Int J Comput Vis (2009) 81: 281–301DOI 10.1007/s11263-008-0172-2

Topology-Invariant Similarity of Nonrigid Shapes

Alexander M. Bronstein · Michael M. Bronstein ·Ron Kimmel

Received: 9 September 2007 / Accepted: 8 August 2008 / Published online: 15 October 2008© Springer Science+Business Media, LLC 2008

Abstract This paper explores the problem of similarity cri-teria between nonrigid shapes. Broadly speaking, such crite-ria are divided into intrinsic and extrinsic, the first referringto the metric structure of the object and the latter to how it islaid out in the Euclidean space. Both criteria have their ad-vantages and disadvantages: extrinsic similarity is sensitiveto nonrigid deformations, while intrinsic similarity is sen-sitive to topological noise. In this paper, we approach theproblem from the perspective of metric geometry. We showthat by unifying the extrinsic and intrinsic similarity criteria,it is possible to obtain a stronger topology-invariant similar-ity, suitable for comparing deformed shapes with differenttopology. We construct this new joint criterion as a tradeoffbetween the extrinsic and intrinsic similarity and use it as aset-valued distance. Numerical results demonstrate the effi-ciency of our approach in cases where using either extrinsicor intrinsic criteria alone would fail.

Keywords Shape similarity · Isometry · Topologicalnoise · Gromov-Hausdorff distance · Generalized MDS ·GMDS · Iterative closest point

1 Introduction

In the childhood game Rock, Paper, Scissors, the playersbend their fingers in different ways to make the hand resem-ble one of three objects: a rock, a sheet of paper and scissors.Looking at these shapes, we can recognize the hand posturesas the objects they intend to imitate. The rock is represented

A.M. Bronstein (�) · M.M. Bronstein · R. KimmelDepartment of Computer Science Technion, Israel Institute ofTechnology, Haifa 32000, Israele-mail: bronstein@ieee.org

by a closed fist, the paper by an open palm and the scissorsby the extended index and middle fingers (Fig. 1). At thesame time, we can say that all these shapes are just differentarticulations of the same hand.

This example demonstrates the difficulty of defining thesimilarity of nonrigid shapes. On one hand, instances of non-rigid objects can be considered as stand-alone rigid shapes.On the other hand, these shapes can be regarded as nonrigiddeformations of the same object. Using geometric terminol-ogy, the first similarity criterion is extrinsic, i.e., considersthe properties of the shape related to the particular way it islaid out. The second criterion, looks at the intrinsic proper-ties of the shape, invariant to the object deformations.

Extrinsic similarity of shapes has been widely addressedin computer vision, pattern recognition, and computationalgeometry literature (Tangelder and Veltkamp 2004; Bron-stein et al. 2008c). Most of the papers in these fields, eitherimplicitly or explicitly, look at the problem of shape similar-ity from the extrinsic point of view (see, for example, Bruck-stein et al. 1992; Latecki and Lakaemper 2000; Jacobs et al.2000a). A classical method for rigid object matching, intro-duced by Chen and Medioni (1991) and Besl and McKay(1992), is the iterative closest point (ICP) algorithm. ICPand its numerous flavors (Zhang 1994; Leopoldseder et al.2003; Mitra et al. 2004; Gelfand et al. 2005) try to find arigid transformation between two shapes, minimizing an ex-trinsic distance between them, usually a variant of the Haus-dorff distance.

Another important class of extrinsic similarity methodsis based on high-order moments (Teague 1979; Horn 1987;Groemer 1996; Tal et al. 2001), where the shape’s extrin-sic geometry is represented by a vector of coefficients ob-tained from the decomposition of shape properties in somebasis. Conceptually, one can think of such methods as of aFourier-like representation (Zhang and Chen 2001a, 2001b),

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Fig. 1 (Color online) Rock, paper, and scissors: this childhood gameis based on the similarity of the postures of the hand to different objects(blue arrows). Though being a valid similarity criterion, in many otherapplications, we would rather be interested in saying that the differentpostures are all similar, being nonrigid deformations of the same object(red arrows)

though other bases like wavelets (Paquet et al. 2000), spheri-cal harmonics (Vranic et al. 2001; Yu et al. 2003; Kazhdan etal. 2003), and Zernike descriptors (Novotni and Klein 2003)have been explored in the literature as well. Besides “global”methods, there exist other families of extrinsic shape simi-larity methods based on histograms of local shape proper-ties like curvatures, distances, angles and areas (Osada etal. 2002; Shum et al. 1995). Such local methods can be, tosome extent, insensitive to shape deformations. For a com-prehensive survey of these approaches, the reader is referredto (Tangelder and Veltkamp 2004).

At the other end, methods for the computation of intrin-sic similarity of shapes started penetrating into the computervision and pattern recognition communities relatively late.As a precursor, we consider the paper by Schwartz et al.(1989), in which a method for the representation of the in-trinsic geometry of the cortical surface of the brain usingmultidimensional scaling (MDS) was presented. MDS is afamily of algorithms (Borg and Groenen 1997) commonlyused in dimensionality reduction and data analysis (Roweisand Saul 2000; Donoho and Grimes 2003; Tennenbaum etal. 2000) and graph representation (Linial et al. 1995). Theidea of Schwartz et al. was extended by Elad and Kimmel(2003), who proposed a nonrigid shape recognition methodbased on Euclidean embeddings. Elad and Kimmel mappedthe metric structure of the surfaces to a low-dimensionalEuclidean space and compared the resulting images (calledcanonical forms) in this space.1 Canonical forms were ap-

1The measurement of pairwise geodesic distances and the solutionof the underlying MDS problem are the two most computationally-intensive components of the canonical forms method. The measure-ment of geodesic distance can be performed very efficiently using the

plied to the problem of three-dimensional face recognition,where this method was proved to be efficient in recognizingthe identity of people while being insensitive to their facialexpressions (Bronstein et al. 2005). Ling and Jacobs usedthis method for recognition of articulated two-dimensionalshapes (Ling and Jacobs 2005b). Other applications were intexture mapping and object morphing (Zigelman et al. 2002;Grossman et al. 2002; Bronstein et al. 2007a), mesh seg-mentation (Katz et al. 2005) and image matching (Ling andJacobs 2005a).

One of the main disadvantages of the canonical formsis the fact that they can represent the intrinsic geometry ofthe objects only approximately, as it is generally impossi-ble to isometrically embed a non-flat surface into a low-dimensional (and in general, finite-dimensional) Euclid-ean space. It was shown empirically in Walter and Ritter(2002), Bronstein et al. (2007b) that using spaces with non-Euclidean geometry, it is possible to obtain more accuraterepresentations. Mémoli and Sapiro (2005) showed how therepresentation error can be theoretically avoided by usingto the Gromov-Hausdorff distance, introduced by MikhailGromov in Gromov (1981). Theoretically, the computationof the discrete Gromov-Hausdorff distance is an NP-hardproblem. In order to overcome this difficulty, Mémoli andSapiro proposed an approximation, related to the Gromov-Hausdorff distance by a probabilistic bound.

Using the fact that the Gromov-Hausdorff distance canbe related to the distortion of embedding one surface intoanother, we proposed in Bronstein et al. (2006b) a re-laxation of the discrete Gromov-Hausdorff distance, yield-ing a continuous optimization problem similar to MDS.This algorithm, named generalized MDS (GMDS) (Bron-stein et al. 2006b, 2006a), can be thought of as a naturalextension of previous works on isometric embedding intonon-Euclidean spaces. GMDS was used for the compari-son of two-dimensional (Bronstein et al. 2008b) and three-dimensional (Bronstein et al. 2006b) shapes, face recogni-tion (Bronstein et al. 2006c), and in a particular setting ofintrinsic self-similarity for symmetry detection (Raviv et al.2007; Bronstein et al. 2008b). It was also shown that GMDScan be used to find deformation-invariant correspondencebetween non-rigid shapes in computer graphics applicationssuch as texture mapping and shape manipulation (Bronsteinet al. 2007a). In these problems, GMDS is closely related tothe method of Litke et al. (2005).

