WESD - Weighted Spectral Distance for Measuring Shape Dissimilarity

Transcription

1 1 WESD - Weighte Spectral Distance for Measuring Shape Dissimilarity Ener Konukoglu, Ben Glocker, Antonio Criminisi an Kilian M. Pohl Abstract This article presents a new istance for measuring shape issimilarity between objects. Recent publications introuce the use of eigenvalues of the Laplace operator as compact shape escriptors. Here, we revisit the eigenvalues to efine a proper istance, calle Weighte Spectral Distance (WESD), for quantifying shape issimilarity. The efinition of WESD is erive through analysing the heat-trace. This analysis provies the propose istance an intuitive meaning an mathematically links it to the intrinsic geometry of objects. We analyse the resulting istance efinition, present an prove its important theoretical properties. Some of these properties inclue: i) WESD is efine over the entire sequence of eigenvalues yet it is guarantee to converge, ii) it is a pseuometric, iii) it is accurately approximate with a finite number of eigenvalues, an iv) it can be mappe to the [0, 1) interval. Lastly, experiments conucte on synthetic an real objects are presente. These experiments highlight the practical benefits of WESD for applications in vision an meical image analysis. Inex Terms Shape Distance, Spectral Distance, Laplace Operator, Laplace Spectrum, Segmentations, Label Maps, Meical Images I. INTRODUCTION Quantifying shape ifferences between objects is an important task for various areas in computer science, meical imaging an engineering. In manufacturing, for example, one may wish to characterize the ifference in shape of two fabricate tools. In raiology, a octor frequently iagnoses a isease base on anatomical an pathological shape changes over time. In computer vision, iscriminative shape moels are use for automate object recognition, [1], [2]. In orer to efine measurements of shape issimilarity, scientists rely on escriptors of objects that capture information on their geometry [1]. These escriptors can be in the form of parametrize moels (e.g. point clous, surface patches, space curves, meial axis transforms) or in the form of geometric properties (e.g. volume, surface area to volume ratio, curvature maps). Once a escriptor is formulate the istance between two shapes can be efine as the ifference between the associate escriptors. The exact efinition of the istance however, is a critical issue. In orer to efine an intuitive an theoretically soun istance, one shoul ensure that it takes Corresponing author: E. Konukoglu ( ener.konukoglu@gmail.com) is with Athinoula A. Martinos Center for Biomeical Imaging, Massachusetts General Hospital an Harvar Meical School, MA 02129, USA. B. Glocker an A. Criminisi are with Microsoft Research Cambrige, CB3 0FB, UK. K.M. Pohl is with University of Pennsylvania, PA 19104, USA. Authors woul like to thank the support in part by Grant Number UL1RR an by the Institute for Translational Meicine an Therapeutics (ITMAT) Transisciplinary Program. into account the nature of the escriptor. For instance, the escriptor might be an infinite sequence of positive values, in which case we shoul be careful not to efine a istance that iverges for every non-ientical pair of shapes. Shape escriptors base on the eigensystems of Laplace an Laplace-Beltrami operators, calle spectral signatures, have recently gaine popularity in computational shape analysis [3], [4], [5], [6], [7], [8]. These escriptors leverage the fact that the eigenvalues an the eigenfunctions of Laplace operators contain information on the intrinsic geometry of objects [9], [10], [11]. A visual analogy useful for an intuitive unerstaning is to think of an object (e.g. in 2D) as the membrane of a rum. In this case the eigenvalues correspon to the funamental frequencies of vibration of the membrane uring percussion, an the eigenfunctions correspon to its funamental patterns of vibration. Both the eigenvalues an the eigenfunctions epen on the shape of the rum hea an thus can be use as shape escriptors for the object. Despite recent progress by [3], [4], [5], [6], [7], [8], esigning meaningful shape istances base on spectral signatures remains challenging. Difficulties arise from the nature of the eigensystems. The eigenfunctions of a shape mostly provie localize information on the geometry of small neighborhoos. Aggregating such local information into an overall shape issimilarity measure is non-trivial. On the other han, the eigenvalues provie information about the overall shape, so they are ieal for efining global istances. However, they form a iverging sequence making it ifficult to efine a theoretically soun metric. Here, we tackle this latter problem an propose a new shape istance base on the eigenvalues, which is technically soun, intuitive an practically useful. In the remainer of this section, we first review in further etail the literature on spectral signatures an shape istances relate to eigenfunctions an eigenvalues. Then, we provie a brief overview of our new shape istance. A. Eigenfunctions The eigenfunctions of an object constitute an infinite set of functions. Each function epens on the shape of the object an is ifferent than the rest of the set. Figure 1 illustrates this for two example objects where a few eigenfunctions are shown. The values these functions attain at each point capture the local geometry aroun the point, i.e. of its neighborhoo. Inspire from this geometric information, methos efine local shape signatures [4], [5], [6], [12] for each point on an object by evaluating a subset of eigenfunctions at that specific location. Global shape istances are then efine using such local

