand anatomy motion.
We performed 12 scans for each subject, which is sufficient to demonstrate stable and precise
common hand motions, including opposition of the thumb to the other four fingers. Accuracy
would be improved with more scans. An interesting research questions is how to automatically
select the next best pose to scan, to maximize coverage of the hand’s range of motion. Some
43
complex hand poses are challenging for our technique, for example, all fingertips touching at a
single point. This is because we cut our molds in two pieces manually using a knife. It would be
interesting to explore how to cut the molds into 3 or more pieces, to improve the ergonomics of
the insertion of the hand into the mold. Although we produced and scanned 12 plastic shapes, we
only used the neutral plastic shape in our simulation pipeline; the other 11 serve to evaluate the
accuracy of our FEM rig (Figure 3.1). It is challenging to register the scanned plastic meshes to
consistent complete mesh connectivity because many of these optical scans are incomplete due to
palm / finger occlusions, and webbing of fingers in close proximity.
We only acquired bone geometry so far; muscles, tendons and subcatenous fat could poten-
tially extracted from MRI too. While our FEM simulations (driven by our skeleton rig) produce
anatomically plausible shapes, accuracy would be improved by modeling muscles and subcate-
nous fat. These structures are available in our MRI scans and could be in principle segmented and
animated; These problems are addressed in the later chapters. We use spatially-varying tissue elas-
tic properties to improve the generation of folds at the joints. Results could be further improved
by optimizing spatially varying material properties. Physically based grasping would also benefit
from good material properties. We do not compute the joint hierarchy motion; but instead rely on
the animator to provide us with the animations of the bone’s joints. Such an approach is standard
in industry today [121], but requires the animator to be careful not to cause self-collisions. A bet-
ter, more principled approach would be for motion to originate from hand muscle activations, with
bones and the rest of the anatomy animating based on a physically based simulation. This poses
interesting neuromechanical control problems, due to the well-known muscle redundancy, and the
specific nature of hands which are mostly controlled by long muscle tendons originating in the
human arm. The understanding of motion of internal hand anatomy as proposed here represents
one step closer to lifelike robots with hands that mimic biological hands. Researchers at Columbia
Univ. recently demonstrated how to make artificial muscles [87]. With recent advances in 3D
printing technology, our work may contribute to the physical replication of anatomically realistic
hands and improved hand prosthetics, in the not so distant future.
44
Chapter 5
Extracting Anatomy using Plastic Strains
In contrast to hand bones, the boundaries of other organs are often blurry. For example, in an MRI
of a human hand, the muscles often “blend” into each other and into fat without clear boundaries;
a CT scan has even less contrast. Thus, the approach used for segmenting the bones is not help-
ful. In this chapter, we want to tackle the problem of how to extract those tissues from volumetric
3D medical imaging (such as MRI or CT scan). To represent such blurry boundaries, we therefore
manually select as many reliable points (”markers”) as possible on the boundary of the organ in the
medical image; some with correspondence (“landmark constraints”) to the same anatomical land-
mark in the template organ, and some without (“ICP constraints”). Given a template volumetric
mesh of an organ of a generic individual, a medical image of the same organ of a new individual,
and a set of landmark and ICP (Iterative Closest Point) constraints, our method asks how to deform
the template mesh to match the medical image.
Given a template tet mesh of a soft tissue organ for a generic individual, as well as known
optional attachments of the organ to other objects, our goal is to deform the tet mesh to match a
medical image of the organ of a new individual. We use the term “medical image” everywhere
in this thesis because this is standard terminology; this does not refer to an actual 2D image, but
to the 3D medical volume. We now describe how we mathematically model the attachments and
medical image constraints.
45
Template
(b) MRI slice
Our output
MRI slice
ICP markers
(d) maximum principal stretch
2.9
0.0
1.0
16.0
1.0
2.0
4.0
8.0
(c) our output shape (a) template shape (e) principal stretch ratio (anisotropicity)
Figure 5.1: We optimized the shape of this hand muscle (Adductor Pollicis) to match the MRI
scan. (a) Template shape [26]; (b) representative MRI slice [134]; we rigidly aligned the template
mesh onto the markers in the MRI scan, producing the white contour that is obviously incorrect
(deeply penetrates the bone and extends out of the volume of the MRI-scanned hand); (c) our
output shape, optimized to the MRI scan; the white contour now matches the scan; (d, e) large
anisotropic spatially varying strains accommodated by our method (d: maximum principal stretch;
e: ratio between maximum and minimum principal stretch).
Figure 5.2: Our shape deformation setup. Our optimization discovers plastic strains F
p
and
vertex positions x so that the model is in elastic equilibrium under the attachments while meeting
the medical image landmark and closest point constraints as closely as possible. The presence of
attachments is optional; our work address both attached and unattached objects.
46
5.1 Attachments and medical image constraints
We start with a template organ tet meshM. We denote its vertex positions by ¯ x∈R
3n
, where n
is the number of tet mesh vertices. In this thesis, we use bold text to represent the quantities for
the entire mesh and non-bold text to represent the quantities for a single vertex or element in the
FEM mesh. We would like to discover vertex positions x∈R
3n
such that the organ shape obeys
the attachments to other organs (if they exist), and of course the medical image. The attachments
are modeled by known material positions X
i
∈M, i= 1,...,t that have to be positioned at known
world-coordinate positions x
i
∈R
3
,i= 1,...t (Figure 5.2). The medical image constraints come in
two flavors. First, there are landmark constraints whereby a point on a template organ is manually
corresponded to a point in the medical image, based on anatomy knowledge. Namely, landmark
constraints are modeled as material positions Y
i
∈M, i= 1,...,q, that are located at known world-
coordinate positions y
i
∈R
3
in the medical image. Observe that landmark constraints are mathe-
matically similar to attachments. However, they have a different physical origin: attachments are
a physical constraint that is pulling the real-world organ to a known location on another (fixed,
un-optimized) object; for example, a muscle is attached to a bone. With landmarks, there is no
such physical force in the real-world; namely, landmarks (and also closest-point constraints) are
just medical image observations.
The second type of medical image constraints are closest-point constraints (“ICP markers”).
They are given by known world-coordinate positions z
i
∈R
3
,i= 1,...r that have to lie on the
surface of the deformed tet mesh. Locations z
i
are easier to select in the medical image than the
landmarks because there is no need to give any correspondence. As such, they require little or no
medical knowledge, and can be easily selected in large numbers. We went through the medical
image slices and selected clear representative points on the organ boundary. We then visually
compared the template and the target shape inferred by the ICP marker cloud. This guided our
positioning of the landmarks, which we place on anatomically “equal” positions in the template
and the medical image. We consulted a medical doctor to help us interpret medical images, such
47
as identifying muscles in the scan, clarify ambiguous muscle boundaries, placing attachments, and
disambiguating tendons.
5.2 Plastic deformation gradients
We model shape deformation using plastic deformation gradients, combined with a (small) amount
of elastic deformation. In solid mechanics, plasticity is the tool to model large shape variations of
objects, making it very suitable for shape deformation with large strains. Unlike using the elastic
energy directly (without plasticity), plastic deformations have the advantage that they can arbitrar-
ily and spatially non-uniformly and anisotropically scale the object. There is also no mathematical
requirement that they need to respect volume preservation constraints. This makes plastic defor-
mations a powerful tool to model shapes. Our key idea is to find a plastic deformation gradient F
p
at each tet ofM, such that the FEM equilibrium shape under F
p
and any attachments matches the
medical image observations. Figure 5.2 illustrates our shape deformation setting. In order to do
so, we need to discuss the elastic energy and forces in the presence of plastic deformations, which
we do next.
Figure 5.3: Comparison to augmented deformation transfer. The beam’s attachments (red)
cause the beam to bend, whereas the ICP markers (blue) cause it to stretch 2x in one of the two
transverse directions. Our method can easily recover such a shape deformation, whereas deforma-
tion transfer [118] cannot, even if augmented with an elastic energy.
Plastic strain is given by a 3× 3 matrix F
p
at each tetrahedron ofM. For each specific de-
formed shape x∈R
3n
, one can define and compute the deformation gradient F between x and x
48
Figure 5.4: Plastic and elastic deformation gradient for a single tet.
at each tet [89]. The elastic deformation gradient F
e
can then be defined as F
e
= FF
− 1
p
[15] (see
Figure 5.4). Observe that for any shape x, there exists a corresponding plastic deformation gradient
F
p
such that x is the elastic equilibrium under F
p
; namely F
p
= F. This means that the space of all
plastic deformation gradients F
p
is expressive enough to capture all shapes x. The elastic energy
of a single tet is defined as
E(F
p
,x)= V(F
p
)ψ(x,F
p
)= V(F
p
)ψ
F(x)F
− 1
p
, (5.1)
where V is the rest volume of the tet under the plastic deformation F
p
, and ψ is the elastic en-
ergy density function. We have V =|F
p
|V
0
, where V
0
is the tet’s volume inM , and|F
p
| is the
determinant of the matrix F
p
. Elastic forces equal f
e
(F
p
,x)= dE(F
p
,x)/dx. When solving our
optimization problem to compute F
p
in Section 5.4, we will need the first and second derivatives
ofE(F
p
,x) with respect to x and F
p
. We provide their complete derivation in Appendix B.
Figure 5.5: Seventeen muscles of the human hand extracted from MRI. Observe that the
template hand is larger than the scanned hand. The pose is also different. Our method solves this
using bone attachments.
49
Our method supports any isotropic hyperelastic energy density function ψ. In our examples,
we use the isotropic stable neo-Hookean elastic energy [110], because we found it to be stable and
sufficient for our examples. Note that we do model anisotropic plastic strains (and this is crucial for
our method), so that our models can stretch by different amounts in different directions. Observe
that plastic strains are only determined up to a rotation. Namely, let F
p
be a plastic strain (we
assume det(F
p
)> 0; i.e., no mesh inversions), and F
p
= QS be the polar decomposition where Q
is a rotation and S a 3× 3 symmetric matrix. Then, F
p
and S are the “same” plastic strain: the
resulting elastic deformation gradients differ only by a rotation, and hence, due to isotropy of ψ,
produce the same elastic energy and elastic forces. Note that it is not required that rotations Q
match in any way at adjacent tets. We do not need to even guarantee that F
p
globally correspond to
any specific “rest shape”, i.e., the F
p
are independent of each other and may be inconsistent. This
gives plastic deformation gradient modeling a lot of flexibility. Hence, it is sufficient to model
plastic strains as symmetric 3× 3 matrices. We can therefore model F
p
as a symmetric matrix and
parameterize it using a vector s∈R
6
,
F
p
=
s
1
s
2
s
3
s
2
s
4
s
5
s
3
s
5
s
6
. (5.2)
We model plasticity globally using a vector s∈R
6m
, where m is the number of tets inM. We
note that Ichim et al. [55] used such a 6-dimensional parameterization to model facial muscle ac-
tivations. In our work, we use it for general large-strain shape modeling. Our application and
optimization energies are different, e.g, Ichim et al. [55] causes muscle shapes to follow a pre-
scribed muscle firing field, and biases principal stretches to be close to 1. Furthermore, we address
the singularities arising with unattached objects.
50
5.3 Shape deformation of attached objects
We now formulate our shape deformation problem. We first do so for attached objects. An object
is “attached” if there are sufficient attachment forces to remove all six rigid degrees of freedom,
which is generally satisfied if there are at least three attached non-colinear vertices. We find the
organ’s shape that matches the attachment and the medical image constraints by finding a plastic
strain F
p
at each tet, as well as static equilibrium tet mesh vertex positions x under the attachments
and plastic strain F
p
, so that the medical image observations are met as closely as possible,
argmin
s,x
||Ls||
2
+αE
MI
(x)+βE
a
(x), (5.3)
subject to: f
e
F
p
(s),x
+ f
a
(x)= 0, (5.4)
where α≥ 0 and β ≥ 0 are scalar trade-off weights, and L is the plastic strain Laplacian. We
define L as essentially the tet-mesh Laplacian operator on the tets, 6-expanded to be able to operate
on entries of s at each tet (precise definition is in Appendix A). The Laplacian term enforces the
smoothness of F
p
, i.e., F
p
in adjacent tets should be similar to each other. The second equation
enforces the elastic equilibrium of the model under plastic strains F
p
and under the attachment
forces f
a
. Attachments pull a material point embedded into a tet to a fixed world-coordinate location
using spring forces; precise definitions of the attachment energy E
a
and attachment forces f
a
are
in Appendix F. It is important to emphasize that our output shapes are always in static equilibrium
under the plastic strains F
p
, and both this equilibrium shape x and F
p
are optimized together; this
is the key aspect of our work. The first equation contains the smoothness and the medical image
(MI) observations; we discuss the rationale for using the attachment energyE
a
in Equation 5.3 in
the next paragraph. The medical image energy measures how closely x matches the medical image
constraints,
E
MI
(x)=
q
∑
i=1
||Sx− y
i
||
2
+
r
∑
i=1
||z
i
− closestPoint(x,z
i
)||
2
, (5.5)
51
where S is the interpolation matrix that selects Y
i
, namely S ¯ x= Y. The function closestPoint(x,z
i
)∈
R
3
computes the closest point to z
i
∈R
3
on the surface of the tet mesh with vertex positions x.
Our treatment of attachments in Equations 5.3 and 5.4 deserves a special notice. Equation 5.4 is
consistent with our setup: we are trying to explain the medical images by saying that the organ has
undergone a plastic deformation due to the variation between the template and captured individual.
The shape observed in the medical image is due to this plastic deformation and the attachments.
We formulate attachment forces in Equation 5.4 as a “soft” constraint, i.e., f
a
(x) is modeled as
(relatively stiff) springs pulling the attached organ points to their position on the external object.
This soft constraint could in principle be replaced for a hard constraint where the attached positions
are enforced exactly. We use soft constraints in our examples because they provide additional
control to balance attachments against medical image landmarks and ICP markers. These inputs are
always somewhat inconsistent because it is impossible to place them at perfectly correct anatomical
locations, due to medical imaging errors. Hence, it is useful to have some leeway in adjusting the
trade-off between satisfying each constraint type. With soft constraints, it is important to keep the
spring coefficient in f
a
(x) high so that constraints are met very closely (under 0.5 mm error in our
examples). We were able to do this using some small amount of spring coefficient tuning.
As per the attachment energyE
a
, we initially tried solving the optimization problem of Equa-
tions 5.3 and 5.4 without it. This seems natural, but actually did not work. Namely, withoutE
a
,
there is nothing in Equations 5.3 and 5.4 that forces the plastic strains to reasonable values. The
optimizer is free to set F
p
to arbitrarily extreme values, and then find a static equilibrium x under
the attachment forces. In our outputs, we would see smooth nearly tet-collapsing plastic strains
that result in a static equilibrium x whereby the medical image constraints were nearly perfectly
satisfied. Obviously, this is not a desired outcome. Our first idea was to add a term that penalizes
the elastic energyE(F
p
,x) to Equation 5.3. Although this worked in simple cases, it makes the
expression in Equation 5.3 generally nonlinear. Instead, we opted for a simpler and more easily
computable alternative, namely add the elastic spring energy of all attachments,E
a
. This keeps
the expression in Equation 5.3 quadratic in x and F
p
, which we exploit in Section 5.4 for speed.
52
Observe thatE
a
behaves similarly to the elastic energy: if the plastic strain causes a rest shape that
is far from the attachment targets, then bothE
a
and the elastic energy will need to “work” to bring
the shape x to its target attachments. Similarly, if the plastic strain already did most of the work
and brought the organ close to its target, then neitherE
a
nor the elastic energy will need to activate
much.
Because our units are meters and we aim to satisfy constraints closely, we typically use weights
close to α = 10
9
and β = 10
8
in our examples. The weightsα and β permit adjusting the trade-
off between three desiderata: make plastic strains smooth, meet medical image observations, and
avoid using too much elastic energy (i.e., prefer to resolve shapes with plastic strains).
Finally, we note that our formulation is different to approaches that optimize the deformation
gradient F directly (i.e., without an intermediary quantity such as the plastic deformation gradi-
ent). In Figures 5.3 and 5.6, we compare to two such approaches: deformation transfer [118] and
variational shape modeling [21]. We demonstrate that our method better captures shapes defined
using our inputs (landmarks, ICP markers, large spatially varying strains). Among all compared
approaches, the variational method in Figure 5.6 came closest to meeting our constraints, but there
is still a visual difference to our method. We provide a further comparison to variational methods
in Section 5.9.5.