Another class of intrinsic similarity methods is based onthe analysis of spectral properties of the Laplace-Beltramioperator of the shape. Reuter et al. used Laplace-Beltrami

recently proposed parallel fast marching method (Weber et al. 2008).The solution of an MDS problem can be carried out efficiently usingthe multigrid framework (Bronstein et al. 2006d) or the vector extrap-olation methods (Rosman et al. 2007). This allows for real-time appli-cations like face recognition.

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Fig. 2 (Color online)Illustration of the differencebetween intrinsic and extrinsicsimilarity

spectra (eigenvalues), referring to them as “shape DNA”, forthe characterization of surfaces (Reuter et al. 2006). Sincethe Laplace-Beltrami operator is an intrinsic characteristicof the surface, it is insensitive to isometric deformations(Chung 1997; Mohar 1991; Lévy 2006). It appears, however,that such criteria are able to identify isospectral rather thanisometric surfaces, and isospectrality is a weaker property(two surfaces can be isospectral but not isometric Kac 1966;Gordon et al. 1992; Berger 2002). Rustamov (2007) intro-duced the global point signature (GPS) embedding, basedon eigenfunctions and eigenvalues of the Laplace-Beltramioperator. Such a descriptor is exact and theoretically al-lows to represent the shape up to isometric deformations.GPS embeddings are intimately related to methods used inmanifold learning and data analysis (see e.g. Roweis andSaul 2000; Zhang and Zha 2002; Belkin and Niyogi 2003;Donoho and Grimes 2003; Coifman et al. 2005) and can bethought of as infinite-dimensional canonical forms.

Though apparently completely different, from the view-point of metric geometry both the intrinsic and extrinsic sim-ilarity criteria can be formalized using the notion of isometryinvariance. Regarding a shape as a metric space, its extrin-sic properties are described by using the Euclidean metric,while intrinsic ones using the geodesic metric, which mea-sures distances between points as the lengths of the short-est paths on the shape. Shape transformations preserving the

metric are called isometries; extrinsic isometries are rigidmotions and intrinsic isometries are inelastic deformations.Two shapes can be thus said to be similar if they are iso-metric. Depending on whether we choose the Euclidean orthe geodesic metric, we obtain the extrinsic or the intrinsicsimilarity, respectively. This perspective is recurrent in thepresent paper, allowing us to consider intrinsic and extrinsicsimilarity using the same framework.

The choice of whether to use intrinsic or extrinsic sim-ilarity depends significantly on the application. The draw-back of extrinsic similarity is its sensitivity to nonrigid de-formations. Using our previous example, a gesture of thehuman hand can be extrinsically more similar to a rock orscissors rather than another hand. This makes extrinsic cri-teria usually unsuitable for the analysis of nonrigid objectswith significant deformations (see Fig. 2, left and middle;see also Fig. 4). On the other hand, extrinsic similarity is in-sensitive to deformation changing the topology of the shape(such as “gluing” the fingers of the hand in Fig. 2, right;see also Fig. 5). The intrinsic similarity criterion, in a sense,behaves as the opposite of the extrinsic one: it is insensi-tive to inelastic deformations, but is sensitive to topologychanges (Fig. 2, right). In practical situation, such changescan arise due to acquisition imperfections (the so-calledtopological noise often encountered in surfaces acquired us-ing three-dimensional scanners, or reconstructed from volu-

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Fig. 3 (Color online) Fitting a glove onto one’s hand may be thoughtof as an optimal tradeoff between the stretching of the rubber surface(intrinsic dissimilarity), and the “amount of air” left between the gloveand the hand (extrinsic dissimilarity)

metric data) or partially missing data (“holes”) (Bronstein etal. 2008a).

In the case shown in Fig. 2, neither extrinsic nor intrinsicsimilarity per se is good enough, as none of these criteriais capable of saying that the three hand shapes are similar,which is the semantically desired result. In this example, wecan distinguish between two types of deformations: geomet-ric and topological. Geometric transformation change thecoordinates of the points of the shape, and include, for exam-ple, rigid motions (to which the extrinsic similarity criterionis invariant) and inelastic deformations (to which intrinsicsimilarity is invariant). Topological transformations, on theother hand, change the “connectivity” of the shape. The ex-trinsic geometry does not change significantly as a result ofsuch transformations, yet, the intrinsic one does: by modify-ing the connectivity, the paths between points can change aswell, which can significantly alter the geodesic metric.

The criterion we need in order to capture correctly thesimilarity of shapes in Fig. 2 must be insensitive to bothtopological and geometric deformations. We call such acriterion topology-invariant similarity. In Bronstein et al.(2007c), we proposed an approach for computing topology-invariant similarity between nonrigid shapes by combiningthe advantages of intrinsic and extrinsic similarity crite-ria, while avoiding their shortcomings. This paper presentsan extended discussion and experimental validation of thismethod. The main idea can be visualized by looking at theexample of fitting a rubber glove onto a hand. The extent towhich the rubber surface is stretched represents the intrin-sic geometry distortion. The fit quality, or in other words,how close the glove is to the hand surface, represents the

extrinsic distance between the two objects (Fig. 3). A “vir-tual” glove fitting is performed by moving the points of theglove with respect to the shape of the hand, trying to simul-taneously minimize the misfit (extrinsic dissimilarity) andthe stretching (intrinsic dissimilarity). Note that unlike realglove fitting, topological changes like glued fingers do notpose an obstacle in our case: the “virtual” glove can inter-sect the hand.

Finding an optimal tradeoff between the misfit andstretching can be posed as a multicriterion optimizationproblem and related to the notion of Pareto optimality. Fromthis point of view, our approach is close in its spirit to themethod of Latecki et al. (2005), Bronstein et al. (2008b),Bronstein and Bronstein (2008) for partial shape matching,using a tradeoff between the size of the parts cropped out ofthe shapes and the similarity between them. In our case, theset of all Pareto optimal solutions can also be represented asa set-valued similarity criterion, which contains much richerinformation than each of the intrinsic and extrinsic criteriaseparately.

Our approach can also be thought of as a generalizationor “hybridization” of ICP and GMDS methods. ICP methodscompute extrinsic similarity of shapes by means of findinga rigid isometry minimizing the extrinsic distance betweenthem. Our method extends the class of transformations, al-lows for nonrigid isometries and near-isometries, thus relat-ing to the recent works on nonrigid ICP (Chui and Rangara-jan 2003; Hahnel et al. 2003; Amberg et al. 2007). UsingGMDS, intrinsic similarity of shapes is computed as the de-gree of distortion when trying to embed one shape into an-other. This problem can be regarded as a particular case ofour approach in which we constraint the extrinsic distancebetween the two surfaces to be zero.

The rest of this paper is organized as follows. In Sect. 2,we introduce the mathematical background and formulatethe shape similarity problem from the perspective of met-ric geometry. We review standard approaches to measur-ing intrinsic and extrinsic similarity. In Sect. 3, we definetopology-invariant similarity and present an approach forcomputing it using a joint intrinsic-extrinsic criterion. Wediscuss this approach using the formalism of Pareto opti-mality. In Sect. 4, we show the numerical framework forcomputing the proposed distance. Section 5 is dedicated toexperimental results. In Sect. 6, we conclude the paper anddiscuss the relation of our approach to recent results in com-puter graphics literature and “as isometric as possible” mor-phing.

2 Isometry-Invariant Similarity

We model a shape as a metric space (X,d), where X is atwo-dimensional smooth compact connected surface (pos-sibly with boundary) embedded in the three-dimensional

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Euclidean space E = R3, and d : X × X → R+, is a (semi)

metric2 measuring the distance between each pair of pointson X. There exist two natural choices for the metric on X.The first one is the restriction of the Euclidean metric de-noted here by dE(x, x′) = ‖x −x′‖2, measuring the distancebetween any x, x′ ∈ X along “straight lines” in E. The sec-ond natural choice is the geodesic metric,

dX(x, x′) = minγ∈�(x,x′)

L(γ ), (1)

measuring the length of the shortest path on the surface con-necting a pair of points x and x′. In order to give a for-mal definition of the path length L(γ ), we express γ asthe limit of piece-wise linear segments connecting the points{x1, . . . , xn} on X,

L(γ ) = limn→∞

n−1∑

i=1

dE(xi, xi+1). (2)

The geodesic metric dX(x, x′) is given as the minimum overthe set of all admissible paths between x and x′, denotedby �(x, x′). A path is admissible if the edges of all the infin-itesimal segments xi, xi+1 are connected. We defer a moreprecise definition to the next section. We broadly refer toproperties described in terms of dE as the extrinsic geome-try of X, and to properties associated with dX as the intrinsicgeometry of X.