2 2 Fig. 1: Starfish an tarantula. The objects represente as binary maps are shown on the left, followe by the 1st, 2n, 5th, 20th, an 100th eigenfunction. The values increase from blue (negative) to re (positive) with green being zero. signatures. Such istance efinitions rely on corresponences. These corresponences shoul hol both in terms of points an the subset of eigenfunctions use in the signatures, a conition har to satisfy in practice [12]. Explicitly searching for such corresponences leas to expensive algorithms [12], [13], [14], [15], [16]. On the other han, computing istances between istributions of local signatures obtaine by aggregating all the points, as in [5], [6], [17], might implicitly construct false corresponences. In summary, efining a global istance base on local signatures is not an easy task. Instea of extracting local information from an eigenfunction, one can also think of capturing its global pattern by looking at regions where its values are all positive or all negative. Such regions are calle noal omains. Different eigenfunctions inuce ifferent patterns an, in turn, have ifferent number of noal omains, calle noal counts [11]. For a given object, the orere sequence of noal counts contain information on its overall geometry [18], [19]. Inspire by these observations, authors in [20] use this sequence as a global shape signature. They further efine the associate shape istance between two objects as the Eucliean norm of the vector ifference between their noal count sequences. However, it is not intuitively clear what the noal counts represent. Furthermore, the entire sequence is iverging so that, in practice, one first chooses a finite subset an then computes the istance for that subset. These ifficulties make it har to efine an intuitive an soun shape istance base on noal counts. B. Eigenvalues Signatures base on eigenvalues, on the other han, have a clearer geometric interpretation. The set of eigenvalues contains information on the overall geometry of the object. Specifically, the orere sequence is analytically relate to the intrinsic geometry by the heat-trace, [21], [22], [23], [24], [25]. Hence, more intuitive istances can be constructe using the eigenvalues. However, similar to the sequence of noal counts, the eigenvalue sequence is also ivergent. This makes the istance efinition theoretically challenging. Inspire by the sequence s link to the geometry, Reuter et al. in [3], use the smallest N eigenvalues as a shape signature, calle shape-dna. As the associate shape istance, the authors propose the Eucliean norm of the vector ifference between the shape-dnas of objects. Although this is a very goo first attempt, the ivergent nature of eigenvalue sequence results in important theoretical limitations for this istance, as also pointe out in [14]. The main problems are i) efining a istance on the entire sequence oes not yiel a proper metric, ii) the ifferences between the higher components of two sequences ominate the final istance value, even though these components o not necessarily provie more information on the geometry, an iii) the istance value is sensitive to the choice of the signature size N. These theoretical problems also cause practical rawbacks as we emonstrate later. This article proposes a new shape istance, calle Weighte Spectral Distance (WESD), using the sequence of eigenvalues of the Laplace operator. We erive WESD from the functional relationship between the eigenvalues an the geometric invariants as given by the heat-trace. This erivation provies WESD a clear geometric intuition as a shape istance. It also links WESD to the istance efine by Reuter et al. in [3] as well as to the local signature efine in [5]. The resulting formulation of WESD iffers from other previously propose scores base on eigenvalues, whether in shape analysis or other fiels [26], both in its formulation an in the fact that it is efine over the entire sequence. As we will show later, the latter point alleviates the critical importance of the choice of the signature. We furthermore analyse an prove theoretical properties of WESD showing that it oes not share some of the funamental problems the shape istance propose in [3] has. Specifically, we prove that WESD: i) converges espite the fact that it is efine over the entire eigenvalue sequence, ii) can be mappe to the [0, 1) interval, iii) is accurately approximate with a finite number of eigenvalues an the truncation error has an analytical upper boun an iv) is a pseuometric. These theoretical properties also yiel important practical avantages such as being less sensitive to the signature size (truncation parameter) N, proviing a principle way of choosing this parameter, proviing more stable low-imensional shape embeing an simplicity in combining with other istances as WESD can be normalise. Applying to synthetic an real objects, we further emonstrate the benefits of WESD in comparison to the other eigenvaluebase istance efine in [3]. The remainer of this article is structure as follows. Section II presents a brief overview of the Laplace operator, the eigenvalue sequence an its role in shape analysis. In Section III we efine WESD an erive its theoretical properties. Section IV presents an extensive set of experimental analysis on 2D objects extracte from synthetic binary maps, shape-base retrieval results for 3D objects using the SHREC ataset [27], low imensional embeings of real 3D ata such as subcortical structures in brain scans an 4D analysis of binary maps extracte from cariac images. II. SPECTRUM OF LAPLACE OPERATOR This section provies a brief backgroun on the Laplace operator, its eigenvalue sequence, calle spectrum, an its role in shape analysis. We first relate an object s intrinsic geometry to the spectrum of the corresponing Laplace operator. We then provie some etails on the previously propose shape-

