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A Linear Elastic Force Optimization Model for Shape Matching

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J Math Imaging VisDOI 10.1007/s10851-014-0520-5
A Linear Elastic Force Optimization Model for Shape Matching
Konrad Simon
·
Sameer Sheorey
·
David Jacobs
·
Ronen Basri
Received: 19 June 2013 / Accepted: 18 June 2014© Springer Science+Business Media New York 2014
Abstract
We employ an elasticity based model to accountfor shape changes. In general, to solve the underlying equa-tions for the deformation, boundary conditions have to beincorporated, e.g., in the form of correspondences betweencontour points. However, exact boundary correspondencesare usually unknown. We propose a method that is able tooptimizepre-selectedboundaryconditionssuchthatexternalforcescausingtheshapechangeareminimizedinsomesense.Thus we seek simple physical explanations of shape changeclose to a pre-selected deformation. Our method decom-posesthefullnonlinearoptimizationproblemintoasequenceof convex optimizations. Artiﬁcial and natural examples of shape change are given to demonstrate the plausibility of thealgorithm.
Keywords
Shape matching
·
Convex optimization
·
Forceoptimization
·
Linear elasticity
·
Finite elements
K. Simon (
B
)
·
R. BasriDepartment of Computer Science and Applied Mathematics,The Weizmann Institute of Science, 76100 Rehovot, Israele-mail: konrad.simon@weizmann.ac.ilR. Basrie-mail: ronen.basri@weizmann.ac.ilS. SheoreyUtopiaCompression Corporation, Los Angeles, CA 90064, USAD. JacobsDepartment of Computer Science, University of Maryland,College Park, MD 20742, USAe-mail: djacobs@cs.umd.edu
1 Introduction
Giventwoshapesofanobject,asseenforexampleinimagesof the same or similar scenes, shape matching denotes theprocess of ﬁnding point correspondences between them.Finding correspondences between shapes yields valuableinformation about their differences and is important in manyapplications such as recognition, medical image registration(whereshapesrepresentsubstructureslikeorgans)ormotioncorrection during surgery. In these and other imaging situa-tionstheobservedshapesoftendepictactualnon-rigidphysi-calbodies.Hence,shapechangetakesplaceduetoandcanbeexplained by forces that cause an elastic deformation. Elas-tic deformable models can be used to describe the motion of animals, people, hands etc. and can even describe changesin shape across instances of the same visual category (e.g.,different faces).1.1 Related Work Shape matching and image registration are very active ﬁeldsof research and many different methods have been presentedinrecentdecades,see[12,21]forsurveys.Acommonscheme
appliedinelasticityorﬂuidbasedmatchingistoderivealocal(dis-)similaritymeasurebetweenthetransformedandthetar-getimage,e.g.,basedonintensitydifference.Thenforcesarederived from this similarity measure, which drive the regis-tration under a certain smoothness prior on the deformationﬁeld.Examplesinclude[2,3,17],whichemployalinearelas-
ticitypriorand[31],whichemploysahyperelasticsmoother.Boundary conditions are usually set to zero on the imageboundary such that deformation only takes place inside theimageandthuspossiblyfarawayfromtheboundary.Alsothederived image force introduces an (additional) nonlinearity
1 3
J Math Imaging Vis
whichmaycomplicatetheenergyfunctionaltobeminimizedand hence requires careful numerical treatment.Elastic models were used in a number of studies to modeldeformation. Ilic and Fua [22] use a nonlinear Timoshenkobeam model to track large deformations of beam like struc-tures in image sequences. There displacements and forcesare computed in an alternate manner and are derived fromimage forces. Actual physical forces that are responsible forthe deformation are then derived from the obtained displace-ments.A parameter-free approach to elastic registration was pro-posed by [29]. In this work the authors employ an elasticity
model with zero boundary conditions on the image bound-ary and constrain displacements in the interior by prescrib-ing correspondences between identiﬁable substructures suchas object boundaries. This approach implicitly incorporatesdriving forces as described above.The authors of [27] use a linear elasticity-based approachto reconstruct a three-dimensional surface from a singleimage by minimizing the stretch energy of a template sur-face. Their optimization is constrained by a set of boundarypoint correspondences whose objective is to limit the searchspacetodeformationssubjecttounknownexternalforces.Assuch unknown external loads usually prevent the deformedsurface from achieving a stress free equilibrium state theyshow that knowing a set of correspondences on the bound-aries is necessary to obtain good results.In[1]theauthorsusealinearelasticframeworkcombined