In order to determine whether two shapes X and Y aresimilar, we compare them as metric spaces. From the pointof view of metric geometry, two metric spaces are equivalentif their corresponding metric structures are equal. Such met-ric spaces are said to be isometric. More formally, a bijectivemap f : (X,d) → (Y, δ) is called an isometry if

δ ◦ (f × f ) = d. (3)

In other words, an isometry is a metric-preserving map be-tween two metric spaces, such that d(x1, x2) =δ(f (x1), f (x2)). We call such (X,d) and (Y, δ) isometricand denote this property by (X,d) ∼ (Y, δ).

This definition of equivalence obviously depends on thechoice of the metric. A bijection f : (X,dX) → (Y, dY ) sat-isfying dY ◦ (f × f ) = dX is called an intrinsic isometry.Saying that (X,dX) and (Y, dY ) are isometric is synony-mous to X and Y being intrinsically equivalent. On the otherhand, if we consider the extrinsic geometry of the shapes(i.e., look at the shapes endowed with the Euclidean ratherthan geodesic metric), we notice that (X,dE) and (Y, dE) aresubsets of the same metric space (E, dE). As a result, an ex-trinsic isometry is a bijection between subsets of the Euclid-ean space rather than between two different metric spaces.

2A semi-metric does not require the property d(x, x′) = 0 if and onlyif x = x′ to hold.

In Euclidean geometry, the only possible isometries are rigidmotions, which include rotation, translation and reflectiontransformations; we denote the family of such transforma-tions by Iso(E). Thus, X and Y are extrinsically isometricif there exists f ∈ Iso(E) such that dE = dE ◦ (f × f ) onX ×X. This means that two shapes are extrinsically isomet-ric if one can be obtained by a rigid transformation of theother, which is often expressed by saying that X and Y arecongruent.

To avoid confusion, in the following, we say that X and Y

are isometric implying intrinsic isometry, and that X and Y

are congruent when referring to extrinsic isometry. The classof intrinsic isometries is usually richer than that of congru-ences, since any congruence is by definition also an intrinsicisometry. However, for some objects these two classes co-incide, meaning that they have no incongruent isometries.Such shapes are called rigid, and their extrinsic geometry iscompletely defined by the intrinsic one.

2.1 Similarity

In practice, perfect equivalence rarely exists, and we are usu-ally limited to speaking about similarity of shapes ratherthan their equivalence in the strict sense. In order to ac-count for this, we need to relax the notion of isometry. Twometric spaces (X,d) and (Y, δ) are said to be ε-isometricif there exists an ε-surjective map f : (X,d) → (Y, δ) (i.e.,δ(y,f (X)) ≤ ε for all y ∈ Y ), which has the distortion

disf = supx,x′∈X

|d(x, x′) − δ(f (x), f (x′))| ≤ ε. (4)

Such an f is called an ε-isometry. ε-isometries are quitedifferent from their true counterparts. Particularly, an isom-etry is always bi-Lipschitz continuous (Burago et al. 2001),which is not necessarily true for an ε-isometry. If we furtherrelax the requirement of ε-surjectivity by demanding that f

has only disf ≤ ε, we refer to such f as an ε-isometric em-bedding.

A way to quantify the degree of shape dissimilarity isby defining a shape distance dshape : S × S → R+ on thespace of shapes S. Note that dshape is a function of (X,d)

and (Y, δ). In the following, we will write dshape(X,Y ) omit-ting explicit reference to the metrics for notation brevity. Itis common to require dshape to satisfy the following set ofproperties for any X,Y , and Z in S and a constant c ≥ 1independent of X,Y and Z:

(I1) Equivalence: dshape(X,Y ) = 0 if and only if (X,d) and(Y, δ) are isometric.

(I2) Similarity: if dshape(X,Y ) ≤ ε, then (X,d) and (Y, δ)

are cε-isometric; and if (X,d) and (Y, δ) are ε-iso-metric, then dshape(X,Y ) ≤ cε.

(I3) Symmetry: dshape(X,Y ) = dshape(Y,X).

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(I4) Triangle inequality: dshape(X,Z) ≤ dshape(X,Y ) +dshape(Y,Z).

The first two properties guarantee that the shape distancedshape reflects the degree of shape dissimilarity, i.e. dshape

is small for similar (isometric) shapes, and large for dis-similar ones. The third property is synonymous to the re-flexivity of the shape similarity relation, while the fourthreflects its transitivity: if X is similar to Y and Y is sim-ilar to Z, then X and Z cannot be too much dissimi-lar. Properties (I1), (I3), and (I4) can be expressed equiv-alently by saying that dshape is a metric on the quotientspace of S under the isometric equivalence relation, de-noted by S\ ∼. We refer to a shape distance satisfying (I1)–(I4) as to isometry-invariant shape distance. Note, however,that in the case of a partial shape similarity relation, met-ric axioms are usually too restrictive (Jacobs et al. 2000b;Bronstein et al. 2008a).

Since the definitions of equivalence and similarity de-pend on the choice of the metrics on the shapes, our no-tion of isometry-invariant distance between shapes also de-pends on them. In the remainder of this section, we willconsider two particular cases of extrinsic and intrinsic shapedistances, defining, respectively, the extrinsic and intrinsicsimilarity.

2.2 Extrinsic Distances

Extrinsic similarity is a simple case of the general prob-lem of metric space comparison, since two shapes with theEuclidean metric, (X,dE) and (Y, dE), are a subset of thesame metric space (E, dE). Consequently, we can use theHausdorff distance measuring the distance between two setsX and Y in E,

dE

H(X,Y ) = max{

supx∈X

dE(x,Y ), supy∈Y

dE(y,X)}, (5)

where dE(y,X) = infx∈X‖y − x‖2 denotes the distancebetween the set X and the point y. In practice, a non-symmetric version of dE

H,

dE

NH(X,Y ) = supx∈X

dE(x,Y ), (6)

is often preferred since it allows for partial comparison ofsurfaces. L2 approximations are also often preferred, as theoriginal L∞ formulation is sensitive to outliers.

Extrinsic equivalence of shapes implies that they are con-gruent, i.e., can be brought into ideal correspondence bymeans of a rigid transformation. In other words, there ex-ists f ∈ Iso(E) such that dE

H(f (X),Y ) = 0. We can use thesame idea in order to measure extrinsic similarity: find theminimum possible value of the Hausdorff distance over allpossible rigid motions,

dext(X,Y ) = inff ∈Iso(E)

dE

H(f (X),Y ). (7)

Fig. 4 (Color online) The weakness of extrinsic similarity. Top rowshown deformations of a three-dimensional shape of the nonrigid hu-man body. Weak deformations (left and middle) can be compared usingICP algorithm, which finds a meaningful alignment between the shapes(bottom row, left), despite topological noise (simulated here by weld-ing hand and leg). However, in the case of strong deformations (toprow, right) ICP produces meaningless matching (bottom row, right)

Since Iso(E) can be easily parametrized, the resulting ex-trinsic distance can be computed as

dext(X,Y ) = infR,t

dE

H(RX + t, Y ), (8)

where R denotes the rotation matrix and t is the trans-lation vector.3 Practical methods to solve problem (7) arethe ICP algorithms, which use an alternating minimizationscheme consisting of two stages. First, the closest-point cor-respondence between X and Y is computed. Next, the rigidtransformation is found between X and Y that minimizesthe Hausdorff distance. The process is repeated until con-vergence. Though theoretically a global minimum shouldbe searched for, in practice, local optimization methods areusually employed, and thus, ICP algorithms are often proneto convergence to a suboptimal solution (local minimum).