3 3 DNA [3] an iscuss the associate issues. For further etails we refer the reaer to [11], [24], [25] an [3]. We enote an object as a close boune omain Ω R with piecewise smooth bounaries. In the case of binary maps, Ω woul correspon to the foregroun representing the object. For a given Ω, the Laplace operator on this object is efine with respect to a twice ifferentiable real-value function f as Ω f 2 x 2 i=1 i f, x Ω where x = [x 1,..., x ] is the spatial coorinate. The eigenvalues an the eigenfunctions of Ω are efine as the solutions of the Helmholtz equation with Dirichlet type bounary conitions 1, [11], f + λf = 0 x Ω, f(x) = 0, x Ω, where Ω enotes the bounary of the object an λ R is a scalar. There are infinitely many pairs {(λ n, f n )} satisfying this equation an they form the set of eigenvalues an eigenfunctions respectively. The orere set of eigenvalues is a positive iverging sequence such that 0 < λ 1 λ 2 λ n.... This infinite sequence is calle the Dirichlet spectrum of Ω, which we simply refer as the spectrum. In aition, each component of the spectrum is calle a moe, e.g. λ n is the calle n th moe of the spectrum The spectrum contains information on the intrinsic geometry of objects. Weyl in [9] showe the first spectrum-geometry link by proving that the asymptotic behavior of the eigenvalues is given as ( ) 2/ n λ n 4π 2, n, B V Ω where V Ω is the volume of Ω an B is the volume of the unit ball in R. Later works, as [21], [22], [23], [24], extene this result by stuying the properties of the Green s function of the Laplace operator, an showe that a more accurate spectrumgeometry link is given by the heat-trace, which in R is given as Z(t) e λnt = a s/2 t /2+s/2, t > 0. (1) s=0 The coefficients of the polynomial expansion, a s/2, are the components carrying the geometric information. These coefficients are given as sums of volume an bounary integrals of some local invariants of the shape, [22], [23], [25]. For instance, as given in [22], the first three coefficients are: 1 a 0 = (4π) V /2 Ω 1 a 1/2 = 4(4π) S Ω, /2 1/2 1 a 1 = κ Ω, 6(4π) /2 where S Ω is the surface area (circumference in 2D) an κ is the mean (geoesic) curvature on the bounary of Ω. 1 Other bounary conitions yiel ifferent eigensystems. Here we are only intereste in the Dirichlet type. Please refer to [11] for the other types. Ω The functional relationship between the eigenvalue sequence an the coefficients a s/2 can be seen in Equation (1). This connection relates the spectrum to the intrinsic geometry, which is the reason why Laplace spectrum is important for the computational stuy of shapes. In aition to the spectrum-geometry link, the eigenvalues of the Laplace operator have two other properties which make them useful for shape analysis, [11]. These are: 1) the Laplace operator is invariant to isometric transformations an 2) the spectrum epens continuously on the eformations applie to the bounary of the object. The avantage of the first property is obvious since isometric transformations o not alter the shape. In aition to this, the secon property states that there is a continuous link between the ifferences in eigenvalues an the ifference in shape, which makes eigenvalues ieal for measuring shape ifferences. Unfortunately, it has also been shown that there exists isospectral non-congruent objects, i.e. objects with ifferent shape but the same spectrum [28]. Therefore, theoretically the Laplace spectrum oes not uniquely ientify shapes. However, as state in [3], practically this oes not cause a problem mostly because the constructe isospectral noncongruent objects in 2D an 3D are rather extreme examples with nonsmooth bounaries. The spectral signature, shape-dna, propose in [3] is inspire from the properties given above. For a given shape Ω, its shape-dna is the first N moes of the spectrum of the Laplace operator efine on Ω: [λ 1, λ 2,..., λ N ]. In aition to the properties the shape-dna inherits from the eigenvalues, the authors also propose several normalisations to obtain almost scale invariance 2. The normalisations use in the experiments in [3], [7], [27], [29] are given as λ n λ n V 2/ Ω an λ n λ n /λ 1. In [3], the authors also efine a shape istance base on shape-dna. Either using the original or its scale invariant version, this istance is given as [ N ] 1/2 ρ N SD(Ω λ, Ω ξ ) (λ n ξ n ) 2, (2) where Ω ξ enotes the object with the spectrum {ξ n }. Using ρ N SD (Ω λ, Ω ξ ), the authors were able to istinguish between istinct shapes [27], construct shape manifols base on the pairwise istances an perform statistical comparisons [7], [29]. However, as also pointe out in [14], ue to the iverging nature of the spectrum, ρ N SD suffers from three essential rawbacks limiting its usability: i) ifferences at higher moes of the spectrum have higher impacts on the final istance value even though they are not necessarily more informative about the intrinsic geometry, ii) the istance is extremely sensitive to the signature size N, while the choice of this parameter is arbitrary, an iii) the istance cannot be efine over the entire spectrum because it oes not yiel a proper metric in 2 We use the term almost because scale invariance is an application epenent concept an the efinition of scale ifference between arbitrary objects is a mathematically vague notion. A further iscussion of scale invariance is outsie the scope of this article an we refer the reaer to [3].