5.4 Solving the optimization problem for attached objects
We adapt the Gauss-Newton method [109] to efficiently solve the optimization problem of Equa-
tions 5.3 and 5.4 (example output shown in Figure 5.5). Although the Gauss-Newton method is
commonly used to solve nonlinear optimization problems, the dimension of our optimization space
is several hundred thousands; contrast this to recent related work in computer graphics [109, 86]
where the dimension is usually less than 50. In our case, a direct application of the Gauss-Newton
method will result in large dense matrices that are costly to compute and store, causing the method
to fail on complex examples. Below, we demonstrate how to avoid these issues, producing a robust
53
Figure 5.6: Comparison to variational shape modeling. In a variational method [21], the
wiggles increase if one imposes a stricter constraint satisfaction. First row: under a small number
of landmarks, variational methods with k = 2,3 produce a smooth and reasonable result, albeit
somewhat smoothening the rest shape. Middle row: under more landmarks, it becomes more
difficult for variational methods to meet the landmarks while avoiding the wiggles. Bottom row:
variational methods produce wavy results. Our method meets the landmarks and produces fewer
wiggles. This is because the plastic deformation field can arbitrarily rotate and non-uniformly scale
to adapt to the inputs; the elastic energy then finally irons out the kinks.
method capable of handling geometrically complex examples involving complex spatially varying
plastic strains. Before settling on our specific Gauss-Newton approach, we attempted to use the
54
Figure 5.7: Elastic energy methods only work with dense markers. In this figure, we compare
to a state-of-the art medical imaging technique [91], whereby the output shape is calculated by
minimizing an elastic energy of a template shape, subject to dense medical image markers. With
dense markers, elastic energy methods work well (left). As the constraints sparsify, elastic energy
produces artefacts (middle). Our plasticity method (right) produces a good shape, even with sparse
markers.
interior-point optimizer available in the state-of-the-art Knitro optimization library [9]. This did
not work well because our problem is highly nonlinear. The interior-point method (IPM) worked
well on simple examples, but was slow and not convergent on complex examples. IPM fails be-
cause it requires the constraint Hessian, which is not easily available (because it involves the third
derivative of the elastic energy). When we approximated it, IPM generated intermediate states too
far from the constraints, and failed. The strength of our Gauss-Newton approach is that we only
need constraint gradients. Our method inherits the convergence properties of the Gauss-Newton
method. While not guaranteed to be locally convergent, Gauss-Newton is widely used because its
convergence can approach quadratic when close to the solution.
We note that our method is designed for sparse medical image landmarks and ICP markers. In
Figure 5.7, we give a comparison to a related method from medical imaging which used an elastic
energy, but with dense correspondences. Our method can produce a quality shape even under
sparse inputs, and can consequently work even with coarser MRI scans (such as our hip bone
example; Figure 5.12). The ability to work with sparse markers also translates to lower manual
processing time to select the markers in the medical image.
55
Because the object is attached, Equation 5.4 implicitly defines x as a function of s. The Gauss-
Newton method uses the Jacobian J= dx/ds, which models the change in the static equilibrium
x as one changes the plastic deformation gradient. It eventually relies on the derivative of elastic
forces with respect to the plastic deformation gradient, which we give in Appendix B. Although
closestPoint(x,z
i
) is a nonlinear function of x, we can treat the barycentric coordinates of the
closest point to z
i
as fixed during one ICP iteration, and therefore closestPoint (x,z
i
) becomes a
linear function of x. Therefore,E
MI
(x) can be seen as a quadratic function of x, and so isE
a
(x).
We can thus rewrite Equations 5.3 and 5.4 as
argmin
x,s
1
2
||Ls||
2
+
q+r+t
∑
k=1
c
k
2
||A
k
x+ b
k
||
2
, (5.6)
s.t. f
net
(s,x)= 0, (5.7)
where f
net
(s,x)= f
e
F
p
(s),x
+ f
a
(x) is the net force on the mesh, and constant matrices, vectors
and scalars A
k
,b
k
,c
k
are independent of s and x (we give them in Appendix F). The integer t
denotes the number of attachments. We now re-write Equations 5.6 and 5.7 so that the plastic
strains are expressed as s+∆s, and the equilibrium x as x+∆x, where ∆x= J∆s. At iteration i
of our Gauss-Newton method, given the previous iterates x
i
and s
i
, we minimize a nonlinearly
constrained problem,
argmin
x
i+1
,∆s
i
1
2
||L(s
i
+∆s
i
)||
2
+
q+r+t
∑
k=1
c
k
2
||A
k
x
i
+ J∆s
i
+ b
k
||
2
, (5.8)
s.t. f
net
(s
i
+∆s
i
,x
i+1
)= 0. (5.9)
After each iteration, we update s
i+1
= s
i
+∆s
i
. Observe that Equation 5.8 does not depend on
x
i+1
, and that the constraint of Equation 5.9 is already differentially “baked” into Equation 5.8 via
∆x= J∆s. We therefore first minimize Equation 5.8 for ∆s
i
, using unconstrained minimization; call
the solution∆s
i
. A naive minimization requires solving a large dense linear system of equations,
which we avoid using the technique presented at the end of this section. We regularize ∆s
i
so
56
that the corresponding F
p
is always positive-definite for each tet; we do this by performing eigen-
decomposition of the symmetric matrix F
p
at each tet, and clamping any negative eigenvalues to
a small positive value (we use 0.01). Our method typically did not need to perform clamping in
practice, and in fact such clamping is usually a sign that the method is numerically diverging, and
should be restarted with better parameter values.
We then minimize the optimization problem of Equations 5.8 and 5.9 using a 1D line search,
using the search direction ∆s
i
. Specifically, for η≥ 0, we first solve Equation 5.7 with s(η) :=
s
i
+η∆s
i
for x= x(η) using the Knitro library [9]. Direct solutions using a Newton-Raphson solver
also worked, but we found Knitro to be faster. We then evaluate the objective of Equation 5.6 at
x= x(η) and s= s(η). We perform the 1D line search for the optimal η using the gradient-free
1D Brent’s method [94].
Initial guess: We solve our optimization problem by first assuming a constant s at each tet,
starting from the template mesh as the initial guess. This roughly positions, rotates and globally
scales the template mesh to match the medical image. We use the output as the initial guess for our
full optimization as described above.
Optimization stages and stopping criteria: We first do the optimization with attachments only.
Upon convergence, we add the landmarks, ignoring any ICP markers. This is because initially, the
mesh is far away from the target and the ICP closest locations are unreliable. After convergence, we
disable the landmarks and enable the ICP markers and continue optimizing. After this optimization
meets a stopping criterium, we are done. Our output is therefore computed with ICP markers
only; landmarks only serve to guide the optimizer. This is because landmarks require a correct
correspondence, and it is harder to mark this correspondence reliably in the scan than to simply
select an ICP marker on the boundary of an organ. We recompute the closest locations to ICP
markers after each Gauss-Newton iteration. We stop the optimization if either of the following
three criteria is satisfied: (i) reached the user-specified maximal number of iterations (typically
20; but was as high as 80 in the liver example), (ii) maximum error at ICP markers is less than
57
Figure 5.8: Convergence plots. The iterations are presented on the X-axis, and the optimization
energy is marked on the Y-axis. The initial optimization energies are normalized to 1.0.
a user-specified value (1mm for hand muscles), (iii) the progress in each iteration is too small,
determined by checking ifη is under a user-defined threshold (we use 0.01). Figure 5.8 shows the
convergence of our optimization.
Avoiding dense large linear systems Because the object is attached,
∂f
net
∂x
is square and invert-
ible. Therefore, one can obtain a formula for J by differentiating Equation 5.7 with respect to
s,
J=−
∂f
net
∂x
− 1
∂f
net
∂s
. (5.10)
The matrix J is dense (dimensions 3n× 6m). Observe that because Equation 5.8 is quadratic in∆s
i
,
minimizing it as done above to determine the search direction∆s
i
is equivalent to solving a linear
58
system with the system matrix H, where H is the second derivative (Hessian matrix; dimension
6m × 6m) of Equation 5.8 with respect to∆s
i
. Because J is dense, H is likewise a dense matrix,
H= L
2
+ J
T
∑
k
c
k
A
k
T
A
k
J= L
2
+ Z
T
Z, where (5.11)
Z=
√
c
1
A
1
√
c
2
A
2
.
.
.
√
c
q+r+t
A
q+r+t
J=−
√
c
1
A
1
√
c
2
A
2
.
.
.
√
c
q+r+t
A
q+r+t
∂f
net
∂x
− 1
∂f
net
∂s
. (5.12)
Therefore, when the number of tet mesh elements m is large, it is not practically possible to com-
pute H, store it explicitly in memory or solve linear systems with it. To avoid this problem, we first
tried solving the system of equations using the Conjugate Gradient (CG) method. This worked,
but was very slow (Table 5.1). The matrix Z∈R
3(q+r+t)× 6m
is dense. In our complex examples,
the number of medical image constraints q+ r+ t is small (typically 10 - 800) compared to the
dimension of s (6m; typically ˜200,000). Our idea is to efficiently compute the solution to a system
Hy= h for any right-hand side h using the Woodbury matrix identity [141], where we view L
2
as
a “base” matrix and Z
T
Z a low-rank perturbation. Before we can apply Woodbury’s identity, we
need to ensure that the base matrix is invertible. As we prove in Appendix A, the plastic strain
Laplacian L is singular with six orthonormal vectorsψ
i
in its nullspace (assuming thatM is con-
nected). Each ψ
i
is a vector of all ones in component i of s, i= 1,...,6 and all zeros elsewhere,
divided by
√
m for normalization. It follows from the Singular Lemma (i) (Section 5.5) that L
2
is
also singular with the same nullspace vectors. Therefore, we decompose
H=
L
2
− 6
∑
i=1
ψ
i
ψ
i
T
+
Z
T
Z+
6
∑
i=1
ψ
i
ψ
i
T
= B+
ˆ
Z
T
ˆ
Z, (5.13)
59
Table 5.1: Solving a single linear system of equations with H, using the conjugate gradients
(CG) and our method. The naive direct solver failed in all cases. Note that H is a dense 6m× 6m
matrix. The column t
prep
gives a common pre-processing time for both CG and our method.
Example 6m 3(q+ r+t) t
prep
CG Ours
Hand muscle 237,954 1,143 17.5s 897.5s 9.5s
Hip bone 172,440 1,497 11.4s 408.9s 7.2s
Liver 259,326 1,272 24.4s 1486.7s 10.0s
where B= L
2
− ∑
6
i=1
ψ
i
ψ
i
T
and
ˆ
Z is matrix Z with an additional added 6 added rowsψ
i
T
. By the
Singular Lemma (iii) (Section 5.5), B is now invertible, and we can use Woodbury’s identity to
solve
y= H
− 1
h=
B
− 1
− B
− 1
ˆ
Z
T
I+
ˆ
ZB
− 1
ˆ
Z
T
− 1
ˆ
ZB
− 1
h. (5.14)
We rapidly compute Z, without ever computing or forming J, by solving sparse systems
∂f
net
∂x
˜ z
k
=
− √
c
k
A
k
T
, and z
k
= ˜ z
T
k
∂f
net
∂s
, for k= 1,...,(q+ r+ t), where z
k
∈R
3× 6m
is the k-th 3-row block
of Z. Observe that this sparse system matrix is symmetric and the same for all k. We factor it once
using the Pardiso solver and then solve the multiple right-hand sides in parallel. Once ˜ z
T
k
has been
computed, z
k
= ˜ z
T
k
∂f
net
∂s
is easy to compute because it is a multiplication of a thin dense matrix
and a sparse matrix. The matrix B is constant, and we only need to factor it once for the entire
optimization. Finally, the matrix I+
ˆ
ZB
− 1
ˆ
Z
T
∈R
3(q+r+t)× 3(q+r+t)
is small, and so inverting it is
fast. We analyze the performance of our algorithm in Table 5.1.
5.5 Singular lemma
In this thesis, there are two occasions where we have to solve a singular sparse linear system
with known nullspace vectors. Such systems occur often in modeling of unattached objects, e.g.,
finding static equilibria, solving Laplace equations on the object’s mesh, animating with rotation-
strain coordinates [53], or computing modal derivatives [12]. Previous work solved such systems
ad-hoc, and the underlying theory has not been stated or developed in any great detail; or it only
60
addressed non-singular systems. For example, in order to constrain vertices in cloth simulation,
Boxerman [25] modified the linear system Ax= b to SAx= Sb, for a properly chosen filtering ma-
trix S. However, A is non-singular to begin with; there is no discussion of singularity in Boxerman’s
work.
We hereby state and prove a lemma that comprehensively surveys the common situations aris-
ing with singular systems in computer animation and simulation, and back the lemma with a math-
ematical proof (Appendix C). Recall that the nullspace of a matrix A∈R
p× p
isN (A)={x∈
R
p
; Ax= 0}, and the range of A isR(A)={Ax; x∈R
p
}. Both are linear vector subspaces ofR
p
.
Figure 5.9:
Illustration of
the Singular
Lemma.
Singular Lemma: Let the square symmetric matrix A∈R
p× p
be singular with
a known nullspace spanned by k linearly independent vectorsψ
1
,...,ψ
k
. Then
the following statements hold:
(i)N (A) andR(A) are orthogonal. Every vector x∈R
p
can be uniquely
expressed as x= n+ r, where n∈N (A) and r∈R(A). Vector r is orthogonal
to n and toψ
i
for all i= 1,...,k (Figure 5.9).
(ii) Let b∈R(A). Then, the singular system Ax= b has a unique solution x
with the property that x is orthogonal toψ
i
for all i= 1,...,k. This solution can
be found by solving the non-singular linear system
A ψ
1
... ψ
k
ψ
T
1
0 ... 0
.
.
.
.
.
.
.
.
.
ψ
T
k
0 ... 0
x
λ
1
.
.
.
λ
k
=
b
0
.
.
.
0
. (5.15)
All other solutions equal x+∑
k
i=1
µ
i
ψ
i
for some scalarsµ
i
∈R.
(iii) For any scalars α
i
̸= 0, the matrix B= A+∑
k
i=1
α
i
ψ
i
ψ
T
i
is invertible. If ψ
i
are orthonormal
vectors, then the solution to By = h equals y = x+∑
k
i=1
λ
i
α
i
ψ
i
, where x and λ
i
are solutions to
61
Equation 5.15 with b= proj
R(A)
h= h− ∑
k
i=1
(ψ
T
i
h)ψ
i
. We give the proof of the singular lemma in
Appendix C.
5.6 Unattached objects
Figure 5.10: Unattached optimization of a hip bone shape to a CT scan. The scanned bone is
smaller and has a substantially different shape from the template.
Figure 5.11: Unattached optimization of a liver shape to a CT scan. Our method successfully
captures the large shape variation between the template and the scan. This figure also demonstrates
that our method makes it possible to transfer the rendering textures and UV coordinates from the
template onto the output.
Figure 5.12: Attached optimization of a hip muscle (gluteus medius) to an MRI scan. Our
method successfully captures the large shape variation between the template and the scan.
62
The difficulty with unattached objects is that we now have f
net
= f
e
, and the equation f
e
F
p
(s),x
=
0 no longer has a unique solution x for a fixed plastic state s. This can be intuitively easily under-
stood: one can arbitrarily translate and rotate any elastic equilibrium shape x under the given
plastic state s; doing so produces another elastic equilibrium shape. The space of solutions x is
6-dimensional. This means that we can no longer uniquely solve Equation 5.7 for x during our line
search of Section 5.4. Furthermore, the square tangent stiffness matrix
K(F
p
(s),x)=
∂f
e
F
p
(s),x
∂x
(5.16)
is no longer full rank. In order to address this, we now state and prove the following Nullspace
Lemma.
Nullspace Lemma: The nullspace of the tangent stiffness matrix of an elastoplastic deformable
object in static equilibrium x under plasticity, is 6-dimensional. The six nullspace vectors areψ
i
:=
[e
i
,e
i
,...,e
i
], where e
i
∈R
3
is the i-th standard basis vector, andψ
3+i
:=[e
i
× x
1
,e
i
× x
2
,...,e
i
× x
n
], for i= 1,2,3.
To the best of our knowledge, this fact of elasto-plasto-statics has not been stated or proven in
prior work. It is very useful when modeling large-deformation elastoplasticity, as real objects are
often unattached, or attachments cannot be easily modeled. We give a proof in Appendix D. To
accommodate unattached objects, it is therefore necessary to stabilize the translation and rotation.