The distance dext satisfies Properties (I1)–(I4), being ametric on the space of shapes modulo Iso(E), and is a goodway to measure extrinsic similarity.

3Reflection is usually excluded as having no physical realization in E.

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2.3 Intrinsic Distances

Computation of intrinsic shape dissimilarity is a more com-plex task compared to its extrinsic counterpart due to twomain reasons. First, intrinsic isometries usually constitute asignificantly richer class of transformations than Euclideancongruences, and there often exists no simple parametriza-tion comparable to the six degrees of freedom that are suffi-cient to represent a rigid motion. Second, unlike in the rigidcase where (X,dE) and (Y, dE) are compared as subsets ofa common metric space (E, dE), the comparison of (X,dX)

and (Y, dY ) does not allow using the Hausdorff distance, andhence dext in (7) cannot be straightforwardly generalized tothe intrinsic case.

Elad and Kimmel (2003) reduced the problem of intrinsicshape similarity to the more tractable problem of extrinsicsimilarity, by first computing an extrinsic representation ofthe intrinsic geometry of the shapes in some common metricspace (Q, dQ). Such a representation, dubbed as the canoni-cal form, is constructed by minimum-distortion embedding,attempting to find two maps ϕ : (X,dX) → (Q, dQ) andψ : (Y, dY ) → (Q, dQ) with minimum distortions dis ϕ anddis ψ . The embedding, in a sense, allows to “undo” all theisometric deformations of the shapes (though, some degreeof ambiguity stemming from isometries in Q still remains).Once the canonical forms ϕ(X) and ψ(Y ) are computed,they are compared extrinsically using, for example, ICP. Inother words, the intrinsic distance between two shapes X

and Y is computed as the extrinsic distance between theircanonical forms,

dint(X,Y ) = dext(ϕ(X),ψ(Y )). (9)

If the Euclidean space (Em,dEm) is used as the embed-ding space, the minimum distortion embedding of (X,dX)

can be found by solving a multidimensional scaling (MDS)problem. Given the shape X sampled at points X = {x1, . . . ,

xN } and the N × N matrix of distances DX

= (dX(xi, xj )),MDS algorithms try to find a configuration of points inR

m such that the Euclidean distances between these pointsare as close as possible in some sense to the elementsof D

X. Typically, the L2 distortion criterion is used, in which

case the MDS problem is referred to as least-squares MDS(LSMDS). The canonical form is given by

Z = argminZ∈RN×m

∑

i>j

|‖xi − xj‖2 − dX(xi, xj )|2, (10)

where Z is an N × m matrix representing the coordinatesof the points in E

m. There exist numerically simple and ef-ficient algorithms for solving problem (10) (Borg and Groe-nen 1997; Bronstein et al. 2006d).

The choice of Q has an important influence on the com-putation of canonical forms. Though the Euclidean space is

the simplest and most convenient metric space for this pur-pose, other choices are possible as well (Elad and Kimmel2002; Walter and Ritter 2002; Bronstein et al. 2007b). Thereare a few criteria for choosing the embedding space. First, itis desired that Q is homogeneous and its isometry group issimple, in order to reduce the number of degrees of freedomin the definition of the canonical forms. Secondly, an ana-lytic expression for dQ allows using MDS algorithms (Borgand Groenen 1997). Finally, it appears that for some classesof shapes, certain embedding spaces are generally more suit-able (see, for example, experimental results in Bronstein etal. 2007b).

However, achieving a true isometric embedding (i.e., theminimum in problem (10) equal to zero) is usually impos-sible (Linial et al. 1995) with a metric space Q satisfyingthe above criteria. Hence, the canonical forms are only anapproximate representation of the intrinsic geometry of theshapes. The problem of inaccuracy introduced by the em-bedding into Q can be resolved if we do not assume a givenembedding space, but instead, include Q as a variable intothe optimization problem. We can always find a sufficientlycomplicated metric space into which both X and Y can beembedded isometrically, and compare the images using theHausdorff distance,

dGH((X,dX), (Y, dY )) = infQ

ϕ:X→Q

ψ :Y→Q

dQ

H (ϕ(X),ψ(Y )), (11)

(here ϕ and ψ are assumed to be isometric embeddings).The resulting distance is referred to as Gromov-Hausdorffdistance (Gromov 1981). This distance was first introducedto surface matching by Mémoli and Sapiro (2005). In thefollowing, we use a brief notation dGH(X,Y ) when the im-plied metrics are clear.

While the computation of dGH as defined in (11) ishardly tractable due to the minimization over all embeddingspaces Q, it appears that for compact surfaces, the Gromov-Hausdorff distance can be expressed in terms of the distor-tion obtained by embedding one surface into the other,

dGH(X,Y ) = 1

2inf

ϕ:X→Y

ψ :X→Y

max{disϕ,disψ,dis (ϕ,ψ)}, (12)

where,

dis (ϕ,ψ) = supx∈X,y∈Y

|dX(x,ψ(y)) − dY (y,ϕ(x))|. (13)

The computation of the distortions is performed usingthe generalized multidimensional scaling (Bronstein et al.2006a, 2006b), a procedure similar in its spirit to MDS, butnot limited to spaces with analytically expressed geodesicdistances. Using the Gromov-Hausdorff distance, the intrin-sic shape distance can be computed as

dint(X,Y ) = dGH((X,dX), (Y, dY )). (14)

288 Int J Comput Vis (2009) 81: 281–301

Fig. 5 (Color online) Theweakness of intrinsic similarity.Shown are deformations of athree-dimensional nonrigidhuman body shape (top row)and the corresponding canonicalforms (bottom row) computedusing classical MDS. Thecanonical forms appear to beinsensitive to near-isometricdeformations of the shape (leftand middle columns). However,topological noise (simulatedhere by welding the hands to thelegs at points indicated by redcircles) results in a completelydifferent canonical form

2.4 A Unified View of Extrinsic and Intrinsic Similarity

It is worthwhile to note that the Gromov-Hausdorff dis-tance is a generic isometry-invariant shape distance satis-fying properties (I1)–(I4) with c = 2. Particularly, this im-plies that if dGH((X,d), (Y, δ)) ≤ ε, then (X,d) and (Y, δ)

are 2ε-isometric and conversely, if (X,d) and (Y, δ) areε-isometric, then dGH((X,d), (Y, δ)) ≤ 2ε (Burago et al.2001). One can think of the Gromov-Hausdorff distance asa general framework for isometry-invariant shape compari-son, unifying both extrinsic and intrinsic similarity. Select-ing d and δ as the geodesic metrics (dX and dY , respec-tively), we obtain the intrinsic shape distance dint. Consider-ing the shapes with the Euclidean metric dE yields an equiv-alent (but not equal) way to express the extrinsic shape dis-tance as

dext(X,Y ) = dGH((X,dE), (Y, dE)), (15)

instead of the definition we have seen before. In this case, weinterpret isometry as congruence, while in the former case, itimplies the existence of a geodesic distance-preserving de-formation of X into Y .

3 Topology-Invariant Similarity

Let us now return to our example of glove fitting. Assumethat Y is the hand surface, and X is the glove we wish to fit.A perfect fit is achieved when X and Y are equivalent. Since

both the glove and the hand are nonrigid shapes, we under-stand equivalence in the intrinsic sense, i.e., as the existenceof an isometry between (X,dX) and (Y, dY ). An alternativeway to express this fact is by saying that there exists an iso-metric deformation f of X, such that f (X) = Y . In otherwords, we may say that the glove X perfectly fits the handY if there exists Z intrinsically equivalent to X and extrinsi-cally equivalent to Y .

Let us now assume that the hand is given in a posturewhere the fingers are glued, while the fingers of the gloveare disconnected. Such a topological difference would makedistinct the intrinsic geometries of the hand and the glove,preventing X and Y from being intrinsically equivalent (foran illustration, see Fig. 5). However, our alternative defini-tion of equivalence would still hold, as we can still find Z

intrinsically equivalent to X, which will fit Y extrinsically(though, unlike the previous case, Z and Y are no longerintrinsically equivalent).