with an extended Kalman ﬁlter to provide a method for non-rigid structure from motion. In this method boundary corre-spondences are not needed since these are computed withintheir framework.The authors of [19] use a linear elastic model to set up aRiemannian structure on a manifold of shapes. After prov-ing the existence and uniqueness of energy minimizers theycompute geodesic paths between shapes to compare shapesthat differ by possibly large deformations.Another approach is to model the template image as aviscous ﬂuid such that pixels are transported by a ﬂowdeforming the template to the desired target image. Asﬂuids do not carry memory about their previous state,these methods are very ﬂexible and therefore suitable forlarge deformations. We refer the reader to [11] for one of
the ﬁrst works in a non-variational framework and to [5]
for a variational setting. Smoothness of the transforma-tion is ensured by requiring the velocity to be smoothenough [15]. Despite impressive registration results this
approach might cause physically unreasonable transforma-tions [36]. For a comparison measure between shapes or
images Beg et al. [5] give a geodesic distance in thespace of diffeomorphisms (which is not invariant underrigid motions of objects observed in the scene). All theseapproaches are in the spirit of Grenander’s deformable tem-plates,see[20],modelingshapedeformationsastheactionof agroupofdiffeomorphismsandhavebeenwellinvestigated.One disadvantage of the above mentioned elasticity andﬂuid methods is their computational complexity. Anotherway to deﬁne a shape is by its outline. This representationis more compact in terms of dimensionality and effectivelyamounts in matching (closed) curves. In [37,38] curves are
modeled by their angle functions using their arc-length para-metrization, and shape change is modeled by a group actionon the shape contours leading to an elastic energy. Corre-spondences and a distance measure are obtained by solvinga nonlinear variational problem derived from that energy.A similar approach was chosen in [28], where curves aremodeled as points on an (inﬁnite dimensional) manifold andelastic properties are incorporated by imposing a suitableRiemannian metric. Curves are matched by computing geo-desics on this manifold, and the geodesic distance is usedas a comparison measure. The representation of shapes asa closed curve does not a priori take into account the inte-rior of the shape. A good survey and references to additionalliterature are given in [39].Other works focus on comparing shapes bymeans of their intrinsic geometric properties [9], e.g., the
Gromov-Hausdorff distance. Also see work on statisticaldynamics shape models [13]. These models capture corre-lations of deforming shapes (silhouettes) over time and arebased on the principle that observations of a certain shapeat a given time may depend on previous observations of thatshape.1.2 Our MethodIn the present work we propose a method for registeringshapes around a pre-selected (possibly user supplied) defor-mation that is able to account for small deformations byemploying a linear elasticity model. This model is invari-antundertranslationsandunderinﬁnitesimalrotations.Oncethe shape boundaries of the template and target objects areextracted our task is to ﬁnd physically reasonable boundarycorrespondences. Exact boundary conditions are in generalunknown [18] but one can often give an initial guess. For
example, the authors of [18,29] employ active contour mod-
els (active surface models respectively) to obtain boundarycorrespondences of objects observed in the images that areto be compared.To measure the complexity of shape change externalforces acting on the shape boundaries can be used. In partic-ular, if these forces are sparse (or small), one can considerthe deformation to be small. Sparse forces, we believe, canprovide a suitable and simple comparison measure for defor-mations. Given two shapes and a guess of boundary corre-spondencesweminimizetheforcesontheboundary(insomenorm) by simultaneously allowing the estimated correspon-
1 3
J Math Imaging Vis
dences to drift after paying a penalty in terms of a restoringforce. The full optimization problem is highly nonlinear butwe present a convex relaxation which is subject to a linearPDE. Nevertheless, there are possibly many local minimaof the optimization problem. Hence, an initial guess of theboundary displacements localizes the search for a plausible(and simple) explanation of the observed deformation. Theinitial guess can be interpreted as a pre-selected deforma-tion. So the convex relaxation of the full optimization thatwe sugguest can be interpreted as a method to provide asimpler explanation (in terms of forces) of the pre-selecteddeformation close to the initial guess.The discretization of the PDE is done by means of adisplacement based ﬁnite element method (FEM), which isa suitable tool for connecting boundary displacements andexternal forces. This is done by a splitting of the stiffnessmatrix. The decomposition of the stiffness matrix used forour purpose was also applied in [8] in the context of surgery