For translations, this could be achieved easily by fixing the position of any chosen vertex. Matters
are not so easy for rotations, however. Our idea is to constrain the centroid of all tet mesh vertices
to a specific given position t, and to constrain the “average rotation” of the model to a specific
given rotation R. We achieve this using the familiar “shape-matching” [90], by imposing that the
63
rotation in the polar decomposition of the global covariance matrix must be R. We therefore solve
the following optimization problem,
argmin
x,s,R,t
||Ls||
2
+αE
MI
(x) (5.17)
s.t. f
e
F
p
(s),x
= 0, (5.18)
∑
j∈D
w
j
x
j
!
− ∑
j∈D
w
j
X
j
= t, (5.19)
Polar
∑
j∈D
w
j
x
j
− t
X
j
−
∑
k∈D
w
k
X
k
T
!
= R, (5.20)
where D is the set of points on the mesh surface where we have either a landmark or an ICP
constraint, w
j
is the weight of a point, X
j
is the position of vertex j inM and Polar(F) is the polar
decomposition function that extracts the rotational part of a matrix F. We set all weights equal, i.e.,
w
j
= 1/|D|. We choose the set D as opposed to all mesh vertices so that we can easily perform
optimization with respect to R and t (next paragraph). We assume that our argument matrices F to
Polar are not inversions, i.e., det(F)> 0, which establishes that Polar(F) is always a rotation and
not a mirror. This requirement was easily satisfied in our examples, and is essentially determined
by the medical imaging constraints; the case det(F)< 0 would correspond to an inverted (or mirror)
medical image, which we exclude.
We solve the optimization problem of Equations 5.17, 5.18, 5.19 and 5.20 using a block-
coordinate descent, by iteratively optimizing x,s while keeping R,t fixed and vice-versa (Fig-
ure 5.10, 5.11). Rigid transformations do not affect smoothness of s so we do not need to consider
it when optimizing R,t. We need to perform two modifications to our Gauss-Newton iteration of
Section 5.4. The first modification is that we need to simultaneously solve Equations 5.18, 5.19
and 5.20 when determining the static equilibrium in the current plastic state s. As with attached
objects, we do this using the Knitro optimizer. In order to do this, we need to compute the first
and second derivatives of the stabilization constraints in Equations 5.19 and 5.20 (Section 5.7).
The second modification is needed because the tangent stiffness matrix K(F
p
(s),x), as explained
64
above, is now singular with a known 6-dimensional nullspace. In order to compute the Jacobian
matrix J using Equation 5.10, we use our Singular Lemma (ii) (Section 5.5). Note that the right-
hand side is automatically in the range of K because Equation 5.10 was obtained by differentiating
a valid equation, hence Equation 5.10 must also be consistent.
5.7 Gradient and Hessian of Polar(F)
Previous work computed first and second-order time derivatives of the rotation matrix in polar
decomposition [14], or first derivative with respect to each individual entry of F [124, 28]. In our
work, we need the first and second derivatives of R with respect to each individual entry of F.
We found an elegant approach to compute them using Sylvester’s equation, as follows. Observe
that Polar(F) = FS
− 1
, where S is the symmetric matrix in the polar decomposition. Because
det(F)> 0, S is positive-definite and uniquely defined as S=
√
F
T
F. To compute the first-order
derivatives, we start from F = RS, and differentiate,
∂F
∂F
i
=
∂R
∂F
i
S+ R
∂S
∂F
i
, hence
∂R
∂F
i
=
∂F
∂F
i
− R
∂S
∂F
i
S
− 1
, (5.21)
Therefore, we need to compute∂S/∂F
i
. We have
F
T
F = S
2
, and thus
∂F
T
F
∂F
i
=
∂S
∂F
i
S+ S
∂S
∂F
i
, (5.22)
i.e., this is the classic Sylvester equation for the unknown matrix
∂S
∂F
i
[119]. The Sylvester equation
AX+ XB= C can be solved as
B
T
⊕ A
− 1
vec(X)= vec(C), (5.23)
where⊕ is the Kronecker sum of two matrices. In our case,
vec(
∂S
∂F
i
)=(S⊕ S)
− 1
vec(
∂F
T
F
∂F
i
). (5.24)
65
The computation of second-order derivatives follows the same recipe: differentiate the polar de-
composition and solve a Sylvester equation. We give it in Appendix E.
We can now compute the gradient and Hessian of our stabilization constraints. The translational
constraint is linear in x and can be expressed as W
1
x− d
1
= 0, where W
1
is a 3× 3n sparse matrix.
Although Polar is not linear, the argument of Polar is linear in x. The rotational constraint can
be expressed as Polar(W
2
x− d
2
)− ¯
R = 0, where W
2
is a 9× 3n sparse matrix. The Jacobian
of the translational constraint is W
1
, and the Hessian is zero. For the rotational constraint, the
Jacobian is
∂R
∂F
: W
2
∈R
9× 3n
and the Hessian is(W
T
2
:
∂
2
R
∂F
2
: W
2
)∈R
9× 3n× 3n
, where : denotes tensor
contraction.
Figure 5.13: Standard volume-based shape deformation methods result in wiggly and spiky
artefacts. The displayed hand Palmar interossei muscle has a tendon on one side and no tendon
on the other; both ends are attached to a bone (light blue). The acronyms are defined in Sec-
tion 5.8. The MRI landmark constraints are shown in dark blue. The shape deformation between
the template muscle and the shape in the MRI consists of large and spatially varying stretches. Our
method successfully models this deformation. We note that spikes cannot be avoided by using,
say, a spherical region for the constraints as opposed to a point; the non-smoothness merely moves
to the spherical region boundary.
66
Figure 5.14: Comparison between surface energy and volumetric energy. In both examples
(bunny and hand), we performed non-rigid iterative closest point alignment between a template
triangle mesh and a collection of target ICP markers (13 for bunny and 456 for the hand). For
the bunny, we manually placed the markers to greatly enlarge the bunny’s left ear. For the hand,
we placed the markers on a pre-existing target hand surface mesh that has different geometric
proportions and mesh topology as the template. The template is a man and target is a woman. We
then solved the ICP problem using the surface-based ARAP energy, volume-based ARAP energy,
and our volumetric plastic strains. Our method produces smooth artefact-free outputs.
5.8 Comparison to standard shape deformation methods
Our shape deformation setup is similar to standard shape deformation problems in computer graph-
ics. In fact, we first attempted to solve the shape deformation problem with as-rigid-as-possible
energy (ARAP) [113], bounded biharmonic weights (BBW) [59], biharmonic weights with linear
precision (LBW) [137], and a Finite Element Method static solver (FEM) [11]. Unfortunately,
none of the methods worked well. Figures 5.13 and 5.14 demonstrate that these methods produce
non-smooth shapes with spikes (ARAP, BBW, FEM), or wiggles (LBW).
67
Mathematically, the reason for the spikes in ARAP, BBW and FEM is that point constraints for
second-order methods are inherently flawed. As one refines the solution by adding more tetrahe-
dra, the solution approaches a spiky point function at each point constraint, which is obviously not
desirable. This mathematical issue is exposed in our work because our shape deformation consists
of large spatially varying stretches. Often, the template mesh needs to be stretched∼ 2x or more
along some (or several) coordinate axes. The medical image constraints are distributed all around
the muscle, pulling in different directions and essentially requesting the object to undergo a spa-
tially non-uniform and anisotropic scale. This exacerbates the spikiness for second-order methods.
We note that these problems cannot be avoided simply by using an elastic energy that permits
volume growth. Namely, Drucker’s stability condition [37] requires a monotonic elastic energy
increase with increase in strain. An elastic energy therefore must penalize strain increases if it is to
be stable; and this impedes large-strain modeling in methods that rely purely on an elastic energy.
Our plasticity method does not penalize large strains and thus avoids this problem. Spikes can
be avoided by using a higher-order variational method such as LBW. However, our experiments
indicate that such methods suffer from wiggles when applied to medical imaging problems (see
also Figures 5.6 and 5.22).
5.9 Results
We extracted muscles of the human hand and the hip muscle from an MRI scan, and a hip bone and
a liver from a CT scan. We analyze the performance of our method in Table 5.2. In Figure 5.15,
we give histograms of the magnitude of the difference between the positions of the medical image
markers and their output positions. It can be seen that our method produces shapes that generally
match the medical image constraints to 0.5mm or better. In Figure 5.16, we demonstrate that
the output quality of our tetrahedra is still good; if needed, this could be further improved by re-
meshing [15]. Figures 5.17 and 5.18 superimpose our output meshes on the CT and MRI scans,
respectively. In Figure 5.20, we compare to a recent implicit point set surface reconstruction
68
Figure 5.15: Output ICP error histograms. Each medical image marker contributes one entry to
the histogram. The hand muscles histogram is a combined histogram for all the 17 hand muscles.
Figure 5.16: Minimum tetrahedral dihedral angles before and after our optimization. It can
be seen that the output angles are still relatively large. As expected, the output angles are somewhat
worse than the initial ones, as the object has undergone a plastic deformation.
69
Table 5.2: The statistics for our examples: #vtx = number of vertices; #ele = number of tetra-
hedra inM ; #iter = number of ICP iterations; “time” = total computation time to compute the
output shape; “attached” indicates whether the object is attached; e
init
= error between our tem-
plate mesh and the ICP constraints; e
final
= error between our result and the ICP markers. The first
and second reported error numbers are the average and maximum errors, respectively. In the hand
example, there are 17 groups of muscles; “min”, “med” and “max” refer to the smallest, represen-
tative median, and largest muscle groups; “max-m” is the example with the largest number of ICP
constraints. Observe that our markers are sparse; the ratio between the number of markers and the
number of vertices is 0.3%–7.3% in our examples.
Example # vtx # ele # markers # iter time [min] attached e
init
[mm] e
final
[mm]
Hand muscle (min) 4,912 20,630 15 8 14.2 yes 0.53 / 2.22 0.06 / 0.14
Hand muscle (med) 6,260 31,737 32 12 12.8 yes 0.62 / 1.56 0.11 / 0.55
Hand muscle (max) 8,951 42,969 96 11 14.2 yes 3.35 / 12.82 0.11 / 0.34
Hand muscle (max-m) 7,552 34,966 151 18 20.3 yes 3.28 / 9.11 0.16 / 0.47
Hip muscle (Fig 5.12) 6,793 34,118 82 21 28.3 yes 7.41 / 21.27 0.39 / 1.85
Hip bone 6,796 28,740 499 34 49.2 no 4.12 / 14.27 0.25 / 1.30
Liver 11,392 43,221 424 80 128.3 no 9.00 / 33.68 0.21 / 4.81
Hand surface (Fig 5.14) 11,829 49,751 456 31 43.8 no 4.87 / 16.78 0.07 / 0.86
method [54]. In Figure 5.19, we evaluate our method in the presence of known ground truth
plastic deformation gradients. Figures 5.21, 5.23, 5.24, 5.25 provide comparisons to surface-based
methods, in addition to Figures 5.6,5.14. In all comparisons, our method outperformed surface-
based methods.
5.9.1 Hand muscles
In our muscle hand example, we extracted 17 hand muscles from an MRI scan (Figure 5.5). We
obtained the scan and the already extracted bone meshes from [134]; scan resolution is 0.5mm x
0.5mm x 0.5mm . We considered two “templates”, the first one from the Centre for Anatomy and
Human Identification at the University of Dundee, Scotland [26], and the second one from Zy-
gote [147]. We used the first one (Figure 5.5, left) because we found it to be more medically accu-
rate (muscles insert to correct bones). Muscle anatomy of a human hand is challenging (Figure 5.5).
We model all muscle groups of the hand, namely the thenar eminence (thumb), hypothenar em-
inence (below little finger), interossei muscles (palmar and dorsal) (between metacarpal bones),
70
Figure 5.17: Our extracted organs match the medical image. The intersection of the output
mesh with the medical image slices is shown in green.
71
adductor pollicis (soft tissue next to the thumb, actuating thumb motion), and lumbricals (on the
side of the fingers at the fingers base). Our template models the correct number and general lo-
cation of the muscles, but there are large muscle shape differences between the template subject
and the scanned subject (Figure 5.1). We solve the optimization problem of Equations 5.3 and 5.4
separately for each muscle, starting from the template mesh as the initial guess. In our results, this
produces muscles that match the attachments and medical image constraints markers at 0.5 mm or
better, which is at, or better than, the accuracy of the MRI scanner.
5.9.1.1 Marking the muscles in MRI scans
During pre-processing, we manually mark as many reliable points as possible on the boundary of
each muscle (∼ 10− 20 landmarks and∼ 50− 100 ICP markers per muscle) in the MRI scans.
This process took approximately 5 minutes per muscle.
5.9.1.2 Attachments to bones
The template muscles are modeled as triangle meshes. We build a tetrahedral mesh for each mus-
cle. Our tet meshes conform to the muscle’s surface triangle mesh; this requirement could be
relaxed. For each muscle in the template, we attach its tet mesh to the bones using soft constraints.
We do this by marking where on one or multiple bones this muscle inserts; to do so, we consulted a
medical doctor with specific expertise in anatomy. For each bone triangle mesh vertex that partic-
ipates in the insertion, we determine the material coordinates (i.e., tet barycentric coordinates) in
the closest muscle tet. We then form a soft constraint whereby this muscle material point is linked
to the bone vertex position using a spring.
5.9.1.3 Direct attempt using segmentation:
We note that we have also attempted to model the muscle shapes directly using segmentation, sim-
ply from an MRI scan. Recent work has demonstrated that this can be done for hand bones [134],
and we attempted a similar segmentation approach for muscles. However, given that the muscles
72
Figure 5.18: Interpenetration removal. The yellow lines are muscle cross-sections in this repre-
sentative MRI slice of the hand. It is clear that our interpenetration removal method successfully
removes penetrations without modifying the interpenetration-free sections of the boundaries of
muscles.
touch each other in many places (unlike bones), the contrast in the MRI scan was simply not suf-
ficient to discern the individual muscles. Our conclusion is that a segmentation approach is not
feasible for hand muscles, and one must use a pre-existing anatomically accurate template as in
our work.
5.9.1.4 Removing interpenetrations of muscles
Many hand muscles are in close proximity to one another and several are in continuous contact.
One strategy to resolve contact would be to incorporate contact into our optimization (Equa-
tions 5.3 and 5.4). This approach is not very practical, as unilateral constraints are very difficult
to optimize. Furthermore, such an approach couples all (or most) muscles, and requires one to
solve an optimization problem with a much larger number of degrees of freedom. It is extremely
73
slow at our resolution; it overpowers our machine. Instead, we optimize each muscle separately.
Of course, the different muscles interpenetrate each other, which we resolve as follows. For each
muscle, we already positioned the MRI constraints so that they are at the muscle boundary. There-
fore, observe that if our marker positioning and the solution to the optimization problem were
perfect, interpenetration would be minimal or non-existent. The marker placement is relatively
straightforward at the boundary between a muscle and another tissue (bone, fat, etc.) due to good
contrast. However, placing markers at the boundary between two continuously colliding muscles is
less precise, due to a lower MRI contrast between adjacent muscles. This is the main cause of the
interpenetrations. We remove interpenetrations with FEM simulation because it produces smooth
organic shape changes; alternatively, one could use geometric approaches [105]. Specifically in
our work, for a pair of interpenetrating muscles, we run collision detection to determine the set of
triangles of each muscle that are inside the volume of the other muscle. On each muscle, we then
determine the set of tetrahedra that are adjacent to the collision area. We then slightly enlarge this
set, by including any tet within a 5-ring of tets. We then run an FEM contact simulation just on
these two tetrahedral sets on the two muscles. FEM simulation pushes the muscle surface bound-
aries apart, without displacing the rest of the muscle (Figure 5.18). We handle contact islands of
multiple muscles by running the above procedure on two muscles, then for a third muscle against
the first two muscles, then the fourth against the first three, and so on. Although in principle this
procedure is order-dependent, the input penetrations were shallow in our examples, making the
procedure relatively unaffected by the processing order.
5.9.2 Hip bone
In our second anatomical example (Figure 5.10), we apply our method to a CT scan of the human
hip bone (pelvis). We obtained the template from the human anatomy model of Ziva Dynam-
ics [146], and the CT scan from the “KidneyFullBody” medical image repository [114]. The
template and the scanned hip bone differ substantially in shape, and this is successfully captured
by our method.