This example visualizes the limitations of the notion ofintrinsic equivalence of shapes, and brings forth the need toconstruct a more general notion of equivalence, insensitiveto topological changes. Such a construction requires severaladditions to our mathematical machinery. We define a topol-ogy T of X as a family of subsets of X (i.e., T is a subset ofthe power set 2X) closed under finite intersection and unionof arbitrarily many elements of T . By definition, both theempty set and X itself are members of T . Conventionally, asubset of X is called open if it belongs to T , and closed ifits complement belongs to T . From now on, when speaking

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about a shape X, we will actually imply the triplet (X,T , d),and omitting T will imply that the topology is induced by themetric. For example, for the choice d = dE, the topology ofthe Euclidean space E is assumed. Topology defines proper-ties of the shape which are “coarser” than metric geometry,such as connectivity of points.

Our glove fitting example suggests that there may existtwo shapes (X,T , dX) and (X,T ′, d ′

X) with identical real-ization X in the Euclidean space, yet different topology andintrinsic geometry. This is possible due to the fact that ourdefinition of the geodesic metric in (1) was based on thelength of the shortest admissible path γ ∈ �(x, x′) connect-ing two points x and x′ on X.

However, the definition of the set of all admissible pathsconnecting x and x′ is itself a function of the topology, as itdepends on the connectivity. For example, in case of a handwith glued fingers, there exists a path between two fingertips crossing the fingers. When the fingers are disconnected,the shortest path between the finger tips goes along the fin-gers, since the path directly connecting the finger tips is nolonger admissible (for a more formal discussion, the readeris referred to Chap. 2 in Burago et al. 2001).

Two topological spaces (X,T ) and (Y,T ′) are said to behomeomorphic or topologically equivalent if there exists acontinuous bijection g : (X,T ) → (Y,T ′), whose inverse isalso continuous (the continuity of g and g−1 is understoodin the sense of T and T ′, respectively). Since isometriesare also homeomorphisms, it follows straightforwardly thatboth T and d remain invariant under an isometry.

In order to distinguish between purely geometric andpurely topological deformations of a shape, we will saythat a map f : (X,T , d) → (Y,T ′, δ) is a topological de-formation if it leaves X unchanged (i.e., X = Y ).4 Usingthis notion, we can finally define two shapes (X,TX,dX)

and (Y,TY , dY ) to be equivalent in the sense of topology-invariant similarity if there exists a topological deforma-tion g : (X,TX,dX) → (X,T ′

X,d ′X), and an isometry f :

(X,T ′X,d ′

X) → (Z,T ′X,d ′

X) such that Y = Z = (f ◦ g)(X).Note that though we gave a simple example of topologi-cal deformations that change point-wise connectivity, other,more generic deformations (such as opening holes) are alsopossible.

Such a definition of topology-invariant equivalence ismore general than extrinsic or intrinsic similarity we hadbefore. In our example, a glove (Y,TY , dY ) shown in Fig. 2(left) and a hand with glued fingers (X,TX,dX) shown inFig. 2 (right) are extrinsically dissimilar since X and Y areincongruent. On the other hand, (X,TX,dX) and (Y,TY , dY )

are intrinsically dissimilar since (X,TX) and (Y,TY ) are nottopologically equivalent and as a result, (X,dX) and (Y, dY )

are not isometric. Yet, using the above criterion, there is a

4Or more generally, X and Y are congruent.

topology-invariant equivalence between shapes (X,TX,dX)

and (Y,TY , dY ): we can first “unglue” the fingers of the handby means of a topological deformation g : (X,TX,dX) →(X,T ′

X,d ′X), and then bend the hand by means of an isom-

etry f : (X,T ′X,d ′

X) → (Z,TZ, dZ), such that Z = Y , T ′X is

equivalent to TZ , and d ′X is equivalent to dZ .

In order to relax this notion of equivalence into atopology-invariant similarity relation, a slightly more com-plicated construction is required. We will say that (X,TX,dX)

and (Y,TY , dY ) are (εint, εext)-similar if there exists a topo-logical deformation g : (X,TX,dX) → (X,T ′

X,d ′X), and

an εint-isometry f : (X,T ′X,d ′

X) → (Z,T ′X,d ′

X) such thatdE

H((f ◦ g)(X),Y ) ≤ εext.In many practical situations, an asymmetric definition of

topology-invariant similarity is sufficient. In such cases, wedistinguish between a probe X that is fit to a model Y . Wesay that (X,TX,dX) and (Y,TY , dY ) are (εint, εext)-similar ifthere exists an εint-isometry f : (X,T ′

X,d ′X) → (Z,T ′

X,d ′X)

such that dE

H((f ◦g)(X),Y ) ≤ εext. In what follows, we willlimit our attention to the latter definition of similarity, andwill construct a distance based on a combination of extrinsicand intrinsic distances to reflect it.

3.1 Topology-Invariant Distance

Another way to express our notion of topology-invariantsimilarity is by saying that the probe X admits a topology-preserving deformation Z such that dint(X,Z) ≤ εint, anddext(Z,Y ) ≤ εext. Since it is usually impossible to say whichof the two criteria is more important, we judge the similarityas a tradeoff between them. Such a joint intrinsic-extrinsicsimilarity between two shapes can be quantitatively repre-sented by the extent to which we have to modify the ex-trinsic geometry in order to make the two shapes intrinsi-cally similar, or alternatively, the extent to which we haveto modify the intrinsic geometry in order to make the twoshapes extrinsically similar. This, in turn, can be formu-lated as a multicriterion optimization problem, in which webring to minimum the vector objective function �(Z) =(dint(Z,X), dext(Z,Y )) with respect to Z.

Unlike optimization with a scalar objective, we cannotdefine unambiguously the minimum of �, since there doesnot exist a total order relation between the criteria—we can-not say, for example, whether it is better to have � = (0.5,1)

or � = (1,0.5). At the same time, we have no doubt that� = (0.5,0.5) is better than � = (1,1), since both criteriahave smaller values. This allows us to define a minimizer ofour vector objective � as a Z∗, such that there is no otherZ for which �(Z) is better that �(Z∗) (which can be ex-pressed as a vector inequality �(Z∗) > �(Z)). Such an Z∗is called non-inferior or Pareto optimal. Pareto optimum isnot unique; we denote by ∗ the set of all Z∗ that satisfythe above relation. The corresponding values of �(∗) are

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Fig. 6 (Color online) Visualization of the concept of set-valued dis-tance. Different points on the Pareto frontier represent different trade-offs between intrinsic and extrinsic similarity

referred to as a Pareto frontier and can be visualized as aplanar curve (Fig. 6).

Bronstein et al. (2008a) proposed considering the entirePareto frontier as a criterion of similarity, of which we canthink as a generalized, set-valued distance.5 In our case, thisdistance measures the tradeoff between the intrinsic and ex-trinsic similarity, which we denote by dP(X,Y ) = �(∗)and refer to as the Pareto distance. The Pareto distancequantifies the degree of asymmetric topology-invariant dis-similarity; particularly, (0,0) ∈ dP(X,Y ) if and only if X

and Y are equivalent in the sense that X has an isome-try congruent to Y . The Pareto distance also generalizesthe similarity criteria based on purely extrinsic or intrin-sic geometry. Asserting dint(X,Z) = 0, the other criteriondext(Z,Y ) in dP(X,Y ) will measure how close a perfectlyisometric deformation of X can bring X to Y . Since in prac-tice we work with meshes which are known to be almostalways rigid, the only possible deformations are congru-ences of X. This means that with dint(X,Z) fixed to zero,dP(X,Y ) is equivalent to ICP. On the other hand, if werequire dext(Y,Z) = 0, the probe is forced to be attachedto the model surface, which boils down to a GMDS prob-lem, where we are trying to find a deformation of X mini-mizing dint(Z,Y ). We conclude that the two extreme cases

5Set-valued distances arise from similarity relations based on morethan one criterion such as in the case of partial similarity, which canbe regarded as the tradeoff between similarity and significance of theparts. We believe that it is more natural to use set-valued distances forsuch relations, rather than trying to fit them into the Procrustean bed ofscalar-valued similarity.

of dP(X,Y ) are (0, dext(X,Y )) and (dint(X,Y ),0). Otherpoints on the Pareto frontier represent different tradeoffs be-tween the extrinsic and the intrinsic criteria, as depicted inFig. 6.