simulation. This decomposition has the advantage that itincorporates also the elastic properties of the interior of theshapealthoughtheforceoptimizationiseffectivelyonlyper-formedatboundarynodes.Thisreducesthedimensionofthefull problem signiﬁcantly. The difference is that in [8] forcesare known and the displacements are computed whereas ourmethod assumes neither is known.A similar idea to our force optimization problem hasbeen introduced in [33] in the context of motion tracking.UnknownforcesareassumedtodrivethemotionofanobjectinasceneaccordingtoNewton’ssecondlaw.Thisisadynam-ical approach and it is used to give a convex formulation of the tracking problem. Their approach is very scene speciﬁc,i.e., the optimization problem has to be adapted to the spe-ciﬁc situation in order to incorporate as much information aspossible about known forces and other problem constraints.Ourforceoptimizationproblem,incontrast,isnonlinear,andits relaxation consists, in contrast, of a sequence of convexproblems and does not depend on the speciﬁc deformationobserved.Our method works on silhouette shapes in
R
2
and ignoresintensities. The idea can easily be generalized to shapes in
R
3
. In our approach we do the actual optimization of thecorrespondences on the boundary curve only but at the sametime we keep information of the interior of the shape. Wedo not employ more information than the elastic proper-ties of the shape. After the optimized boundary conditionsare computed one can solve the elasticity PDE, post-processthe solution, and obtain various valuable (comparison) mea-sureslikeexternalforces,strain,stressorstoredenergyofthedeformed shape. Knowing such quantities can help to assesshow severe a given deformation is and hence enables us tocompare different deformations.As a motivating example imagine a physician who wantsto compare different CT-scans of a patients liver. Taking ascanofthehealthyliverwecancompareittootherscansandcompare the different deformations to decide whether or nota pathophysiological deformation is progressing by looking,for example, at the external forces that are responsible forthis deformation.However, note that assessing an observed deformation isan (in the sense of Hadamard) ill-posed inverse problem andobservingasmalldeformationdoesnotnecessarilymeanthatforcesaresparse(orsmall).Thesituationisevenworsesincethe deformation can be nonlinear. In a forthcoming paper wewill investigate a generalized nonlinear elasticity setting inorder to be able to treat rotations and larger deformations notcovered by linear elasticity.In the following we will give a short overview of the rele-vant elasticity model and the FEM we employed. A detaileddescription of the decomposition of the optimization pro-cedure is given in Sect. 2.3 and experimental examples areshown in the following section. We end the paper with adiscussion of the method and of our results.
2 Description of Methods
2.1 The Physical Model of a Linear Elastic BodyThis section intends to give a concise overview of the basicprinciples of elasticity theory that are relevant for our model.More material on elasticity can be found, e.g., in [4,6]
and [32]. We will follow the notation introduced in [10].
Elasticity theory regards the state of a body submitted toforces. An elastic body is a body that reacts to applied forceswith a deformation, but returns to its srcinal shape afterremoving the forces. Furthermore, an elastic body does notmemorize previous deformations.
Strain, Stress and Equilibrium.
Our starting point is theassumption that there is a known reference conﬁguration
B
⊂
R
3
of the deformable body that will serve as a templatebody in the template image. This set
B
shall be bounded,and it describes the subset in
R
3
which is occupied by thebody when no forces are applied, i.e., the body is free of stress. The shape of the deformed body will be describedby a vector ﬁeld
Φ
:
B
→
R
3
, i.e.,
Φ(
x
)
is a point of thedeformed body corresponding to a point
x
in the referenceconﬁguration. We use the decomposition
Φ
=
I
+
u
, where
I
denotes the identity transformation and
u
the displacementﬁeld. Now,
Φ
is a deformation if subsets of the body aremapped to sets of positive volume, i.e., the determinant of the deformation gradient det
(
∇
Φ) >
0.A Taylor approximation of
Φ
shows that the quan-tity that is responsible for a local change of length isthe (right) Cauchy-Green strain tensor
C
=
(
∇
Φ)
T
∇
Φ
.Its half deviation from the identity
E
=
1
/
2
(
C
−
I
)
1 3
J Math Imaging Vis
Fig. 1
Left
an illustration of asurface force
t
(
x
,
n
)
at somepoint
x
in an arbitrary crosssection with normal
n
of a body
B
.