74
5.9.3 Liver
In our third anatomical example, we apply our method to a CT scan of the human liver (Fig-
ure 5.11). We purchased a textured liver triangle mesh on TurboSquid [122]. We subdivided it
and created a tet mesh using TetGen [51]. This serves as our “template.” We used a liver CT scan
from the “CHAOS” medical image repository [65]. We then executed our method to reshape the
template tet mesh to match the CT scan. Much like with the hip bone, our method successfully
models large differences between the template and the scanned liver. Finally, we embedded the
TurboSquid triangle mesh into the template tet mesh, and transformed it with the shape defor-
mation of the tet mesh. This produced a textured liver mesh (Figure 5.11) that matches the CT
scan.
5.9.4 Hip muscle
In our fourth anatomical example, we apply our method to a MRI scan of a female human hip
muscle (gluteus medius) (Figure 5.12). We obtained the data from The Cancer Imaging Archive
(TCIA) [31]. The image resolution is 384× 384× 240 with voxel spacing of 1mm, which is 2x
coarser to the hand MRI dataset. We use the template mesh from the human anatomy model of Ziva
Dynamics [146]. We subdivided it and created a tet mesh for it using TetGen [51]. Because the
muscle is attached to the hip bone and the leg bone, we needed to first extract the bones from the
MRI scan; we followed the method described in [134]. Note that the subject in the Ziva Dynamics
template is male. The template and the scanned hip muscle differ substantially in shape, and this
is successfully captured by our method.
5.9.5 Comparison to variational methods
We compare our method to variational shape modeling methods on an illustrative 1D example.
Note that Figure 5.6 gave a comparison on 3D muscle geometry. Consider an elastic 1D line
segment whose neutral shape is the interval[0,1], and study its longitudinal 1D deformation under
75
Figure 5.19: Quantitative evaluation on ground truth. We first performed a dynamic FEM
simulation with plasticity [57], whereby the back of the dragon is fixed (red), and the head was
pulled to the left with uniform force. This produced our ground truth plastic deformation gradients.
We then selected 488 sparse triangle mesh vertices as landmarks and ICP markers, and ran our
method to compute F
p
and shape x. It is evident that F
p
and x closely match the ground truth.
the following setup. Let us prescribe hard attachments whereby we attach endpoint 0 to position
0, and endpoint 1 to position 2. Furthermore, assume landmarks whereby point 1/4 is at 1/4, and
point 1/2 is at 3/2. Effectively in this setup, we are specifying that the subintervals [0,1/4] and
[1/2,1] do not stretch, whereas the subinterval[1/4,1/2] stretches from its original length of 1/4
to 5/4, (5x stretch). A variational formulation of order r is,
min
x(t)∈C
r
Z
1
0
d
r
x
dt
2
dt+α
x(
1
4
)− 1
4
2
+
x(
1
2
)− 3
2
2
, (5.25)
76
(a) Our method (b) [Huang et. al 2019]
Figure 5.20: Comparison to [54]. Top row: hip bone. Bottom row: liver. Red points are the
markers from the CT scan. We used the publicly available implementation of [54] to compute the
normals, followed by screened Poisson surface reconstruction using the points and computed nor-
mals [67]. We used this combination because it produced better results than running [54] directly.
Our method produces shapes that more closely match the ground truth data.
Figure 5.21: Comparison to surface-based methods. We compare to the “PRIMO” method [22]
because this method is a good representative choice. It also has fewest artefacts in Figure 10 of the
shape deformation survey [20]. Our method produces a clearly superior result in the challenging
scenario where the ICP markers were placed to anisotropically stretch the beam in one transverse
direction. Our method also passes the standard shape deformation benchmark [20].
whereC
r
denotes all functions[0,1]→R whose derivatives exist and are continuous up to order r.
We solved these problems analytically in Mathematica for r= 1,2,3, each time for 3 representative
valuesα, and compared them (Figure 5.22) to our method in 1D,
min
f
p
,x(t)∈C
r
Z
1
0
¨
f
p
2
dt+α
x(
1
4
)− 1
4
2
+
x(
1
2
)− 3
2
2
+β
Z
1
0
˙ x
f
p
− 1
2
dt (5.26)
s. t. x(t)= argmin
x(t)∈C
r
Z
1
0
˙ x
f
p
− 1
2
dt. (5.27)
77
Observe that, in the same vein as in 3D, we can decompose ˙ x= f
e
f
p
, whereby scalar functions f
e
and f
p
and theβ term are the equivalents of the elastic and plastic deformation gradients, and the
elastic energy, respectively. It can be seen that our method produces a better fit to the data and a
significantly less wiggly solution, compared to variational methods (Figure 5.22).
Figure 5.22: Comparison to variational methods. A 1D string of length 1 is attached on both
ends. The left attachment is fixed. The right attachment is moved to coordinate 2. Thus, the spring
stretches longitudinally while attempting to obey the two landmarks at t= 1/4 and t= 1/2. Y-axis
gives the deformed 1D position of the corresponding material point on the X-axis. Big and small
dots denote the attachments and landmarks, respectively. For each method, we plot the result under
a few representative parameters. The blue, red, and green curves represent the different strengths
of the landmarks. With variational methods, one has to either give up meeting the landmarks, or the
curve becomes wiggly. Similar behavior can also be observed in 3D variational results (Figure 5.6).
Our method produces a deformation profile whereby the slope (total 1D deformation gradient) on
all three subintervals [0,1/4],[1/4,1/2],[1/2,1] approximately matches the slope implied by the
attachments and landmarks (1, 5, 1, respectively). Note that the shown output curves are onlyC
r− 1
and not inC
r
at the two juncture points 1/4,1/2; however, their r-th derivative is integrable and
they are the optimal Cauchy limit of a score-decreasing sequence of curves inC
r
.
5.9.6 Comparison to iterative closest point methods
Penalizing the differences of affine transformations between neighboring vertices is one of the
commonly used smoothness terms in “classical” ICP algorithms [5, 6, 77]. Namely, these methods
78
Figure 5.23: Comparison to methods that penalize the difference of neighboring affine trans-
formations[F b]. These methods produce a result similar to the variational method and suffer from
wiggle artefects.
optimize for a 3× 4 affine transformation A=[F b] (x→ Fx+ b) at each vertex, and penalize
an energy that is the sum of the differences in the [F b] matrices between adjacent vertices, plus
the ICP energy. This is similar to variational methods [21, 118], which penalize the difference
in F (as opposed to [F b]). We provide a comparison on our hand muscle example; results are
similar to variational methods (Figure 5.23). Note that “classical” ICP algorithms assume dense
correspondences; they produce artefacts under sparse correspondences (Figure 5.23).
5.9.7 Comparison to ShapeOP
As shown by previous work, variational methods suffer from artefacts under large rotations [20].
One can enhance variational methods by adding local rotations to each vertex, producing the energy
E
bend
=
1
∑
i
A
i
∑
i
A
i
∥∆x
i
− R
i
∆¯ x
i
∥
2
, (5.28)
used by [24, 1, 42]. Here, A
i
is the local V oronoi area of vertex i. We used the publicly available
ShapeOp library [23, 24] to implement E
bend
. We then use E
bend
for shape deformation by adding
79
Figure 5.24: Comparison to E
bend
. We deformed the surface using the ShapeOp library. Here,
α is the strength of the marker constraints (attachments and ICP markers) relative to the bending
energy. In the beam example, our method is presented in green wireframe and E
bend
is shown as
solid. Low values ofα (such as, for example,α= 10
4
) cannot satisfy the constrains; therefore, we
useα = 10
7
andα = 10
8
to meet the constraints in the beam and muscle examples, respectively.
It can be seen that the E
bend
method suffers from volume loss in the beam example. In addition,
it fails to grow the beam (the middle is collapsed). Wiggle artefacts can be observed in the E
bend
method on the muscle.
the ICP energy to it, and compare it to our method (Figure 5.24). The E
bend
energy produces
visible artefacts: when the beam is bent, there is a visible volume loss. When the beam is bent
and forced to grow by the ICP markers, E
bend
cannot grow the volume properly (the middle of the
beam collapses). Similarly, wiggling artefacts can be observed in our hand muscle example. This
80
is not surprising as rotations are relatively small in this example, and therefore E
bend
becomes the
variational biharmonic energy.
5.9.8 Comparison to incremental plasticity
One idea explored in prior work is to incrementally convert “elastic” deformations into “plasticity”
after each ICP iteration [101, 42]. However, doing so introduces an unwanted hysteresis-like effect.
To demonstrate this, we compare our method to [42] who used two different energies combined
with the “incremental plasticity” strategy. For the skull, they used an energy that penalizes volume
change (Equation 1 in [42]). In the beam example, we can see that as one incrementally converts
“elastic” deformation into “plastic” deformation, the bending beam undergoes a large spurious
volume change. When imposing constraints that should cause the beam to inflate in its central
region, the incremental plasticity method produces spikes (Figure 5.25, top left). This is because
the constraints require volume to grow, but the energy penalizes volume growth. For the skin,
Gietzen et al. used an energy similar to Equation 5.28 (Equation 5 in [42]), combined with “incre-
mental plasticity.” On the bending beam, the method suffers from a loss of volume and still causes
the middle of the beam to collapse (Figure 5.25, top right). In the muscle example (Figure 5.25,
bottom), the wiggle artefacts can be reduced by applying “incremental plasticity.” However, the
method introduces a vertical sharp edge at the middle of the muscle (clearly seen in the side view);
our method has no such artefacts. The key difference to our method is that in the incremental
plasticity method, the elasticity is explicitly converted to plasticity without any controlling criteria.
Our method, however, finds optimal plastic strains, via optimization. For example, this makes it
possible to employ minimal plastic strains when elastic energy can already do a good job.
We also compare to ShapeOp and “incremental plasticity” on the benchmark examples of [20]
(Figure 5.26). The ShapeOp cylinder suffers from the same volume loss problem as the beam. In
addition, E
bend
causes self-collisions on the cactus example. “Incremental plasticity” improves the
results somewhat, but it still cannot pass the benchmark.
81
Figure 5.25: Comparison to [42]. We compare our method to two different deformation energies
used in [42]. Equation 1 penalizes the volume change relative to the previous ICP iteration. Equa-
tion 5 penalizes E
bend
relative to the previous ICP iteration (“incremental plasticity”). On the beam
example, our method is depicted via a green wireframe, whereas the compared method is depicted
as a solid. It can be seen that the “incremental plasticity” method suffers from similar artefacts
as other prior methods (but to a lesser degree). However, “incremental plasticity” introduces its
own problems, namely volume loss and waviness (observable during bending the beam) and sharp
creases (that occur while growing the muscle). To eliminate any effects of incorrect correspon-
dence from the experiment, both examples use landmark constraints (depicted as blue spheres);
the results are even more favorable to our method when using ICP constraints.
82
Figure 5.26: Comparison to E
bend
on the shape benchmark of [20].
5.10 Discussion and conclusion
We gave a shape deformation method that can model objects undergoing large spatially varying
strains. Our method works by computing a plastic deformation gradient at each tet, such that the
mesh deformed by these plastic deformation gradients matches the provided sparse landmarks and
closest-point constraints. We support both constrained and unconstrained objects; the latter are
supported by giving a numerical method to solve large sparse linear systems of equations with
a known nullspace. We applied our method to the extraction of shapes of organs from medical
images. Our method has been designed to extract as much information as possible from MRI
images, despite the soft boundaries between the different organs. We extracted hand muscles, a
liver, a hip bone and a hip muscle.
83
Our method does not require a dense target mesh; only a sparse set of observations is needed.
If a dense target mesh is available, the problem becomes somewhat easier, as one can then use
standard ICP algorithms. However, medical images contain many ambiguities and regions where
there is not a sufficient contrast to clearly disambiguate two adjacent medical organs; making it
impractical to extract dense target meshes. We apply our method to solid objects, but our plastic
strain shape deformation method could also be used for shells (cloth). Doing so would require
formulating the elastic energy of a plastically deformed FEM cloth, and computing the energy
gradients and Hessians with respect to the plastic parameters. The size of the small square dense
matrix that we need to invert in our incremental solve is three times the number of markers. While
we easily employed up to a thousand of markers in our work, our method will slow down under
a large number of markers. We do not re-mesh our tet meshes during the optimization. If the
plastic strain causes some tetrahedra to collapse or nearly collapse, this will introduce numerical
instabilities. Although not a problem in our examples (see Figure 5.16), such situations could be
handled by re-meshing the tet mesh during the optimization [15]. Our method requires a plausible
template with a non-degenerate tet mesh. Re-meshing is important future work as it could extend
the reach of our method, enabling one to start the optimization simply from a sphere tet mesh.
84
Chapter 6
Comprehensive Modeling of the Human Hand
Using MRI techniques, we were able to capture the interior of the hand. As can be seen from the
MRI presented in Figure 6.1, a human hand contains multiple different organs. We cannot only
find the bones, but also other organs such as muscles, tendons, and fat. We were able to model all
soft tissues in the hand as a single soft tissue in the previous chapter. However, both functionalities
and mechanical properties of these organs are rather different. For example. a fat tissue is very soft
and is a passive tissue. On the contrary, the muscle tissue is much stiffer compared to the fat tissue
and is an active tissue. Therefore, to improve the accuracy of our hand model, we model different
organs separately. In addition, we have captured MRIs in multiple example poses. By using this
data, we can improve the accuracy of our hand model.
We model tendons, fat, muscles, bones, and ligaments separately. As presented in Figure 6.2,
we model each organ differently. In our simulation hierarchy, we first animate the bones. The
method to animate bones is described in Chapter 4. On top of the bones, we perform bone fascia
simulation to fill out the valleys between the bones. The simulation is achieved using a cloth
solver, which is described in Section 6.3. Thereafter, we perform tendon simulation based on the
bones, which is explained in Section 6.2, and we run muscle simulation, which is explained in
Section 6.1. After the simulation, we treat muscles, tendons, and the bone fascia as kinematic
objects. Similarly to the bone fascia, we do muscle fascia simulation. Then, we perform the joint
ligament simulation. The ligaments are treated the same as muscles, which is controlled by plastic
strains. Finally, we perform fat simulation. When the fat is being simulated, all other objects are
85
Figure 6.1: MRI slices in the coronal plane and transversal plane. Here, we show two MRI
slices of human hand. It is evident that there are several different tissues inside the hand represented
by different intensities in the MRI image.
Figure 6.2: The overview of our human hand rig. We model tendons (dark purple), fat (flesh
color), muscles (light pink), bones (light yellow), and ligaments (green) around joint regions. To
improve the simulation quality, we also model two additional fascia layers: bone fascia (light blue)
and muscle fascia (light pruple). The fat tissue is directly attached to fascia layers and ligaments.
kinematic. The fat layer is directly constrained to tendons, the muscle fascia, the bone fascia, and
86
joint ligaments. The fat layer simulation will be explained in Section 6.4. The remainder of the
pipeline from simulation results to the rendering results has been discussed in Chapter 4.
6.1 Hand muscle
Figure 6.3: Muscle attachments. The muscle attachment vertices are depicted as red dots. To
show the attachments clearly, the muscles are visualized using dark wireframes in the left two fig-
ures. Further, to present the relative locations to the bones, we show the bones in dark wireframes
in the right two figures.
We now describe our muscle preprocessing pipeline and simulation model. We first extract the
muscles at the neutral shape using the approach described in Chapter 5. To do so, we specify land-
marks, attachments and ICP markers manually based on the MRI images and guided by medical
literature and a medical doctor. For landmarks, we try to identify the anatomical correspondences
between the MRI and the input template mesh. Further, we place ICP markers at the locations of
the muscle boundary in the MRI with high confidence. For the attachments, we first specify the
attachment vertices at the template mesh. Then, we extrapolate the vertex positions based on the
deformation from the template bone meshes to our bone meshes. Specifically, for each attachment
vertex, we find the closest patch on the template bone meshes. The patch is defined as a set of
connected triangles. Then, we extract the transformation based on the deformation of the same
patch from the template bone meshes to our bone meshes. Finally, the attachment vertex is trans-
formed based on the extracted transformation. After that, we further check whether the attachment
87
vertex positions correctly correspond to the MRI. If not, we manually adjust them using modeling
tools. Applying above steps yields the final results, as depicted in Figure 6.3. After we obtained
the markers and attachment locations, we successfully extracted all hand muscles in the neutral
pose, as shown in Figure 5.5. As is well known, human muscles are not passive tissues and muscle
deformation is governed by Hill’s model [144]. Hill’s model takes muscle activation into account
and combines it with passive elastic deformation. Given the muscle elastic material properties,
fiber directions, and activation parameters, the model is able to deform the muscle accordingly.