In general, by saying that dP(X,Y ) ≤ (εint, εext) (the vec-tor inequality implies that the point (εint, εext) is above oron the Pareto frontier), we mean that there exists a deforma-tion Z of X distorting its intrinsic geometry by less than εint,while making its extrinsic geometry less than εext-dissimilarfrom that of Y . Saying that dP(X,Y ) < dP(X,Y ′), impliesthat bringing X to a certain proximity of Y requires a smallerintrinsic distortion than bringing X to the same proximityof Y ′. Using our glove fitting example, we would say thatthe glove X fits the hand Y better than the hand Y ′. Geomet-rically, this fact is manifested by the curve dP(X,Y ) beingentirely below dP(X,Y ′).

3.2 Joint Intrinsic and Extrinsic Similarity

The set-valued distance dP may be inconvenient to usesince only a partial order relation exists between the Paretofrontiers—given two set-valued distances, we cannot in gen-eral say which of them is “smaller”, unless one curve is en-tirely above or below the other. In order to compare shapedistances, we have to convert them into a single scalar value.

Intuitively, the “speed of decay” of the Pareto frontier in-dicates how similar two shape are (the faster, the more sim-ilar). There are multiple ways to represent this informationas a single number. First, one can measure the area underthe Pareto frontier. Smaller area corresponds to higher sim-ilarity. Second, it is possible to select a single point on thePareto frontier by fixing a pre-set value of one of the dissimi-larities. For example, if we know that the shapes are inelasticto a certain degree, i.e. they can stretch or shrink by no morethan ε, we will fix the intrinsic dissimilarity dint = ε and willuse the value extrinsic dissimilarity at this point as the shapedistance. A third alternative is to require that both dissimi-larities are equal and compute the point at which dint = dext.This way, we obtain a shape distance similar in its spirit tothe equal error rate (EER) used in receiver operating char-acteristic analysis.

A more generic approach to converting a set-valued dis-tance into a scalar-valued one comes from the multicrite-rion optimization theory (Salukwadze 1979). Among all thePareto optima, we cannot prefer any since they are non-comparable: we cannot say which Pareto optimum is bet-ter. However, ideally we want to bring both of our criteriato zero, that is, achieve the “utopia point” (0,0). We canchoose a single point on the Pareto frontier the closest to theutopia point, in the sense of some distance.

Using this idea, we define the joint distance as

djoint(X,Y ) = inf(εint,εext)∈dP(X,Y )

‖(εint, εext)‖R2+ , (16)

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where ‖ ·‖R

2+ is some norm on R2+. In the following, we will

consider a family of norms ‖(εint, εext)‖λ = εint + λεext, forλ > 0. The joint distance in this case can be written as

djoint(X,Y ) = minZ

dint(X,Z) + λdext(Z,Y ). (17)

Different selections of the multiplier λ attribute importanceeither to dint or dext, and give us different points on thePareto frontier.

3.3 Simplification

The intrinsic distance term dint(X,Z) can be simplified byobserving that the deformation Z = f (X) gives a one-to-one correspondence between X and Z. We can therefore fixϕ and ψ in the Gromov-Hausdorff distance to f and f −1,respectively, obtaining the following intrinsic distance,

dint(X,Z) = disf

= supx,x′∈X

|dX(x, x′) − dZ(f (x), f (x′))|. (18)

Note that the correspondence between the surfaces is nowfixed and does not participate anymore in the minimization.dint(X,Z) defined this way measures the distortion in theintrinsic geometry of X introduced by the extrinsic defor-mation f . The main difficulty in its computation stems fromthe fact the geodesic metric dZ has to be re-computed everytime the deformation changes. We will defer the discussionof this issue to the following section.

A simplification of the extrinsic distance term dext(Z,Y )

in djoint(X,Y ) is possible due to the fact that the deforma-tion Z of X already accounts for all possible congruences.Consequently, there is no need to minimize over Iso(E)

when computing dext(Z,Y )—we can simply use dE

H(Y,Z)

or dE

NH(Y,Z) instead. Like in ICP, the main computationalchallenge is the need to re-compute the set of closest pointsevery time the deformation changes.

4 Numerical Framework

For practical computations, we work with discretized shapes.The surface X is sampled at N points X = {x1, . . . , xN } ⊆X, constituting an r-covering, i.e., X = ⋃N

n=1 BX(xn, r)

(here, BX(x, r) is a metric ball of radius r around x). Theextrinsic coordinates of X can be represented as an N × 3matrix X, each row of which corresponds to xi ∈ E. The dis-crete shape is represented as a triangular mesh; each triangleis a triplet of indices of vertices belonging to it. The maxi-mum length of an edge is r . Vertices connected by an edge(or in other words, belonging to the same triangle) are saidto be adjacent; we describe the adjacency by the set E of alladjacent pairs of vertices in X. The geodesic distances on X

are approximated using the fast marching method (Kimmeland Sethian 1998) or the Dijkstra’s algorithm, forming anN × N matrix D

X.

Assuming the deformed surface Z = f (X) maintains theconnectivity of X, we can formulate the following mini-mization problem with respect to the N × 3 matrix Z of theextrinsic coordinates of Z:

djoint(X, Y ) = minZ

1

N2

N∑

i,j=1

(dX(xi, xj ) − dij (Z))2

+ λ

N

N∑

i=1

d2(zi, Y ) (19)

where dij (Z) = dZ(zi, zj ) denote the geodesic distances on

Z, and d(zi, Y ) denotes the Euclidean distance from thepoint zi to the discretized surface Y . The first term of theabove cost function is the discretization of dint, whereas thesecond term is the discretization of dext. In the sequel, weshow how to efficiently compute these two terms and theirderivatives with respect to Z, required for the minimizationof (19).

4.1 Intrinsic Distance Computation

The main challenge in the computation of the intrinsic dis-tance term dint is the need to evaluate the geodesic distanceson Z and their derivatives with respect to the extrinsic geom-etry of Z changing at each iteration of the minimization al-gorithm. The simplest remedy would be to modify the in-trinsic term by restricting i and j to the neighboring pointsonly,

dint ≈ 1

|E|∑

(i,j)∈E

(dX(xi, xj ) − d

Z(zi, zj )

)2. (20)

This way, we use only the local distances on Z, which canbe approximated as the Euclidean distances d

Z(zi , zj ) =

‖zi − zj‖. However, such a modification makes dint sig-nificantly less sensitive to large deformations of X. Indeed,many deformations change the local distance only slightly,while introducing large distortion to larger distances.

In order to penalize for such deformations of X, we needto approximate the full matrix of geodesic distances on Z.We first define the matrix D(Z) of local distances, whoseelements are, as before,

dij (Z) ={‖zi − zj‖ (i, j) ∈ E

0 (i, j) /∈ E.(21)

Using the Dijkstra algorithm,6 we compute the set of short-est paths between all pairs of points (i, j). For example,

6Dijkstra’s algorithm (unlike e.g. Fast Marching) is known to producea triangulation-dependent approximation of the geodesic distances due

292 Int J Comput Vis (2009) 81: 281–301

let Pij = {(i, i1), (i1, i2), . . . , (in−1, in), (in, j)} ⊂ E be theshortest path between the points i and j . Its length is givenby L(Pij ) = di,i1 +di1,i2 +· · ·+din,j , which is a linear com-bination of the elements of D(Z). We can therefore “com-plete” the missing entries in the matrix D(Z) by defining thematrix of global distances

D(Z) = I(Z)D(Z), (22)

where I is a sparse fourth order tensor, with the elementsIijkl = 1 if the edge (k, l) is contained in the shortest pathPij , and 0 otherwise. Note that I depends on the connectiv-ity E, which is assumed to be fixed, and the matrix of localdistances D, which, in turn, depends on Z.