Right
illustration of the threecomponents of the stress tensorwith respect to the canonicalcoordinate planes. The normalstress is orthogonal to theconsidered plane shear stresseslie within the plane. The ﬁgurewas created using Incscape
is the Green-Saint-Venant strain tensor or simply strain.Intuitively, strain is a measure that locally represents achange of distance of nearby points relative to each other.In terms of the deformation ﬁeld
u
Cauchy-Green strainreads
E
=
12
∇
u
+
(
∇
u
)
T
+
(
∇
u
)
T
∇
u
. Neglecting thequadratic nonlinearity leads to the linear theory of elasticity(small deformations) and
E
simply becomes the symmetricpart of the derivative of the displacement ﬁeld. We denote itas
ε
=
12
∇
u
+
(
∇
u
)
T
.The next important quantity is the stress tensor or simplystress. A given deformation induces, as described above, acertain strain. Stress is a measure that quantiﬁes how difﬁ-cult it is to achieve a certain strain, i.e., stress is a functionof strain (see next section). Intuitively it is clear that rigidtransforms do not change stresses in a body. One can provethat if
B
is connected, a deformation
Φ
is a rigid transform(rotationandtranslation)iftheCauchy-Greentensorsatisﬁes
C
(
x
)
=
I
for all
x
∈
B
. More generally, one can prove thattwo deformations corresponding to the same strain tensor
C
can be obtained by a composition with a rigid transform,see[10].Asaconsequencewecanidentifytwodeformations
if they have the same strain tensor. Hence, the strain
E
canbe viewed as a measure of the deviation of a deformation
Φ
from a rigid transformation. However, for the measure
ε
thisis not true. As
ε
is a linear approximation to
E
it is invariantunder translations and inﬁnitesimal rotations but not undergeneral rotations.This causes some difﬁculties in comparingshapes through forces applied to the template body. But if the shapes that are to be compared differ from each otherwith no or minimal global rotation one can still expect a rea-sonable comparison by strain or stress or any quantities thatare derived from them like external forces. The latter will beexplained in the following.Forces acting on a body can either be body forces orsurface forces. Body forces are proportional to mass andare related to an outside source (e.g., gravity or magneticforces).These forcesarevolume forcesand can bedescribedby a force density
F
:
B
→
R
3
(measured per unit vol-ume). The surface forces will be particularly interesting inour formulation and are described in the following. Let ustake a point
x
∈
B
and an arbitrary cross-section within thebody, described by its (unit) normal
n
∈
R
3
. A surface force
t
(
x
,
n
)
∈
R
3
, called traction and measured per unit area,acts on the cross-section in
x
and is described by the Cauchystress tensor
σ
∈
R
3
×
3
. The rows of this stress tensor rep-resent the three tractions (normal and shear stresses) withinthree canonical coordinate planes, usually the three differentorthogonally intersecting cartesian coordinate planes. Onecanthenwritethesurfaceforceat
x
∈
B
as
t
(
x
,
n
)
=
σ(
x
)
n
,see Fig. 1 for an illustration and [32] for more details. In par-
ticular, if
x
∈
∂
B
and if
n
is its corresponding outward unitnormal the surface force
t
(
x
,
n
)
is the traction on the body’sboundary in
x
.Theequationsofstaticequilibrium(Cauchyprinciple)canbe formulated as a consequence of Newton’s second law: Inanyclosedsubvolume
V
thebodyforcesonitsboundaryandinside
V
have to balance, meaning
∂
V
t
(
x
,
n
)
d
S
+
V
F
(
x
)
d
V
=
0
.
(1)The divergence theorem then yields
V
(
div
σ
+
F
)
d
V
=
0
,
(2)andsincethisshallholdforanysubvolumetheintegrandmustvanish. Hence, the equations of equilibrium can be writtenasdiv
σ
+
F
=
0
.
(3)By the same principle for the momenta one shows that
σ
is symmetric. The divergence of the tensor is to be takenrow-wise. Note that the equations of equilibrium, althoughwe did not point that out clearly, must be formulated in
1 3
J Math Imaging Vis
the (unknown) deformed conﬁguration
Φ(
B
)
. Transformingthese back to the reference conﬁguration would introducea nonlinearity, namely an additional factor that depends on
∇
Φ
.However,inthesettingofsmalldeformationsthisfactoris close to the identity and will be neglected, so that it doesnot make a difference to formulate the equations in terms of the reference conﬁguration. We will deal with this nonlin-earity in a separate paper that addresses the case of largerdeformations.
Constitutive Equations for Linear Elastic Materials.