However, in practice, these parameters are difficult to obtain. For example, extracting the fiber
directions of a muscle from MR images is usually not feasible or reliable. It requires imaging
techniques like diffuse tensor imaging, which is prohibitively expensive and rarely available. It is
only possible to approximate them based on the shape of the muscle [102]. These facts make Hill’s
model difficult to apply. Instead, we use 6-DOF plastic model from [55] as our muscle activation
model.
Unlike [55], we do not embed all tissue into a single tetrahedral mesh. Instead, we model each
muscle into a separate tetrahedral mesh. The first reason for doing this is that tissues are very
often sliding and contacting against each other in addition to attaching to certain locations. If a
single tetrahedral mesh were used, such effects are neglected. Moreover, it is much more reliable
to define the pose space for each muscle individually than to define it for the entire hand. Unlike
human faces, a human hand has a highly concave shape and has a large amount of flexible joints.
If a single tetrahedral mesh were used, each joint would contribute a few DOFs into the pose
space, thereby making the final dimension of the pose space large. Admittedly, we could define a
spatially-varying pose space, meaning that different parts of the hand have different pose spaces,
but an accurate and smooth separation of the human hand is not easy. Moreover, given the fact that
our example poses are sparsely distributed in the pose space, the large dimension of the pose space
implies lower accuracy of the interpolation.
As shown in Figure 6.4, a muscle is attached to several rigid bones. As described earlier in
Chapter 4, bones undergo rigid motions around their parent bones. The attachment vertices (in
88
Figure 6.4: Overview of muscle simulation. A muscle is attached to more than one bone (or
tendons). The muscle surface is embedded into a tetrahedral mesh. The muscle contraction is
controlled by the plastic strain of each tetrahedron.
red) move rigidly with the attached bones. The muscle tetrahedral mesh contains plastic strains
that are also controlled by the joint transformations.
6.1.1 Muscle pose space
Without loss of granularity, consider a single hand muscle. We assume we have N
p
example poses.
For example pose i, denote the surface mesh by S
i
, and the plastic strain by P
i
. Then, the next
question we solve is to define the pose space for the muscle. A single muscle usually attaches
to several bones and tendons at the ends of the muscle. The tendons, as described subsequently,
are also controlled by the motion of the bones. Thus, we can consider that the muscle is solely
controlled by the motion of the bones. The dominant motion of the bones are rotations around the
parent bones and, thus, we define the pose space of the muscle by the rotations of the related bones.
Take a muscle with two attached ends as an example in which each attached bone contributes one
rotation. Since we do not care the global rigid transformation of the muscles, we can eliminate
one of the two rotations. Consequently, the pose space of a muscle with two insertions is only
controlled by a single rotation.
89
To represent a rotation, there are several choices: 1) a quaternion; 2) a 3D rotation matrix;
3) a 3D axis-angle vector. Using a 3D rotation matrix leads to a high dimension of the pose
space. In addition, when using a 3D rotation matrix or a quaternion, it is difficult to measure the
distance to the rotation matrix/quaternion of the example pose and, thus, it is difficult to generate
interpolation weights for example poses. In contrast, a 3D axis-angle vector is defined in Euclidean
space, so it is simple to compute the distance to each example pose. Moreover, compared to other
options, a 3D axis-angle vector has the least number of dimensions. We still need to address the
problem that multiple 3D vectors correspond to the same rotation. To solve this problem, we
define two additional conditions when computing the corresponding 3D vector to a rotation. The
first condition is that the rotation axis must form a angle of less than 90 degree with the primary
rotation axis of the joint. The primary rotation axis has been computed when we generated the joint
hierarchy of the skeleton. We define the rotation axis with the largest angle ranges as the primary
rotation axis. By doing so, all rotation axes of the rotations in all example poses will be similar to
each other. This avoids the flip of the rotation axes. The second condition is that the rotation angle
must be between− 2π and 2π, because any angle +/- 2π is equivalent to itself.
With these definitions, we are able to define the pose space of a muscle. Assume a muscle
has N
e
insertions attaching to the bone. Then, the dimension of the pose space is 3(N
e
− 1). If a
muscle is attached to the tendon, then we treat the bones that control the tendon as the bones that
also control the muscle in addition to the bones that the muscle is directly attached to. Once the
bones are determined, we first extract the current rotation of the joint that each bone is rotating
around. Then, we remove one joint that is the common ancestor of all joints to eliminate the
global rotation. Finally, we convert all rotations into 3D vectors using the method mentioned in the
previous paragraph, and combine them into a pose-space vector a
i
for example pose i. We repeat
this process for all example poses for all muscles.
During the simulation, we encounter an arbitrary pose-space vector a. Then, the interpolation
weights w
i
for example pose i are defined as
90
w
i
=
φ(a,a
i
)
∑
N
p
j
φ(a,a
j
)
(6.1)
φ(a,b)=
1
∥a− b∥
. (6.2)
Thus, the final plastic strain P of the muscle is computed as
P=
N
p
∑
i
w
i
P
i
. (6.3)
We do not use radial basis functions to interpolate the plastic strains, because this introduces neg-
ative weights, which in turn causes the determinant of the plastic strain to be negative or close to
zero, thereby leading to simulation instabilities.
6.1.2 Muscle plastic strain extraction
As we use plastic strains to control the muscle activation, we need to have plastic strains for all
example poses defined on a single tetrahedral mesh for each muscle. To obtain the plastic strain P
i
at example pose i, we start from the volumetric mesh at the neutral pose. We could simply apply
the method proposed in Chapter 5 for each example pose separately, as it can directly output the
plastic strain. However, such an approach has several issues. First, the method does not provide the
smoothness of the plastic strain between example poses. Our objective is to achieve the property
that if the pose-space vectors of two example poses are close to each other, the corresponding plas-
tic strains must be similar to each other. As the muscle boundaries are sparse markers, the surface
areas where there are no markers are only deformed in accordance with the spatial smoothness
but not the pose-space smoothness. We observed that if we did not apply the pose-space smooth-
ness, the muscle simulation produced non-smooth motions when transitioning from one example
to another one. Second, when optimizing the shape to match the sparse markers, the optimization
parameters—like attachment strengths—may not always be the same for a single muscle across
91
all example poses, for modeling purposes. Therefore, applying the plastic strains obtained sep-
arately using the method described in Chapter 5 to each example pose, the runtime simulation
may not reproduce the same shape. In general, such coupling between simulation methods and
modeling/segmentation methods is not desirable. Ideally, we should be able to obtain the muscle
shape using any applicable method, and then generate smooth pose-varying plastic strains across
the entire hand range of motion using such data.
Figure 6.5: The muscle and joint ligament surface meshes. The muscles are presented in red
and the joint ligaments are presented in green.
To obtain smooth muscle plastic strains across the range of the motion of the hand, we proposed
the following procedure. Assume that the plastic strains for the neutral pose are identity matrices.
At the beginning of the procedure, only the neutral pose is the processed example pose.
1: Select a non-processed example pose. Select the non-processed example pose that is the
most different from all already processed example poses.
2: Prepare the initial guess for the method of Chapter 5. If there are two or fewer processed
example poses, use the neutral shape as the initial guess for the selected example pose (ex-
plained later in Section 6.1.3). Otherwise, use the processed example poses to initialize the
92
muscle simulation model (explained later in Section 6.1.4). Then, simulate the muscle to the
selected example pose (explained later in Section 6.1.5), which yields the initial tet mesh of
the muscle.
3: With the initial shape, apply the method of Chapter 5 to generate the tet mesh of the
muscle. During this process, we manually place markers on the boundary of the muscle in
MRI based on the medical knowledge, as explained in Chapter 5. These markers will be
reused in further iterations of the algorithm.
4: Add the selected pose to the processed example poses and go back to step 1.
It can be seen that each iteration (four steps) involves simulating the muscle to the example
pose, and then optimizing the muscle shape, which is time consuming. To extract the muscle
shapes in all N
p
example poses, at least N
p
iterations are required, because each pose must at least
be processed once. After N
p
iterations, we can continue repeating it by flagging all processed
poses as unprocessed. This time, we start from the first example pose to the last example pose.
Each time we select a pose, we treat the other ones as processed poses when generating the initial
guess. We can continue repeating these iterations as much as we want. In our demos, we use six
example poses. We apply 1× 6 iteration for nine muscles and six ligaments and 2× 6 iterations
for the remaining six muscles. At the end of this procedure, we obtain the muscle simulation
rig for the hand, which includes the plastic strains for each muscle in each example pose and
the corresponding pose-space vector. By applying our procedure, we successfully processed 15
muscles (red) and 6 joint ligaments (green), as illustrated in Figure 6.5.
6.1.3 Initial guess of muscle tet mesh in non-neutral example poses
When the number of processed example poses is two or fewer, we create the initial guess directly
from the shape in the neutral pose. To do so, we first create a muscle fiber field by solving a Laplace
equation [102]. We assume that a muscle only contracts along the fiber field when it switches the
pose from the neutral pose to any other pose. Then, we define R diag(a,1,1) R
T
as the plastic
93
strain of each tetrahedron for the muscle, where R is the local frame defined by the muscle fiber
direction and a is the muscle contraction parameter [102]. By applying the same idea as described
in Chapter 5, we first solve a simple optimization problem
argmin
a,x
∥La∥
2
+ c
1
E
elastic
(a,x)+ c
2
E
att
(x), (6.4)
where a is the muscle contraction parameters for all tetrahedra, x is the shape of the muscle, E
elastic
is the elastic energy, and E
att
is the attachment energy. This problem is a simplified version of the
optimization problem defined in Chapter 5, because (1) it uses a much smaller space for the plastic
strains, (2) it does not attempt to match the sparse markers, and (3) we treat the static equilibrium
constraint as a penalty term in the objective function. The reason we can do (3) is that there
are no sparse markers. Therefore, the unconstrained optimization problem can be solved simply
by using Newton’s method. Intuitively, this objective function attempts to ascertain the muscle
contraction along the transversal direction, so that, in the static equilibrium under the contraction,
the attachment energy to the bones is small. The DOFs provided by the contraction a are necessary,
because the shape of a straight muscle would bend, as opposed to contracting, when two insertions
come closer to each other. This is due to volume preservation terms in the elastic energy.
To make the problem even simpler, we perform model reduction over vector a. We manually
select a few tetrahedron (usually 3-4) on the tetrahedral mesh and treat them as “handles,” meaning
that the contractions of these tetrahedra will control the entire a. Then, we generate the subspace
basis U by calculating bounded bi-harmonic weights [59]. Then, the contraction a= Uˆ a, where ˆ a
is the subspace quantity, and our optimization problem becomes
argmin
ˆ a,x
∥LUˆ a∥
2
+ c
1
E
elastic
(Uˆ a,x)+ c
2
E
att
(x). (6.5)
After we solve it, the initial guess of the muscle shape is given by x.
94
Figure 6.6: Muscle preprocessing results. In step 2, we remesh the muscle surface mesh. As
demonstrated on the left, remeshing improves the triangle quality. On the right of the figure, the
target attachment locations are depicted as red dots, and the green dots are constraints to preserve
the shape of the muscle. The target attachment locations are obtained by rigidly transforming the
positions of the attachment vertices at the neutral pose to the target example pose. By applying
step 3, we make the surface vertices better match the attachment locations.
6.1.4 Generating muscle volumetric plastic strains in example poses
Before we can run muscle simulation with pose-varying plastic strains, we need to generate vol-
umetric plastic strains from the example muscle surface meshes that have been registered to each
other across example poses. The following are the steps to achieve it.
Step 1: Compute pose-space vectors for all existing example poses. This step has been de-
scribed in Section 6.1.1.
Step 2: Remesh example surface meshes (if needed) for improving the mesh quality. During
the muscle modeling, the mesh quality may decrease on account of matching the sparse markers.
This step improves the mesh quality, which leads to a better-quality tetrahedral mesh and simula-
tion results (Figure 6.6 left). To keep the surface meshes registered across all example poses, we
only remesh the surface mesh at the neutral pose. Then, we embed the remeshed surface to the
input surface mesh and deform it to all other non-neutral example poses. This creates registered
remeshed surfaces in all example poses.
95
Step 3: Eliminate muscle-bone attachment mismatches incurred during the muscle model-
ing process. During the muscle modeling, the attachment vertices may deviate from the target
attached locations in order for the surface to match the markers better (Figure 6.6 right). Even
though these errors are very small (usually about 1mm), they must be eliminated to the furthest
extent possible. Note that if the error is zero, there is no attachment force in the example pose.
Under static equilibrium, the elastic forces will also be equal to zero because there are no other
forces. Therefore, the resulting shape under static equilibrium is only deformed by the example
plastic strain.
To achieve this zero error, we keep all muscle surface vertices that are further than d edges from
the attachment vertices fixed (green dots in Figure 6.6). We use d= 4 for our muscle simulation. In
addition, we constrain all attachment vertices to the exact target locations (red dots in Figure 6.6)
obtained by rigidly transforming the neutral pose positions of the attachment vertices. Then, we
define a smoothness energy and solve for positions of the other vertices,
argmin
x
cE
fixed
(x)+ cE
att
(x)+ E
smooth
(x), (6.6)
where c is a large constant, E
fixed
(x) and E
att
(x) penalize the mismatch of fixed vertices and attach-
ment vertices, and E
smooth
(x) defines the smoothness of the shape. We typically use bi-Laplacian
smoothness of the surface positions or bi-Laplacian smoothness of the deformation gradients.
Step 4: Resolve the interpenetrations between muscles. As we extract the muscles separately,
it is inevitable that muscles may be in contact with each other. Section 5.9.1.4 gives a method
to procedurally resolve the contacts muscle by muscle. However, we found that it is too slow to
process all muscles in all example poses. It takes hours to resolve the contact in a single example
pose. This is because the volumetric meshes of muscles have many DOFs, and the effect of contact
propagates volumetrically, thereby imposing a large computational burden. In contrast, the surface
of the muscle is easier to deform under contact if a surface-based method is used. Therefore,
we prefer a surface-based deformation method. In the end, we found that using ShapeOp as a
96
replacement of the elastic energy dramatically improves performance [23]. Specifically, we use
ShapeOp bending energy as our smoothness energy [24], and then we add a sliding constraint for
each vertex-triangle colliding pair. In our experiments, it takes less than 10 minutes to resolve the
inter-contacts between muscles for each example pose.
Step 5: Given the cleaned-up and registered surface meshes for all example poses, find volu-
metric plastic strains to satisfy both spatial smoothness and pose-space smoothness. At first
glance, it appears that we can modify our method from Chapter 5 in the following manner
argmin
s
i
,x
i
N
p
∑
i
||Ls
i
||
2
+αE
MI
(x
i
)+βE
a
(x
i
)
+γ||L
ps
s||
2
, (6.7)
st. f
e
F
p
(s
i
),x
i
+ f
a
(x
i
)= 0,for each i= 1,2,...,N
p
(6.8)
where L
ps
is a pose-space Laplacian matrix. We can build a graph in the pose space. For
each example pose-space vector, we find m closest example pose-space vectors as the neigh-
bors. Parameter m is set to be equal to the number of example poses - 1. Then, we can build
a graph and define L
ps
as the graph Laplacian matrix. Therefore, ||L
ps
s||
2
can be rewritten as
∑
N
p
i
||s
i
− ∑
j∈N(i)
w
i j
s
j
||
2
. To solve this, we adopt the idea of the block-gradient descent method.
In each iteration, we randomly select an integer k∈[1,N
p
] and fix all s
j
,x
j
for j̸= k. Then, the
problem in each iteration becomes
argmin
s
k
,x
k
||Ls
k
||
2
+αE
MI
(x
k
)+βE
a
(x
k
)+γ||s
k
− ¯ s
k
||
2
, (6.9)
subject to: f
e
F
p
(s
k
),x
k
+ f
a
(x
k
)= 0, (6.10)
where ¯ s
k
=∑
j∈N(k)
w
k j
s
j
. Since s
j
are fixed, ¯ s
k
is fixed. Closely examining the objective function,
we can see that this formulation produces artifacts. Remember thatE
MI
(x
k
) andE
a
(x
k
) are sparse
observations. They sparsely specify that certain points on the surface of the muscle must reach
the target positions. On the other hand, ||s
k
− ¯ s
k
||
2
implies that the objective function attempts
97
to match the plastic strains of the entire mesh to given target plastic strains. This implies that it
seeks a concrete target shape, which is a dense observation. Therefore, the dense term||s
k
− ¯ s
k
||
2
conflicts with the sparse terms E
MI
(x
k
) andE
a
(x
k
). We have implemented this approach and
verified experimentally that the method does not work. In order for the pose-space smoothness
to be effective, the weight γ has to be sufficiently large (¿= 1.0). The weight α must also be
sufficiently large compared with γ because our top priority is to match the ICP markers. We
observed that there are local tiny bumps around the sparse ICP markers. This happens because
the ICP marker positions conflict with the target shape represented by the dense plastic strain ¯ s
k
.