It is straightforward to verify that the entries of D(Z) andD(Z) coincide for all (i, j) ∈ E. The global distance matrixD(Z) constitutes an approximation of the geodesic distanceson the surface Z, dij ≈ d

Z(zi , zj ), while having a simple

linear form in terms of the local distances. A consistent andmore accurate estimate of d

Zcan be produced by replacing

the Dijkstra algorithm with fast marching and allowing theshortest paths to pass on the faces of the mesh represent-ing Z. However, such an approach results in more elaborateexpressions and will not be discussed here.

In order to compute the derivative of D(Z) with respectto Z, we assume that a small perturbation dZ of Z does notchange the connectivity of the points on Z, and as a result,the trajectory of the shortest paths between the points on Z

remains constant (though their length may change). Thus,we may write

D(Z + dZ) = I(Z + dZ)D(Z + dZ) = I(Z)D(Z + dZ),

(23)

and compute the derivative of D(Z) as the derivative of thelinear form I(Z)D(Z). If I(Z) �= I(Z + dZ), the assump-tion does not hold, and the derivative of D usually does notexist. Yet, the derivative of I(Z)D(Z) belongs to the sub-gradient set of D at the point Z. This is sufficient for manyminimization algorithms to work correctly.

The intrinsic distance term can be readily written in termsof D as the Frobenius norm

dint(Z) = 1

N2‖D(Z) − D

X‖2

F

= 1

N2trace((D(Z) − D

X)T(D(Z) − D

X)). (24)

Its derivative with respect to Z is given by

∂dint(Z)

∂Z= 2

N2(D(Z) − D

X)T ∂D(Z)

∂Z, (25)

to the so-called metrication error. However, since in our case we arecomparing the shape to a deformed version of itself, the triangulationremains the same and thus triangulation-independence is not required.

where

∂dij (Z)

∂Z=

∑

k,l

Iijkl

∂dkl(Z)

∂Z, (26)

and the elements of ∂dkl (Z)∂Z are given by

∂dkl

∂zmn

= 1

dkl

⎧⎨

⎩

zmk − zm

l n = k

zml − zm

k n = l

0 n �= k, l

(27)

for m = 1,2,3.

4.2 Extrinsic Distance Computation

The computation of the extrinsic distance is similar to theone used in ICP algorithms, where the main difficulty arisesfrom the need to re-compute the closest points each timethe extrinsic geometry of Z changes. The extrinsic distanceterm can be written as

dext(Z) = 1

Ntrace((Z − Y∗(Z))(Z − Y∗(Z))T) (28)

where Y∗(Z) denotes the N × 3 matrix, whose i-th row y∗i

is the closest point on Y corresponding to zi . The closestpoints y∗

i are computed as a weighted average of the pointson Y , which are the closest to zi . The weights are selectedin inverse proportion to the distance from zi .

In ICP algorithms, it is common to assume that Y∗(Z +dZ) ≈ Y∗(Z). By fixing Y∗, dext(Z) becomes a simplequadratic function, and its derivative can be written as

∂dext(Z)

∂Z= 2

N(Z − Y∗(Z))T. (29)

4.3 Iterative Minimization Algorithm

For Z′ in the neighborhood of some Z, the cost function thatneeds to be minimized can be approximated as

σ(Z′) ≈ 1

N2trace (D(Z′)TD(Z′) − 2DT

XD(Z′) + DT

XD

X)

+ λ

Ntrace (Z′Z′T − 2Y∗(Z)Z′T + Y∗T(Z)Y∗(Z)),

(30)

where D(Z′) = I(Z)D(Z′). Like in ICP, after finding a newZ′ which decreases σ(Z′), the closest points Y∗ and the op-erator I are updated. The iterative minimization algorithmcan be summarized as follows:

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Compute the closest points Y∗(Z).1

Compute the shortest paths between all pairs of points2

on Z and assemble I .Find Z′ sufficiently decreasing σ(Z′) in (30).3

If the change ‖Z − Z′‖ is small, stop. Else, set Z = Z′4

and go to Step 1.

The update of Z in Step 3 can be safeguarded by evalu-ating the true cost function (with I(Z′) and Y∗(Z′) insteadof I(Z) and Y∗(Z)). In our implementation, no safeguardwas used, and the minimization in Step 3 was done usingconjugate gradients.

The initialization of the algorithm can be done in severalways, the simplest of which is Z = X. This choice workswell when the extrinsic dissimilarity between X and Y is nottoo large; for large dext(X,Y ), the algorithm will suffer frompoor convergence similarly to most ICP methods. Anotherchoice is to initialize Z by the corresponding points on Y re-sulting from the solution of the GMDS problem. This choiceis suitable for objects having sufficiently similar intrinsicgeometries, making the intrinsic correspondence computedby GMDS meaningful. A more robust initialization schemecan be constructed based on branch-and-bound global opti-mization proposed in Gelfand et al. (2005) for the initializa-tion of ICP algorithms, and adopted in Raviv et al. (2007),Bronstein et al. (2008b) for the initialization of GMDS. Thisfamily of approaches consists of computing local descrip-tors (in our case, reflecting intrinsic geometric properties)for a set of prominent feature points on both shapes, fol-lowed by finding the best correspondence between the twosets of features. The branch-and-bound technique is used forfast pruning of the search space, guaranteeing global opti-mality of the found solution on one hand, while maintainingreasonably low complexity on the other.

As both in ICP and GMDS, a multi-resolution schemecan improve significantly the convergence speed of our min-imization algorithm. In a multi-resolution approach, an ini-tial solution is found by first computing djoint between coarseversions of X and Y , and subsequently interpolating it tohigher resolution levels. This allows to practically eliminatemany spurious local minima.

5 Results

In order to assess the proposed approach, three experimentswere performed. In the first experiment, we show the com-putation of the set-valued Pareto distance between differ-ent nonrigid shapes. In the second experiment, we evaluatethe discriminative power of the joint similarity criterion andcompare its performance to extrinsic and intrinsic similarity

criteria. In the third experiment, we compare joint similaritywith state-of-the-art shape matching methods.

The shapes in our experiments were taken from the non-rigid objects database available online at tosca.cs.technion.ac.il. The joint similarity computation was implemented inMATLAB. The Dijkstra algorithm was written in C. Nocode optimization was performed.

5.1 Pareto Distance

In the first experiment, we compared three different objects:a man, a woman and a gorilla. As a model, we used theshape of aiming man (shown in red in Fig. 8), sampled at1000 points. Three probes were compared to the model: anear-isometric deformation of the man shape (a man withopen hands), a woman and a gorilla. Each probe was rep-resented by 100 points. Subsampling was performed usingthe farthest point sampling method (Hochbaum and Shmoys1985b, 1985a; Gonzalez 1985). The Pareto frontier was ob-tained by computing the joint distance for thirteen differentvalues of λ. The computation of each point on the Paretofrontier took several minutes.

Figure 7 shows the Pareto distances between the modelshape and the probe objects. The Pareto frontiers clearly in-dicate that the man shape is more similar to its deformedversion than to a woman and even less to the gorilla.

Figure 8 shows the matching obtained as a result of com-putation of the joint distance. Small values of λ give largerweight to intrinsic similarity, which results in the probe (de-picted in blue) remaining almost rigid. Increasing λ, moreweight is given to the extrinsic similarity, which results inthe probe bent to better fit the model. Thus, “traversing” thesolutions on the Pareto frontier, one can obtain a continu-ous morphing between the probe and the model. Note thatunlike many morphing methods used in computer graphicswhich assume compatible meshing of the source and the tar-get shapes, here the two shapes have arbitrary number ofsamples, and arbitrary triangulations.