Forelastic materials stress will depend on strain and materialproperties. Below we brieﬂy explain the relation betweenstrain and stress.Intuitively it is clear that forces that are of the same mag-nitude will cause different deformations when acting on dif-ferent physical materials, i.e., stress depends on the physicalproperties of the material. It measures how difﬁcult a defor-mation is to achieve since it describes the internal force dis-tribution in the body, i.e., it has to balance the external forceaccording to Newton’s second law (1). On the contrary, any
deformationﬁeldinabodyinducesacertainstrain
ε
=
ε(
u
)
.The cause of the deformation is an external force, gravity forexample. Again, stress measures how much force at eachpoint inside the body is necessary to balance the externalcause of the deformation and this will depend on the mater-ial properties.The relation between stress and strain is expressed by so-called constitutive laws. This is a priorijustany function thatrelates strain and stress. The assumption of frame invariancestates that a physical quantity observed is independent of theobserver, which is a common physical principle. Using thisonecanrestricttheclassofsuitablefunctions.Furtherrestric-tion can be made by assuming that the material is homoge-nous, i.e., the material properties are the same at any pointinside the body. Another restriction can be deduced from theassumptionofisotropy.Thismeansthatthereactionofabodyto an applied external force at any point will essentially bethe same in any direction. Steel, for example, is an isotropicmaterial in contrast to wood. Now, one can mathematicallyprove thatanyfunctionrelatingstressandstrainthatsatisﬁesthese assumptions has a ﬁrst order approximation that onlyinvolves two physical constants. The ﬁrst order approxima-tion assumes small strains (and this is what we do in linearelasticity).ThissimpleconstitutivelawiscalledHooke’slawfor linear isotropic materials and can be written as
σ
=
λ
tr
(ε)
I
+
2
µε
(4)with only two independent positive constants. Here tr
(
·
)
denotes the trace and
ε
is the linearized strain tensor. Theinterested reader is referred to [10]. The material constantsinvolved are the well-known Lamé constants
λ
and
µ
. In theliterature sometimes several other constants are used to for-mulatethislawandtheycanbetransformedintooneanother.Often used are, for example, Young’s modulus
E
and Pois-son’s ratio
ν
which relate to the Lamé constants by
E
=
µ(
3
λ
+
2
µ)λ
+
µ, ν
=
λλ
+
µ.
(5)
Plane Strain.
While our formulation can readily be appliedto the three-dimensional case, this paper deals with defor-mations of two-dimensional shapes, in which case we do nothave to take into account the full three-dimensional model.Denote
x
=
(
x
1
,
x
2
,
x
3
)
T
the three spatial coordinates. If weassume that the body forces and the tractions on the body’sboundary are independent of
x
3
and do not have a com-ponent pointing out of the
(
x
1
,
x
2
)
-plane, we can assume
u
=
(
u
1
,
u
2
,
0
)
T
,
u
1
=
u
1
(
x
1
,
x
2
)
and
u
2
=
u
2
(
x
1
,
x
2
)
.This leads to a reduced 3-by-3 strain tensor in which thethird row and column are zero. Nevertheless, by Hooke’sgeneralized law the materials response leads to a stress ten-sor in which the third row and column also vanish apart from
σ
33
=
λ(ε
11
+
ε
22
)
. Here one can see that even for two-dimensionalforcesanddisplacementsthatstressisingeneralstillathree-dimensionalquantity,buttakingintoaccountthatall these quantities only depend on
x
1
and
x
2
the equilibriumequations reduce to a two-dimensional system. Combiningthestress-strainrelation(4)andtheequilibriumequation(3),
this system, the well-known Navier-Lamé equation, reads
µ
u
+
(λ
+
µ)
∇
div
u
+
F
=
0
.
(6)2.2 The
P
k
-FEMAs an elliptic partial differential equation (PDE) the modelfor small deformations can be handled comfortably by ﬁniteelement methods (FEMs). This shall be explained brieﬂy.The interested reader is referred to literature on FEMs andPDEs, [6,10,16,24].
In order to ensure unique solvability of the problem, onehas to impose boundary conditions so that the PDE can bewritten as
µ
u
+
(λ
+
µ)
∇
div
u
+
F
=
0 in
B
,
u
=
u
0
on
Γ
D
,σ(
u
)
n
=
g
on
Γ
N
(7)where
Γ
D
and
Γ
N
isadecompositionofthebody’sboundary
∂
B
intoapart
Γ
D
onwhichDirichletboundaryconditions
u
0
are given and a part
Γ
N
with Neumann boundary conditions,i.e., a force distribution
g
. The outward normal on
Γ
N
isdenotedby
n
.Forthesakeofsimplicityweusezeroboundaryconditions on the Dirichlet boundary, that is
u
0
=
0. Thegeneralization to non-zero conditions is just technical. Alsonote that (7)
1
is equivalent to
1 3

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