The conflict is unavoidable. To avoid this problem, we have to use a dense observation of the
ICP markers. However, our method only supports sparse ICP markers due to the time complexity.
Therefore, this approach is not feasible.
The next idea (which we also abandoned) is to use deformation transfer [118]. This is also
commonly used for anatomy transfer [102]. In this method, we use the resulting muscle surface in
each example pose generated by previouse steps. The formulation is essentially the same as that in
Equation 6.7 except that there is no elastic deformation.
argmin
x
i
N
p
∑
i
||LF(x
i
)||
2
+αE
MI
(x
i
)
+γ||L
ps
F(x))||
2
, (6.11)
where F(x) is the deformation gradients of x, andE
MI
(x
i
) are now dense correspondences. Here,
F(x)= G(x− ¯ x), where G is the gradient operator matrix and ¯ x is the rest position of the volumetric
mesh. As we can see, the deformation gradient F is a linear function of the position x. The resulting
x can then be converted to plastic strains. Because we are using isotropic hyper-elastic materials,
the plastic strain is essentially the symmetric matrix of the polar decomposition of the deformation
gradient F for each tetrahedron. However, this method cannot handle rotations well, leading to a
large amount of inverted deformation gradients. To solve this problem, we could add a non-linear
inequality constraint that guarantees that the determinant F of each tetrahedron is positive, that
is, det(F)>ε, where ε is a very small positive number such as 0.02. Then, we can solve the
98
optimization using the interior-point method. Nonetheless, this method creates extreme plastic
strains whose determinants constantly hit the boundaryε and whose eigenvalues span widely from
ε to 15.2. This causes problems when interpolating the plastic strains during simulation. Moreover,
the interior-point method cannot always find a solution for all the muscles. To address the problems
due to the rotations, we replaced the Laplacian smoothing energy from deformation transfer with
a standard elastic energy [110]. By tuning the resolution of the tetrahedral mesh, the interior-point
method could find a solution for every muscle. Nonetheless, this method still created extreme
plastic strains. This is because an elastic energy penalizes the growth of the object, while the
muscles are growing/shrinking in different example poses.
To address all mentioned issues, we define the following energy:
argmin
x
i
N
p
∑
i
||LS(x
i
)||
2
+αE
MI
(x
i
)
+γ||L
ps
S(x))||
2
, (6.12)
where S(x) is a function that extracts a symmetric matrix by performing polar decomposition on
each deformation gradient. Under this definition, the objective function does not penalize the
growth of the object and is not affected by the rotation. Note thatE
MI
(x
k
) uses a dense corre-
spondence to the input shape, which is the result of Step 4. We use the same idea to compute L
ps
as the first method, that is, a graph Laplacian matrix. To solve the optimization problem, we use
block-gradient descent. In each iteration, we randomly select an integer k∈[1,N
p
] and fix all s
j
,x
j
for j̸= k. Then, the problem becomes
argmin
x
k
||LS(x
k
)||
2
+αE
MI
(x
k
)+γ||s
k
− ¯ s
k
||
2
(6.13)
at each iteration. The resulting x can then be converted to plastic strains, as explained earlier.
To perform optimization efficiently, we first solve N
p
separate optimization problems without the
pose-space smoothness term. The solution for each example pose is used as the initial x to our
optimization problem. Then, we begin our block-gradient descent. We do 15 iterations for all our
muscles. Energy weight γ is tuned based on the maximum distance between the output surface
99
and the input surface. We use 0.5mm as the maximum allowed distance. By using the strategy of
bisection, we can automatically determine the value ofγ for each muscle. Note that we did not add
an explicit constraint that guarantees that the deformation gradients corresponding to x are positive
definite. This is because even without the constraint the eigenvalues of our resulting plastic strains
for all muscles in all example poses were between 0.08 and 2.3 in our experiments.
6.1.5 Muscle simulation
We perform dynamic simulation for our muscles. We do not need the dynamic motion of the
muscle, so we use a simple and stable integrator, namely implicit backward Euler. During each
timestep, we first calculate the bone rotations and convert them to the pose-space vector for each
muscle. Based on the pose-space vector, we compute the plastic strains for each muscle using
Equation 6.3. In addition to the plastic strains, we apply four different types of constraints during
the simulation to mimic the muscle biomechanical behaviors. They are (1) muscle attachments to
tendons and bones; (2) muscle contacts to bones; (3) muscle inter-contacts and self-contacts; and
(4) muscle inter-slidings. The first three are easily understandable. Muscles are attached to the
bones and tendons. Muscles also cannot penetrate the bones, other muscles, and themselves. In
addition to these facts, we found that muscles are also sliding against neighboring muscles. To
model this effect, we add the sliding constraints between muscles. We also tested what happens if
there are no sliding constraints: large empty space appears between neighboring muscles, which
is unrealistic. Due to these constraints, we treat all muscles as a single simulation system and
simulate them together.
For joint ligaments, we process them using the same pipeline as muscles. For simulation, they
are also treated the same as muscles, except that we do not model contacts and sliding between
them. Therefore, we simulate each joint ligament separately.
6.2 Hand tendons
Tendons are important not only because they control the motion of the hand, but also because
they affect the appearance of the hand. In this thesis, we only model tendons that are attached
100
to the muscle, because they drive the muscle deformations. Tendons are often beam-shaped and
are always black in the MRI image, thereby giving them good contrast to the neighboring tissues.
However, tendon extracts is difficult in practice because of the limited resolution of the MRI. For
such reason, thin tendons are often impossible to extract. We only model four tendons from the
end of the flexor digitorum superficialis muscles for index finger, middle finger, ring finger, and
pinky finger. because some muscles are attached to these tendons. Because they are thick and the
contrast is clear, they are easily extracted from MRI.
6.2.1 Tendon simulation
Sachdeva et al. modeled hand tendons biomechanically [100], but the complex model requires
many biomechanical constraints, such as attachment to the bones and sliding through the pulleys.
In their work, these structures are obtained manually by referencing the medical literature, but they
are difficult to discern from MRI; hense we did not adopt their method.
Figure 6.7: Tendon simulation model. The left of the figure presents our rod simulation model.
We break the rod into linked rigid segments. Each segment is represented by the positions of two
end vertices and a normal showing the orientation. The normal (orange) is perpendicular to the
segment. The tendon in our system is simulated using such a rod. The rod goes through a few
locations (red circles) that are skinned to the closest bones. One end of the tendon is attached to
the bone with a fixed constraint. In addition, we apply a constant force (purple) at the end of the
tendon to stretch it as much as possible without invalidating the constraints.
Similarly to [100], we model tendons as rods. We could use a complex Lagrangian coordinate-
based rod simulation model used in [100], but we found that it is much simpler and sufficient to
use a Euclidean coordinate-based model. Our tendon model captures the following effects: (1) the
length of the tendon is constant and (2) the tendon does not twist. As illustrated in Figure 6.7 (left),
we first discretize the tendon into N small rigid segments. Each rigid segment i is controlled by the
101
vertex positions x
i
,x
i+1
of two ends and the normal of the segment n
i
. Normal n
i
is perpendicular
to the segment i. The simulation flexible DOFs are x
i
and n
i
.
As presented in Figure 6.7 (right), one end of the tendon (blue) is attached to the closest bone
and is rigidly transformed with the attached bone (bone 1). The other end of the tendon is pulled
by a constant external force (purple) in the transverse direction to mimic the fact that the muscle
of the forearm is pulling the tendon. The tendon is constrained by a series of points (shown in
red circles) near the bones. The tendon passes through these points, that is, the tendon is sliding
against them. Given the k sliding locations, constant external forces, and the fixed attachments, we
define our tendon static simulation as the following optimization problem:
argmin
x,n,t
sliding constraints
z }| {
∑
i∈S
((1− t
i
)x
i
+t
i
x
i+1
− ¯ x
i
)
2
− pulling force
z }| {
c
1
x
T
N+1
f − twisting
z }| {
c
2
N− 1
∑
i=1
n
T
i
n
i+1
− c
3
n
T
1
¯ n
1
(6.14)
subject to(x
i+1
− x
i
)
2
− ℓ
2
= 0, ∀i∈[1,N+ 1], (6.15)
n
T
i
(x
i+1
− x
i
)= 0, ∀i∈[1,N], (6.16)
n
T
i
n
i
− 1= 0, ∀i∈[1,N], (6.17)
0≤ t
i
≤ 1, ∀i∈ S, (6.18)
x
1
= ˆ x
1
, (6.19)
where S is the set of indices of segments with sliding constraints, t
i
∈[0,1] is the line segment
barycentric coordinates giving the closest position to the given sliding constrained location ¯ x
i
,
c
1
,c
2
are parameters,ℓ is the length of the rigid segments, ˆ x
1
is the fixed attachment location, and
f is the constant pulling force. We define the sliding constraint as the sum of the squared distance
from each sliding location to the closest point on the closest segment. For the twisting, we define it
as the negative dot product between two neighboring normals. The bigger the dot product, the less
twisting. Nevertheless, it is not sufficient to determine all n
i
because the tendon could be globally
rotated by the same rotation. Therefore, we use an input parameter ¯ n
1
to regularize the normals.
In the constraints, we guarantee that n
i
is perpendicular to the segment i and is a unit vector. The
102
optimization problem is solved using the interior-point method. The method can easily find the
optimal solution.
Figure 6.8: Tendon simulation. Here, we show the rods representing tendons during simulation.
The red lines are the rigid segments and their normals. The blue dots are the sliding locations.
Given the simulation formulation, the only thing left is how we obtain the sliding locations,
fixed attachment location, and the constant force. For the fixed attachment location ˆ x
1
, it is com-
puted by rigidly transforming its initial position using the transformation of the closest bone. For
the pulling force, we define its direction as the opposite of the primary direction of the last few
segments in the rest configuration, because they are usually at the wrist region of the hand and
move along their primary direction in the real world. For each sliding location, we compute it us-
ing skinning + pose-space correction using bone transformations. This gives the location based on
the rotations of bones. After skinning and pose-space correction, we obtain new sliding locations
in each frame. These sliding locations, however, may not be evenly distributed. We observed that
some points may be very close to each other. To solve the problem, we filtered out points that
are very close to each other, because these sliding constraints would introduce noises. Figure 6.8
presents the tendon sliding locations (blue) and the tendon simulation results. The process to obtain
skinning weights and the pose-space corrections are explained in the next section (Section 6.2.2).
103
6.2.2 Tendon preprocessing
To perform the simulation, we need to know the tendon sliding locations. They are determined
from MRI in multiple example poses. Therefore, we first extract the tendons from MRI.
6.2.2.1 Tendon extraction
Although we model tendons as rods, real tendons are volumetric objects. To extract tendons, we
first treate them as cylinder tubes whose centerlines are our simulation rods. We assume that the
radius of the tendon does not change during the simulation and, thus, the surface shape of the
cylinder is skinned by the centerline.
We create a template cylinder manually. We then extract the tendon mesh from the neutral pose
MRI image using classic computer vision techniques. Next, we non-rigidly deform the template
cylinder tube to match the MRI mesh. The center rod is deformed according to the surface mesh
as we can consider it to be embedded into the tube volume. In this manner, we create our neutral
tendon simulation rod and the surface mesh. For non-neutral poses, we only extract the surface of
the tendon from MRI using classic computer vision techniques.
6.2.2.2 Tendon registration
Now, we have the tendon rod and the surface mesh in the neutral pose, and the extracted surface
mesh in non-neutral poses. We need the tendon rod in these non-neutral poses. We want our tendon
skinning surface mesh driven by the tendon rod to match the MRI mesh as closely as possible in
each example pose. This is the standard non-rigid registration problem, but occurring on a skinned
surface controlled by the rod. As is well-known, non-rigid registration requires a good initial
guess to produce a high-quality result. Therefore, the registration of example poses is done in
three stages. The first two stages create a good initial guess and the last stage performs the actual
non-rigid registration.
104
Figure 6.9: Tendon stages. In stage 1, we simulate the tendons based on the sliding constraints
that are only rigidly transforming with the closest bones. This gives a relatively correct position
of the tendon. However, it does not match the MRI shape (blue wireframe). To make it match as
close as possible, we first create an initial guess for our non-rigid registration (stage 2). As shown
in (2), the resulting mesh matches the MRI shape better. In stage 3, we do non-rigid registration to
match the MRI shape.
Stage 1: Deform tendon to the target example pose. We manually select a few points on the
tendon rod, which we think are rigidly transforming with the closest bones. These points are
considered as sliding locations. We then slowly deform our bone rig from the neutral pose to the
target example pose. In each iteration, we compute the positions of the selected points. Since
they are rigidly transforming with the closest bones, the positions are easily obtainable. Then, we
perform our tendon simulation to obtain the shape of the tendon using Equation 6.19. The result is
depicted in Figure 6.9(a).
105
Stage 2: Deform tendon to match the centerline of the extracted MRI mesh. After the first
stage is completed, the position of the resulting tendon is close to the MRI mesh. Because the
tendon is thin, we need to further match the MRI mesh to guarantee the quality of the final non-
rigid registration. To do so, we first extract the centerline of the MRI mesh and sparsely sample
a few points (10 points). Then, we use them as our sliding locations and run our simulation,
beginning from the result of stage 1. The result of stage 2 is depicted in Figure 6.9(b).
Stage 3: Deform tendon to match the extracted MRI mesh. In the last stage, we perform
non-rigid registration by solving the following optimization problem:
argmin
x,n
ICP constraints
z }| {
∑
i
ˆ n
T
i
∑
j∈M
i
w
i j
T
j
(x
j
,x
j+1
,n
j
) ¯ p
i j
!
− ˆ p
i
!!
2
+E
pulling
(x)+ E
twisting
(n) (6.20)
subject to (x
i+1
− x
i
)
2
− ℓ
2
= 0, ∀i∈[1,N+ 1], (6.21)
n
T
i
(x
i+1
− x
i
)= 0, ∀i∈[1,N], (6.22)
n
T
i
n
i
− 1= 0, ∀i∈[1,N], (6.23)
x
1
= ˆ x
1
, (6.24)
where ¯ p
i j
is the unskinned rest position of surface vertex i to tendon segment j; T
j
(x
j
,x
j+1
,n
j
) is
the skinning transformation for segment j obtained using x
j
,x
j+1
,n
j
; w
i j
is the skinning weight
of surface vertex i to the tendon segment j; M
i
is the set of skinning tendon segments for surface
vertex i; ˆ p
i
is the ICP target position on the MRI mesh; and ˆ n
i
is the ICP target normal. Again, we
solve it using the interior-point method. The result of stage 3 is shown in Figure 6.9(c). As can be
seen, it is smooth and closely matches the MRI mesh compared to previous stages.
6.2.2.3 Tendon sliding locations
Using the method described above, we obtain all example tendon rods. Then, we sparsely sample
the tendon rod in the neutral pose; these sample points are our neutral sliding locations. For each
sliding location i, we assume that it is driven by N
s
closest bones (we use N
s
= 2). We then find
106
the skinning weights w by minimizing the distance of the skinned position to the example tendon
rod in each example pose.
argmin
w
N
p
∑
j=1
N
s
∑
k=1
w
k
T
jk
¯ p
ik
− ˆ p
j
!
2
, (6.25)
st.
N
s
∑
k=1
w
k
= 1, (6.26)
0≤ w
k
≤ 1,∀k∈[1,N
s
], (6.27)
where T
jk
is the example transformation for bone k in pose j, ¯ p
ik
is the unskinned rest position
of sliding location i to bone k, and ˆ p
j
is the closest position on the rod to the skinned point in
example pose j. The example rod meshes are obtained from our non-rigid registration. To solve
this optimization problem, we first initialize w to(1,0), where the weight of the closest bone is 1
and the weight of the other bone is 0. Then, we compute the skinned positions in the example poses.
Thereafter, we find the closest position ˆp
j
in each example pose. Next, we run our optimization
to determine the skinning weights w. We repeat the previous steps until the change of w between
two consecutive iterations is close to zero. After the optimization problem is solved, we consider
the vector between the skinning position and the closest point on the tendon rod as our pose-
space correction in each example pose. Similarly to muscles, the pose-space vector consists of
rotations of bones that drive the tendon. The representation of a rotation is the same as the muscle
(Section 6.1.1).