5.2 Comparison of dH, dGH and djoint

In the second experiment, we compared different distanceson a data set of six shape classes. Each object appeared ina variety of instances, obtained by near-isometric deforma-tions. In addition, for each of the deformations, a versionwith different topology was created by welding the shapesat a set of points (marked by red circles in Fig. 9). In to-tal, the data set contained the following shapes: 5 cats, 9dogs, 13 gorillas, 7 lions, 13 males and 13 females (total of60 shapes; 27 of them with welding).

We compared three shape distances: the Gromov-Haus-dorff distance dGH (representing intrinsic dissimilarity), ex-trinsic dissimilarity computed using an ICP algorithm, and

294 Int J Comput Vis (2009) 81: 281–301

Fig. 7 (Color online) Paretodistances between a manshape and its deformed version(green curve), man and woman(red curve) man and gorilla(blue curve)

Fig. 8 (Color online) Matchinga man shape to different shapes(another version of the man, awoman and a gorilla) producedby joint similarity usingdifferent values of λ. Graduallyincreasing the value of λ resultsin a morphing effect

the proposed joint distance djoint. For the approximationof the Gromov-Hausdorff distance, we used GMDS. The

distance was computed using 100 points on the embeddedshape and 1000 points on the shape into which embedding

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Fig. 9 (Color online) The set ofobjects used in the secondexperiment. Topological noisewas modeled by welding themeshes at points indicated byred circles

was performed. For the ICP method, meshes with 1000 sam-ples were used. For the computation of the joint distance,the model and the probe were represented by 1000 and 100points, respectively. The typical computation time of the ex-trinsic, intrinsic and joint similarity between a pair of shapeswas a few seconds, 30–60 seconds, and a few minutes, re-spectively.

The recognition accuracy was assessed both qualitativelyand quantitatively. The first assessment consisted of present-

ing the shapes as points in the Euclidean space, with theEuclidean distance representing the distances between theshapes (Figs. 10–13). Such plots are straightforwardly ob-tained using MDS and allow to visually represent the ap-proximate similarity relations between the shapes.

The second, quantitative assessment consisted of com-puting the receiver operating characteristic (ROC) curvesfor each similarity criterion, representing a tradeoff betweenthe false acceptance rate (FAR) and the false rejection rate

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Fig. 10 (Color online)Visualization of the intrinsicsimilarity (Gromov-Hausdorffdistance) between nonrigidshapes. No topological noise ispresent

Fig. 11 (Color online)Visualization of the intrinsicsimilarity (Gromov-Hausdorffdistance) between nonrigidshapes. Shapes with differenttopology created by welding aremarked by hollow circles

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Fig. 12 (Color online)Visualization of the extrinsicsimilarity (computed using ICP)between nonrigid shapes.Shapes with different topologycreated by welding are markedby hollow circles

Fig. 13 (Color online)Visualization of the jointintrinsic-extrinsic similaritycomputed using the proposedmethod. Shapes with differenttopology created by welding aremarked by hollow circles

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(FRR). Each ROC curve was computed as follows: thematrix of distances between different shapes was thresh-olded by a value ranging from zero to the maximum dis-tance value. Shapes with distances falling below the thresh-old were regarded similar (i.e., instances of the same ob-ject); those with distances above the threshold were regardeddissimilar (different objects). The FAR was computed asthe percentage of dissimilar shapes wrongfully identified assimilar. The FRR was computed as the percentage of similarshapes wrongfully identified as dissimilar. For small valuesof the threshold, the FAR is small (two shapes must havea very small distance in order to be considered similar),while the FAR is large. For large values of the threshold,the FAR is large and the FRR is small. Ideally, both shouldbe as small as possible, meaning that the recognition is ac-curate. A single number capturing the recognition error wascomputed as the point at which the values of FAR and FRRcoincide (referred to as equal error rate or EER).

Figure 10 visualizes the Gromov-Hausdorff distance be-tween the shapes in the absence of topological noise (us-ing a subset of the database of 33 shapes without weld-ing). The Gromov-Hausdorff distance appears insensitive todeformations, which is seen as tight clusters in the figure,and allows to distinguish between different objects almostperfectly (with EER of 1.14%, see Fig. 14). This idealisticpicture changes dramatically when shapes with topologicalnoise are added. Figure 11 shows that shapes with weldedpoints (represented as hollow circles in the figure) are sig-nificantly different from the original ones. This is confirmedby a significant drop in the recognition rate (EER of 7.68%).

Figure 12 visualizes the ICP distance between the shapes.While the extrinsic distance is insensitive to topologicalnoise (which is clearly seen from the hollow circles, rep-resenting shapes with welding, coinciding with points, rep-resenting shapes without welding), it is sensitive to nonrigiddeformations. Overall, the recognition rate is poor (EER of10.34%).

Finally, Fig. 13 visualizes the joint distance. The jointsimilarity criterion combines the advantages of the two dis-tances, and is insensitive to both isometries and topologicalchanges. It significantly outperforms the extrinsic and intrin-sic distances used separately, achieving an EER of 1.61%.

5.3 Comparison to Other Methods

In the third experiment, we compared the joint similar-ity criterion with two state-of-the-art methods: shape DNA(Reuter et al. 2006; Lévy 2006; Rustamov 2007) and D2shape distribution (Osada et al. 2002). The shape DNAmethod is based on comparison of the Laplace-Beltramispectrum of the shapes. The discrete Laplace-Beltrami op-erator was computed on shapes subsampled at 500 points.We used the approximation described in Rustamov (2007).

Fig. 14 (Color online) ROC curves describing the recognition powerof intrinsic, extrinsic and joint similarities in the second experiment

Twenty largest eigenvalues were used as shape descriptors.The descriptors were compared using Euclidean norm. Inthe D2 shape distribution method, the shape is described bya histogram of Euclidean distances between uniform sam-ples. We used uniform subsampling based on farthest pointsampling. We used histograms with 125 bins. Comparisonof histograms was performed using the Earth Moving dis-tance (EMD).

Figure 15 shows the performance of these method onthe same database used in the second experiment. The EERof the shape DNA and the shape distribution methods is8.02% and 11.8%, respectively. One can see that the pro-posed joint similarity significantly outperforms the shapedescriptor methods achieving the EER of 1.61%.

The disadvantage of our method is the computationalcomplexity and comparison efficiency: each comparison re-quires solving a complicated optimization problem (in cur-rent implementation, each comparison takes a few minutes).For comparison, the computation of the Laplace-Beltramispectrum took 4.6 sec in our implementation and the com-parison of two spectra using the Euclidean norm was neg-ligible. The computation of the D2 shape distribution took4.2 sec and the computation of the EMD between two dis-tribution was below one second.

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Fig. 15 (Color online) ROC curves describing the recognition powerof joint similarity, shape DNA and D2 shape distribution methods inthe third experiment

6 Conclusions

We presented a new approach for the computation of non-rigid shape similarity as a tradeoff between extrinsic andintrinsic similarity criteria. Our approach can be illustra-tively presented as deforming one shape in order to make itthe most similar to another from an extrinsic point of view,while trying to preserve as much as possible its intrinsicgeometry. The joint intrinsic and extrinsic similarity appearsto be advantageous over traditional purely extrinsic or intrin-sic similarity criteria. While extrinsic similarity is sensitiveto strong nonrigid deformations and intrinsic similarity issensitive to topology changes, our joint similarity criterionallows to gracefully handle both geometric and topologicaldeformations. Experimental results prove that it can be usedin situations where intrinsic and extrinsic similarities fail.

The numerical framework presented in this paper ex-tends beyond shape similarity problems. As a byproduct ofour similarity computation, we obtain nonrigid alignmentor correspondence of two shapes. This potentially allows toemploy the proposed framework for morphing problems incomputer graphics in a way similar to Eckstein et al. (2007),Kilian et al. (2007). Additional potential applications are in-verse problems arising in shape reconstruction (Anguelov etal. 2005; Salzmann et al. 2007). Our intrinsic and extrinsic

distances can be used as priors for regularization of suchproblems.

Acknowledgements This research was supported by the Israel Sci-ence Foundation grant No. 623/08 and by the United States-Israel Bi-national Science Foundation grant No. 2004274.

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