6.3 Hand fascia
The gaps and valleys around bones and muscles cause problems when we simulate the fat layer
directly on top of the bones and muscles. Therefore, we create fascia layers wrapping muscles
and bones, respectively. Each single fascia is a single surface mesh. For bone fascia, we merge
our bone meshes with the convex hull meshes generated in Section 4.5. This gives us the surface
mesh of the bone fascia. Then, we shrink the rest material coordinates of the bone fascia mesh
107
Figure 6.10: Fascia meshes. (a) Bone meshes, (b) corresponding bone fascia mesh, (c) muscle
meshes, and (d) muscle fascia mesh.
Figure 6.11: Bone fascia. (a) The constraints applied to bone fascia simulation. Vertices in green
are attached to the closest bones using fixed constraints. Vertices in red are in contact with the
bone meshes. (b, c) Bone and bone fascia meshes in the fist pose.
so it tightly wraps the bone geometry, which is already in the rest state. In our experiment, we
shrink them by 0.35x. Similar to bones, we merge all muscle surface meshes together and generate
the muscle fascia mesh. For muscles, we shrink the rest material coordinates by 0.6x. In each
simulation timestep, we first rigidly transform the bones and then simulate the bone fascia using a
108
cloth solver [129]. After that, we perform muscle simulation and then simulate the muscle fascia
using the same cloth solver.
Figure 6.12: Muscle fascia constraints. Here, we show the constraints applied during muscle
fascia simulation. Vertices in green are attached to the muscles using fixed constraints. Vertices in
red are sliding against the muscle surface meshes.
Figure 6.13: Fascia animation. Top: a few frames of an animation sequence of bones and muscles.
Bottom: the fascia simulation results.
109
For bone fascia, the vertices in the area around bone bodies of the bones are attached to the
bone surfaces and move with the bone rigidly. On the other hand, the fascia vertices around joints
are in contact with the bone surfaces. They are selected manually, the same way as for creating
the convex hull in Section 4.5. We select the heads of the bones, as shown in Figure 6.11. The
bone fascia surface nicely shrinks/expands as the bones move (Figure 6.11), partially due to the
shrinking of the rest material coordinates. For example, when the fascia mesh around the joints
shrinks because of the bone movement, the fascia mesh does not fold. Instead, it shrinks nicely and
keeps the tension of the surface. For muscle fascia, the vertices are sliding and contacting against
the muscle surface except in the areas of attachment vertices (Figure 6.12). The animation of the
bone and muscle fascia corresponding to the deformations of the bones and muscles is shown in
Figure 6.13.
6.4 Hand fat
Figure 6.14: The exterior and interior surface of the fat tissue. (a) The exterior surface of the fat
tissue is the skin. (b) The interior surface of the fat tissue is composed of the bone fascia, muscle
fascia, and joint ligaments.
110
Because fat is a passive tissue, our fat simulation is similar to the method proposed in Chapter 4
but with a few changes. Our fat tissue has two surfaces, the skin (Figure 6.14(a)), and the interior
surface (Figure 6.14(b)) that contacts with the bones, muscles, and joint ligaments. The skin
is obtained using the same method as that mentioned in Chapter 4. The skin and the interior
surface of the fat are embedded into a tetrahedral mesh. We can significantly reduce the number
of tetrahedra because the tetrahedral mesh does not have to conform the surface. We first create
a single tetrahedral mesh that represents the skin of the hand. Then, we perform inside/outside
test for all tetrahedra against interior surfaces. We keep tetrahedra that are intersected with muscle
fascia, bone fascia, or joint ligaments. We assign a single material property to the fat. After
creating the tetrahedral mesh, we nevertheless need to assign spatially varying materials, because
some tetrahedra are only partially occupied by the fat tissue. Therefore, we weight the Young’s
modulus and mass density by the volume occupation.
Figure 6.15: The constraints for simulating the fat. The fat is attached to bone fascia, muscle
fascia, ligaments, and tendons. Green dots are the fixed constraints to the bone fascia. Yellow dots
are the contact constraints to the ligaments. Pink dots are the sliding constraints against the muscle
fascia, while the red dots are contact constraints against the muscle fascia. In addition, the purple
dots are the constraints to mimic the rigidity of the nails.
111
After we obtain the simulation mesh, we build the constraints between the interior surface of
the fat and the muscle fascia, bone fascia, and joint ligament layers. There are three types of
constraints in our model: fixed constraints, sliding constraints, and contact constraints, as shown
in Figure 6.15. A fixed constraint anchors a point to a target location. A sliding constraint anchors
a point to a target plane. A contact constraint penalizes penetration of a point beneath a target
plane. Inspired by biomechanics, for bone fascia, we create fixed constraints. As the bone fascia
already slides against the bones, we do not need to apply another layer of sliding constraints. For
muscles, we found that the sliding constraints + contact constraints yield the best results for all
vertices, except the attachment vertices to the bones. If fixed constraints were used, it would result
in a bumpy skin shape. We manually determined the weights for sliding constraints and contact
constraints. Even though such a process is manual, it is simple and usually takes less than 10
minutes to do it once. Moreover, the weights are the same for all example poses, so we only need
to do it once. The procedure for the joint ligaments is similar to the muscles. We apply sliding
constraints to all vertices except the attachment vertices to the bones. Finally, as mentioned in
Chapter 4, we also apply fixed constraints to the nails of the fingers, to keep them rigid. In addition
to the constraints on the fat surface, we also apply self-collision handling on the skin, as described
in Chapter 4.
6.5 Results
The MRI data is obtained using the approach described in Chapter 3. As shown in Chapter 4, we
acquired 12 example poses. Among them, we selected six example poses for our muscles, tendons,
and joint ligaments, as shown in Figure 6.16. We selected them to maximize the exertion of these
tissues as much as possible. We first compared our simulation results with [134] quantitatively in
all example poses. Then, we compared our simulation results visually with [134]. Both compar-
isons demonstrated that our hand model is superior to the previous work and matches the ground
truth better.
112
Figure 6.16: Hand poses used for joint ligaments, muscles, and tendons.
We deformed our entire simulation model to each target example pose. For each example
pose, we used the ground truth transformation for each bone. This guarantees that the bones are
at the correct locations. We first began with the muscles. The extracted meshes of the muscles
are presented in Figure 6.17. We then pre-processed these muscles using our pipeline and build
our muscle simulation model. Therefore, we simulated muscles to reach the example poses and
compare the results with the ground truth meshes extracted from MRI. As shown in Figure 6.18, we
selected a few example poses that we consider are the most extreme poses among all six example
poses. It is evident that the simulation results closely overlap with the ground truth meshes. Still,
tiny errors remain. These errors are expected as our model contains inter-muscle sliding forces
and contact forces. We did not consider these forces when creating the example muscle surfaces
and corresponding example plastic strains. Nonetheless, our pre-processing pipeline guarantees
that these forces are minimized at example poses, because contact resolving cleans up the example
surface meshes. In addition, we performed tendon comparisons at the same example poses as those
with the muscles. As depicted in Figure 6.19, the simulation results closely match with the ground
113
Figure 6.17: Extracted muscle meshes in example poses.
truth meshes. These errors are expected as we only control the sliding locations. We did not
attempt to match the orientations of the tendon rod vertices (i.e., normals) in the example poses.
After the comparing the interior structure, we would like to measure the accuracy improve-
ment between modeling each organ as a separate object and simulating all soft tissues using a
single tet mesh. For each vertex on the ground truth surface mesh matching matches the MRI in
each example pose, we compute the closest distance to the simulation mesh. Table 6.5 presents
114
Figure 6.18: Comparisons between muscle simulation results and the ground truth meshes in
example poses. We simulated hand muscles using our model to reach a few example poses that
we believe are the most extreme poses among all six example poses. Simulation results (green
wireframe) closely overlap with the ground truth meshes (red wireframe).
Figure 6.19: Comparisons between tendon simulation results and the ground truth meshes
in example poses. We simulated hand tendons using our model to reach a few example poses
(same as in the muscle experiment). Simulation results (green wireframe) closely overlap with the
ground truth meshes (red wireframe).
115
average distance, median distance, and max distance for each example pose. We can see that the
separate-mesh reduces the error by 24% for average distance, 21% for median distance, and 24%
for maximal distance. For all 12 poses, this is better in average, median and max distance in ma-
jority of the cases. There are only 4 cases (marked in bold) out of 36 where our method produces
slightly inferior numbers.
Table 6.1: Comparisons between simulated and ground truth skins. Model m
1
simulates each
organ separately (as proposed in this Chapter), whereas model m
2
simulates all soft tissues using
a single mesh (as proposed in this Chapter 4). Column “%” denotes the ratio between the value in
m
1
and the value in m
2
. The values in bold face are cases where our model is slightly worse than
m
2
. The first five rows are the example poses used for our simulation, whereas the last six rows are
the unseen poses. All distances are represented in millimeter units.
pose
average median max
m
1
m
2
% m
1
m
2
% m
1
m
2
%
2 0.94 1.35 69.63% 0.76 1.22 62.30% 4.81 4.70 102.34%
3 0.72 1.19 60.50% 0.58 0.88 65.91% 3.89 6.34 61.36%
4 0.93 1.31 70.99% 0.77 0.86 89.53% 3.34 7.09 47.11%
5 1.02 1.21 84.30% 0.83 1.00 83.00% 4.60 5.61 82.00%
12 0.72 1.15 62.61% 0.59 0.90 65.56% 4.89 7.86 62.21%
6 1.01 1.05 96.19% 0.86 0.78 110.26% 4.50 5.46 82.42
7 0.63 0.84 75.00% 0.50 0.66 75.76% 2.85 2.70 105.56%
8 0.89 1.02 87.25% 0.69 0.8 86.25% 5.20 4.67 111.35%
9 0.97 1.08 89.81% 0.85 0.92 92.39% 4.15 4.80 86.46%
10 1.09 1.43 75.22% 0.80 1.08 74.07% 4.89 6.30 77.62%
11 0.88 1.24 70.97% 0.62 0.87 71.26% 5.63 7.69 73.21%
Average 0.89 1.17 76.22% 0.71 0.91 78.74% 4.38 5.75 76.23%
We employed the same sequence as used in [134] and compared the final rendering results
with it. We observed a clear improvement over the palm area, as shown in Figures 6.20, 6.21.
Compared to [134], the muscles are modeled using plastic strains in our model, and are controlled
by the bone transformations. In other words, we “activate” the muscles based on the hand pose,
which improves the simulation results.
Using our model, the joint regions are also improved, as shown in Figure 6.21. This is because
we model the joint ligaments using plastic strains controlled by the bone transformations, where
previously, these tissues were not modeled.
116
Figure 6.20: Visual comparisons to Wang et al. [134]. The palm area bulges more under our
simulation model, which is closer to the behavior in the real world.
We also performed a few experiments to see how well the muscle shapes are reproduced in both
example poses and non-example poses. Figure 6.22 indicates that our simulation result closely
matches the MRI data in example poses. For the non-example poses, we can see that our model
produces reasonable results (Figure 6.23).
117
Figure 6.21: Visual comparisons of joint regions to Wang et al. [134] in example poses. The
ground truth mesh is depicted in red wireframe, whereas the simulation results are shown in green
wireframe. The palm and joint areas bulge more correctly in our simulation model compared to
the single-layer soft tissue model.
118
Figure 6.22: Visual comparisons to MRI slices in example poses. We compare our simulated
skin and muscle surfaces with the MRI images in example poses 1, 2 and 12. Pose 1 is the neutral
pose. Pose 2 (fist) and 12 (thumb moving to the most opposite location) are extreme example
poses. We intersect our surface meshes with the MRI slices. The intersections between skin and
the slices are presented in green. Further, the intersections between muscles and the slices are
depicted in yellow. We observe a very close match to the MRI data for both skin and muscles.
119
Figure 6.23: Visual comparisons to MRI slices in non-example poses. We compare our simu-
lated skin and muscle surfaces with the MRI images in non-example poses (poses 6 and 10). The
intersections between skin and the slices are shown in green, and the intersections between mus-
cles and the slices are depicted in yellow. We can observe a reasonably close match to the MRI
data for both skin and the muscles.
120
Chapter 7
Conclusion and Future Work
This thesis proposed a pipeline for personalized human hand modeling and simulation. The
pipeline begins from the acquisition of personalized MRIs and surface scans in multiple exam-
ple poses, and then a realistic and animation-ready personalized simulation rig is built. We used
a novel molding procedure to stabilize hands during MRI scanning. We not only captured MRI
in multiple poses but also acquired the high-resolution skin geometry that matches the MRI si-
multaneously. Then, we demonstrated how to acquire highly accurate human skeleton geometry
in multiple poses using MRI. We registered all poses to the same bones mesh connectivity, and
built a skeleton mesh kinematic model (“skeleton rig”) to interpolate or extrapolate the acquired
pose to the entire range of motion of the hand. We demonstrated that our skeleton rig can be used
to drive soft-tissue FEM simulation to produce anatomically plausible organic hand shapes that
qualitatively match photographs.
In addition, we improved our model by capturing the deformation of interior structures includ-
ing muscles, tendons, joint ligaments, and fat independently. To extract these tissues from the MRI,
we proposed a novel shape deformation method that can model objects undergoing large spatially
varying strains, such as muscles. Our method works by computing a plastic deformation gradient
at each tet, such that the mesh deformed by these plastic deformation gradients matches the pro-
vided sparse landmarks and closest-point constraints. We applied our method to the extraction of
shapes of organs from medical images. Further, our method has been designed to extract as much
information as possible from MRI images, despite the soft boundaries among the different organs.
121
Our method not only applies to hand muscles, but also to other organs such as liver, hip bone and
hip muscle. Finally, we gave a complete system for hand animation by modeling bones, muscles,
tendons, joint ligaments, and fat separately. We designed a pipeline to pre-process MRI and optical
scans into a simulation-ready hand rig. We demonstrated that our model is accurate. Our model not
only matches the ground truth skin, with average errors less than 1mm in all example poses, but it
also produces matching interior structures in all example poses. Furthermore, our model produces
realistic hand motions. The simulation results qualitatively match the hand in the real world.
We performed 12 scans for each subject, which is sufficient to demonstrate stable and precise
common hand motions, including the opposition of the thumb to all the other four fingers. Accu-
racy would be improved with more scans. An interesting research question is how to automatically
select the next best pose to scan, in order to maximize coverage of the hand’s range of motion.
Some complex hand poses are challenging for our technique, for example, all fingertips touching
at a single point. This is because we cut our molds into two pieces manually using a knife. It would
be interesting to explore how to cut the molds into three or more pieces, to improve the ergonomics
of the insertion of the hand into the mold. Our accurate hand model can potentially benefit virtual
hands in games / film / VR, robotic hands, grasping, and medical education, such as visualizations
of internal hand anatomy motion. The understanding of the motion of internal hand anatomy as
proposed here takes us one step closer to life-like robots with hands that mimic biological hands.
Modeling the human hand accurately is a very broad area and there are still many aspects in
our model that can be improved and explored in the future. First of all, we only capture 12 poses
due to our limited resources. As several components of our model depend on captured data, the
more data is captured, the more accurate the results would be. Moreover, capturing more hand
poses implies that the ranges of motions of the hand are explored better. For each example pose,
our capturing procedure is more complex compared to optically-based scanning methods, such as
motion capture and 3D reconstruction, but our model requires much fewer number of example
poses compared to them. Still, automating our pipeline would improve the efficiency of capturing
data. In addition, improving the MRI resolution would also help the fitting procedure, as there
122
are many small structures in the hand. For example, we observed that the thickness of one of
the extensor tendons is occasionally represented by only 1-2 voxels. The distal phalanges of our
female subject are also represented by only a few voxels.
Further, we could also explore more types of joints. In our model, we only consider each hand
joint as either a 1D joint or a 2D joint: either a hinge joint or a condyloid joint. Other types
of joints such as saddle joints or ball joints could also be considered and these may lower the
fitting errors in our skeleton model. Our bone rig is a kinematic model: the transformations are
controlled by joint angles. Therefore, our model is artist-friendly and easy-to-use. Nonetheless, it
is difficult to validate the input joint angles in our model. When we captured hand motions from
LeapMotion or other motion capture systems like Google Mediapipe, it was difficult to know if the
captured motion is valid. An invalid input sequence of joint angles can cause contact of the skin,
or unrealistic deformations of the hand. Unless we perform our full-hand simulation, it is not easy
to validate the input motion. Moreover, if repair is required, it needs to be done manually. This
problem could be addressed by capturing many hand poses to build a comprehensive and valid
pose space.
Figure 7.1: The mismatches between simulation results and the MRI. The silhouettes of mus-
cles are marked in white. Mismatches are highlighted by red rectangles. We show two typical
types of mismatches that are not handled by our model.
To use our muscle, ligament, and tendon simulation models, we must preprocess the MRI
and optical scans at multiple example poses. We need to separately extract every tissue in the
scans of multiple example poses. Even with our proposed shape modeling method, this is still a
123
considerable amount of manual work. Reducing this manual work would be beneficial and could
be left for future work. Our model is based on the example pose data. Accordingly, if there is an
unseen pose that is close to the example poses, the prediction of the muscle shape is reasonable,
as demonstrated by experiments depicted in Figure 6.23. On the other hand, if the unseen pose is
far from the examples, it may not be predicted accurately. For example, we found that the shape of
first dorsal interosseus muscle is not well matched with the MRI in our example pose 6, as shown
in Figure 7.1(left). Using more example poses would alleviate this issue.
Another cause of this mismatch is that we did not model the interactions between fat tissue,
fascia, and the muscles. In the real world, fat tissue and fascia are pushing muscles inwards so
that muscles are aggregated together. Without modeling these forces, the muscles may not be
positioned correctly, as depicted in Figure 7.1(right). The fat layer is also affected by the same
problem, although we rarely find such artefacts. It is difficult to introduce this coupling into our
system due to the large amount of computation. We would have to perform our optimization and
simulation with all tissues—including all muscles, fat, and fascia—coupled together. This does not
only increase the time cost tremendously, but it also brings a lot of difficulties in identifying the
optimal solution during the optimization, such as the optimization that extracts the muscle shapes.
We did not model the muscle activation forces. Although our muscles match the ground truth
shape at example poses, they do not produce correct forces. To address this, we could fit the scans
to a muscle model that models muscle activation, such as Hill’s model [144].
Finally, although we modeled key structures underneath the hand skin, there remain additional
tissues that could be modeled in the future. It can be seen in Table 6.5 that, while the average and
median errors are small, the maximal errors are not negligible. One of the possible reasons for
this is that we did not model the extensor tendons on the back side of the hand. This could cause
a mismatch on the back side of the hand, as shown in Figure 6.21. Thus, modeling all important
tissues of the hand is an important avenue for future work.
124
Acknowledgments
This research was sponsored in part by NSF (IIS-1911224), USC Annenberg Fellowship to Bohan
Wang, Bosch Research and Adobe Research.
Conflict of interest statement
The hip bone and hip muscle template meshes were obtained at Ziva Dynamics. Jernej Barbiˇ c is
a shareholder, CTO and board member of Ziva Dynamics. Ziva Dynamics was not involved in
this research. Nothing in this thesis is to be understood as endorsement of Ziva Dynamics or its
products.
125
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135
Appendices
A Plastic strain Laplacian and its nullspace
Let L
sc
∈R
m× m
denote the discrete mesh Laplacian for scalar fields on mesh tetrahedra [145],
L
sc
i, j
=
#adjacent tets if i= j,
− 1 if i̸= j, and i, j are adjacent,
0 otherwise.
(A.1)
Given a plastic strain state s=[s
1
,s
2
,s
3
,s
4
,s
5
,s
6
]∈R
6m
, where s
i
∈R
m
, define the plastic strain
Laplacian
Ls=[L
sc
s
1
,
√
2L
sc
s
2
,
√
2L
sc
s
3
,L
sc
s
4
,
√
2L
sc
s
5
,L
sc
s
6
], (A.2)
where the
√
2 were added to account for the fact that s
2
,s
3
,s
5
control two entries in the symmetric
matrix F
p
∈R
3× 3
.
Lemma: Assume that the tet meshM has a single connected component. Then, the nullspace of
L is 6-dimensional and consists of vectorsψ
i
=[s
1
,...,s
6
] where s
j
∈R
m
is all ones when j= i,
and all zeros otherwise.
Proof: First, observe that L
sc
is symmetric positive semi-definite with a single nullspace vector,
namely the vector of all 1s. This follows from the identity
x
T
L
sc
x=
∑
i and j adjacent
(x
i
− x
j
)
2
, (A.3)
136
i.e., x
T
L
sc
x= 0 is only possible if all x
i
are the same.
We have 0= s
T
Ls=∑
6
i=1
ξ
i
s
i
T
L
sc
s
i
, whereξ
i
=
√
2 for i= 2,3,5; and 1 otherwise. Because
L
sc
is symmetric positive semi-definite, each s
i
must either be 0 or a non-zero nullspace vector
of L
sc
, i.e., a vector of all 1s. A linearly independent orthonormal nullspace basis emerges when
we have a vector for all 1s for exactly one i. There are 6 such choices, giving the vectors ψ
i
; we
normalize them by dividing with
√
m. ■ B First and second derivatives of elastic energy with respect to
plastic strain
For convenience, we denote F
p,i
as i− th entry of the vector vec(F
p
)∈R
9
. The first-order deriva-
tives are
∂E
∂x
i
= V
∂ψ
∂F
e
:
∂F
e
∂x
i
= V P:
∂F
e
∂x
i
, (B.1)
∂E
∂F
p,i
= V
∂ψ
∂F
p,i
+
∂V
∂F
p
i
ψ = V P:
∂F
e
∂F
p,i
+
∂V
∂F
p
i
ψ, where (B.2)
∂F
e
∂x
i
=
∂F
∂x
i
F
− 1
p
,
∂F
e
∂F
p,i
= F
∂F
− 1
p
∂F
p,i
, and (B.3)
∂V
∂F
p,i
=
∂
F
p
∂F
p
i
V
0
. (B.4)
Here, P is the first Piola-Kirchhoff stress tensor and ∂F/∂x is a constant matrix commonly used
in the equations for FEM simulation. For the second-order derivatives, we first compute ∂
2
E/∂x
2
.
This is the tangent stiffness matrix in the FEM simulation under a fixed F
p
. It is computed as
∂
2
E
∂x
i
∂x
j
= V
∂F
e
∂x
j
T
:
∂P
∂F
e
:
∂F
e
∂x
i
. (B.5)
137
Here, ∂P/∂F
e
is a standard term in FEM nonlinear elastic simulation; it only depends on the
strain-stress law (the material model). Next, we compute∂
2
E/(∂x∂F
p
),
∂
2
E
∂x
i
∂F
p, j
=
∂V
∂F
p, j
P:
∂F
e
∂x
i
+ (B.6)
V
∂F
e
∂F
p, j
T
:
∂P
∂F
e
:
∂F
e
∂x
i
+V P
∂
2
F
e
∂x
i
∂F
p, j
, where (B.7)
∂
2
F
e
∂x
i
∂F
p, j
=
∂F
∂x
i
∂F
− 1
p
∂F
p, j
. (B.8)
Finally, we have
∂
2
E
∂F
p,i
∂F
p, j
=
∂
2
V
∂F
p,i
∂F
p, j
ψ+ (B.9)
V
P:
∂
2
F
e
∂F
p,i
∂F
p, j
+
∂F
e
∂F
p, j
T
:
∂P
∂F
e
:
∂F
e
∂F
p,i
!
+ (B.10)
∂V
∂F
p, j
∂ψ
∂F
p,i
+
∂V
∂F
p,i
∂ψ
∂F
p, j
, where (B.11)
∂
2
V
∂F
p,i
∂F
p, j
=
∂
2
F
p
∂F
p,i
∂F
p, j
V
0
, (B.12)
∂
2
F
e
∂F
p,i
∂F
p, j
= F
∂
2
F
− 1
p
∂F
p,i
∂F
p, j
. (B.13)
The quantitiesψ,P and∂P/∂F
e
are determined by the chosen elastic material model. After com-
puting the above derivatives, there is still a missing link between F
p
and s. Because we want to
directly optimize s, we also need the derivatives ofE(F
p
(s),x) with respect to s. From Equa-
tion 5.2 we can see that F
p
is linearly dependent on s. Therefore, so we can define a matrix Y
such that vec(F
p
)= Y s. Then all the derivatives can be easily transferred to derivation by s by
multiplying with Y .
138
C Proof of singular lemma
Statement (i) follows from well-known linear algebra factsR(A)=N (A
T
)
⊥
and dim(N (A))+
dim(R(A))= p, and the symmetry of A. As per (ii), A mapsR(A) into itself, and no vector from
R(A) maps to zero, hence the restriction of A toR(A) is invertible, establishing a unique solution
to Ax= b with the property that x⊥ψ
i
for all i= 1,...,k. This unique solution is the minimizer of
min
x
1
2
x
T
Ax− b
T
x (C.1)
s.t. ψ
T
i
x= 0 for all i= 1,...,k. (C.2)
When expressed using Lagrange multipliers, this gives Equation 5.15. Suppose x= n+r is another
solution and n∈N (A) and r∈R(A). Then b= Ax= Ar and hence r is the unique solution from
Equation 5.15. The vector n can be an arbitrary nullspace vector, proving the last statement of
(ii). As per (iii), suppose we have 0= B(n+ r)= Ar+∑
k
i=1
λ
i
α
i
(ψ
T
i
n)ψ
i
. Observe that the first
summand is inR(A) and the second inN (A). Hence, B(n+r) can only be zero if both summands
are zero. Ar= 0 implies r= 0. The second summand can only be zero if n⊥ψ
i
for each i, which
implies that n= 0. Hence, B is invertible. The last statement of (iii) can be verified by expanding
A+∑
k
i=1
α
i
ψ
i
ψ
T
i
x+∑
k
i=1
λ
i
α
i
ψ
i
. ■ D Proof of nullspace lemma
We are trying to prove that K(F
p
(s),x) is 6-dimensional for any x that solves f
e
(s,x)= 0. First,
if x is a solution, then translating all vertices of the object by the same constant 3-dimensional
vector is also a solution. This means that vectorψ
i
:=(e
i
,e
i
,...,e
i
) is in the nullspace of K, where
e
i
∈R
3
is the i-th standard basis vector, for i= 1,2,3. Now, suppose we rotate the object with an
infinitesimal rotation X7→ X+ e
i
× X.
139
Figure D.1: Illustration of
the nullspace proof. The
original forces F
i
sum to
zero. We have G
i
= RF
i
;
note that R is the same for all
tets. Therefore, the rotated
forces G
i
also sum to zero.
Hence, there is no change in
the internal elastic force un-
der a rotation—that is, in-
finitesimal rotations are in the
nullspace of K.
Observe that for general plastic strains s, the elastic forces in
each individual tet are not zero even in the equilibrium x; but the
contributions of elastic forces on a tet mesh vertex from all ad-
jacent tets sum to zero. As we rotate the object, the forces con-
tributed by adjacent tets to a specific tet mesh vertex rotate by
the same rotation in each tet. Therefore, as these forces sum to
zero, they continue to sum to zero even under the rotation (see Fig-
ure D.1). This means that the vector of infinitesimal displacements
ψ
3+i
:=[e
i
× x
1
,e
i
× x
2
,...,e
i
× x
n
] induced by the infinitesimal ro-
tation is in the nullspace of K, for each i= 1,2,3. Here, x
i
are the
components of x=[x
1
,x
2
,...,x
n
]. The vectorsψ
i
, i= 1,2,3,4,5,6,
form the nullspace of K. ■ Finally, we inform the reader that the nullspace of K(F
p
(s),x)
is only 3-dimensional if x is not an elastic equilibrium. In this case,
only translations are in the nullspace. Infinitesimal rotations are not
in the nullspace because under an infinitesimal rotation, the non-
zero elastic forces f
e
rotate, i.e., they do not remain the same. The
assumption of x being the equilibrium shape is therefore crucial (and is satisfied in our method).
E Second derivative of polar decomposition
To compute the second-order derivatives, we differentiate
∂F
∂F
i
=
∂R
∂F
i
S+ R
∂S
∂F
i
, (E.1)
∂
2
F
∂F
i
∂F
j
=
∂
2
R
∂F
i
∂F
j
S+
∂R
∂F
i
∂S
∂F
j
+
∂R
∂F
j
∂S
∂F
i
+ R
∂
2
S
∂F
i
∂F
j
, (E.2)
∂
2
R
∂F
i
∂F
j
=
− R
∂
2
S
∂F
i
∂F
j
− ∂R
∂F
j
∂S
∂F
i
− ∂R
∂F
i
∂S
∂F
j
S
− 1
. (E.3)
140
To compute ∂
2
R/(∂F
i
∂F
j
), we need to compute ∂
2
S/(∂F
i
∂F
j
) first. This can be derived in the
same way as for∂S/∂F
i
. Starting from Equation 5.22, we have
∂
2
F
T
F
∂F
i
∂F
j
=
∂
2
S
∂F
i
∂F
j
S+
∂S
∂F
i
∂S
∂F
j
+
∂S
∂F
j
∂S
∂F
i
+ S
∂
2
S
∂F
i
∂F
j
. (E.4)
We can now solve a similar Sylvester equation
vec(
∂
2
S
∂F
i
∂F
j
)=(S⊕ S)
− 1
vec(C), (E.5)
C=
∂
2
F
T
F
∂F
i
∂F
j
− ∂S
∂F
i
∂S
∂F
j
− ∂S
∂F
j
∂S
∂F
i
. (E.6)
F Formulas for A
k
,b
k
,c
k
(Equation 5.6)
Let the attachment k be embedded into a tetrahedron t
k
with barycentric weights
w
k
1
,w
k
2
,w
k
3
,w
k
4
.
We have
A
k
=
w
k
1
I
3
w
k
2
I
3
w
k
3
I
3
w
k
4
I
3
S
k
∈R
3× 3n
(F.1)
b
k
=− y
k
∈R
3
, (F.2)
where y
k
∈R
3
is the attachment’s target position, I
3
∈R
3× 3
is the identity matrix, and S
k
∈R
12× 3n
is a selection matrix that selects the positions of vertices of t
k
. The scalar c
k
is the weight of the
attachment k. Equivalent formulas apply to landmarks and ICP markers. The attachment energy
and forces are,
E
a
(x)=
1
2
∑
k
c
k
∥A
k
x+ b
k
∥
2
, (F.3)
f
a
(x)=
dE
a
dx
=
∑
k
c
k
A
T
k
(A
k
x+ b
k
). (F.4)
141
G Meeting constraints by scaling the elastic stiffness
In this section, we demonstrate that scaling the elastic stiffness cannot be used to “better” meet the
constraints. We illustrate this on a simple toy problem of linear elasticity with linear constraints.
Denote the vertex displacements by u∈R
3n
and the stiffness matrix by K∈R
3n× 3n
.
First, consider scaling the stiffness (K→αK) when using hard constraints:
min
u
< Ku,u> (G.1)
s.t. Cu= b (G.2)
v.s.
min
u
<(αK)u,u> (G.3)
s.t. Cu= b (G.4)
produces same u for any scalarα> 0.
Similarly, consider scaling stiffness (K→αK) when using soft constraints:
min
u
<(αK)u,u>+β||Cu− b||
2
(G.5)
produces same u as
min
u
< Ku,u>+β/α||Cu− b||
2
. (G.6)
Therefore, scaling the stiffness does not provide any more expressive shape deformation power
than tweakingβ in:
min
u
< Ku,u>+β||Cu− b||
2
. (G.7)
142
As one tweaks β in the above, one can choose between (A) meeting the constraints well (when
β>> 1), or (B) produce smooth deformation (whenβ<< 1). In option (A), one obtains sharp non-
smooth deformation (”spikes”), and in option (B) one does not meet the constraints. In the middle
territory, one doesn’t meet either of these goals well, as seen in Figure 1.3,f. So, in summary,
merely increasing the elastic stiffness does not help with better satisfying the constraints.
143
Abstract (if available)
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Asset Metadata
Creator
Wang, Bohan
(author)
Core Title
Anatomically based human hand modeling and simulation
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Computer Science
Degree Conferral Date
2021-12
Publication Date
09/30/2021
Defense Date
07/16/2021
Publisher
University of Southern California
(original),
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(digital)
Tag
Anatomy,animation,biomechanics,CT,deformable objects,FEM,finite element method,human hand,large strain,Modeling,MRI,OAI-PMH Harvest,optical scanning,optimization,plastic strain,shape deformation,simulation
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Advisor
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committee chair
), Fiume, Eugene (
committee member
), Gupta, Satyandra K. (
committee member
), Pollard, Nancy (
committee member
), Sukhatme, Gaurav (
committee member
)
Creator Email
bohanwan@usc.edu,wangbh11@gmail.com
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Tags
biomechanics
CT
deformable objects
FEM
finite element method
human hand
large strain
MRI
optical scanning
optimization
plastic strain
shape deformation
simulation