4 4 that case. Therefore, efining a soun an intuitive istance base on the spectrum is still an open question for which we propose a solution in the next section. III. WEIGHTED SPECTRAL DISTANCE - WESD This section presents the propose spectral istance, WESD, the analysis of the heat-trace leaing to its efinition an its theoretical properties. The structure of presentation aims to separate the efinition of the istance, which is essential for its practical implementation, from the etails relate to its erivation an theoretical properties. In this light, we first present the efinitions an mention the associate properties with appropriate references to the following subsections, which contain further etails. We efine the Weighte Spectral Distance - WESD - for two close boune omains with piecewise smooth bounaries, Ω λ, Ω ξ R as [ ( ) ] p 1/p λn ξ n ρ(ω λ, Ω ξ ), (3) λ n ξ n with p R an p > /2. Unlike the istance given in Equation (2), WESD is efine over the entire eigenvalue sequence an the factor p is not fixe to 2. In aition, the ifference at each moe contributes to the overall istance proportional to λ n ξ n /λ n ξ n instea of λ n ξ n. The aitional λ n ξ n factor (seeming like a simple aition to Equation 2) actually arises from analysing the relation between the n th moe of the spectrum an the heat-trace, which will be presente in Section III-A. This analysis also provies WESD with a geometric intuition. Furthermore, for p > /2 the infinite sum in the efinition is guarantee to converge to a finite value for any pair of shapes. Hence, WESD exists. In aition to its existence, WESD also satisfies the triangular inequality making it a pseuometric. These points are proven in Section III-B. Moreover, the pseuometric WESD has a multi-scale aspect with respect to p. In Section III-C we show that ajusting p controls the sensitivity of WESD with respect shape ifferences at finer scales, i.e. with respect to geometric ifferences at local level such as thin protrusions or small bumps. Thus, for higher values of p the istance becomes less sensitive to finer scale ifferences. In aition to WESD, we efine the normalise score for shape issimilarity nwesd as ρ(ω λ, Ω ξ ) ρ(ω λ, Ω ξ ) [0, 1), (4) W(Ω λ, Ω ξ ) which maps ρ(ω λ, Ω ξ ) to the [0, 1) interval using the shapeepenent normalisation factor W(Ω λ, Ω ξ ) { C + K [ ζ ( ) 2p 1 ( 1 2 ) 2p ]} 1 p. The factors C an K are the shape base coefficients efine in Corollary 1, an ζ( ) is the Riemann zeta function [30]. Being confine to [0, 1), nwesd allows us to i) compare issimilarities of ifferent pairs of shapes an ii) easily use the shape issimilarity in combination with scores quantifying other type of ifferences between objects such as volume overlap in case of matching or Jacar s inex in case of accuracy assessment. One important issue in efining a istance or a score using the entire eigenvalue sequence is computational limits. In practice we can only compute a finite number of eigenvalues an therefore, can only approximate such istances. Consiering this, here we efine the finite approximations of WESD an nwesd using the smallest N eigenvalues as [ N ( ) ] 1/p p ρ N λn ξ n (Ω λ, Ω ξ ) (5) λ n ξ n ρ N (Ω λ, Ω ξ ) ρn (Ω λ, Ω ξ ) [0, 1), (6) W(Ω λ, Ω ξ ) where N is a truncation parameter. Previous works, such as [3], [5], [17], [20], [26], also efine istances base on finite number of moes. However, their view on the istance efinition was first to construct finite shape signatures an then to efine a istance on the signatures. Therefore, the signature size was a critical component of the efinition itself. Furthermore, the effects of the choice of the signature size on the istance values have not been carefully analyse in these works. The view presente here efines the istance irectly using the entire sequence without constructing a finite signature. This alleviates the importance of the signature size on the istance. The finite computation given in Equations 5 an 6 are viewe as approximations to the istance an N as the truncation parameter. In this conceptually ifferent setting, unlike previous works, we provie in Section III-D a careful analysis of the choice of N on the spectral istance. Specifically, we prove that lim N ρ(ω λ, Ω ξ ) ρ N (Ω λ, Ω ξ ) = 0 an lim N ρ(ω λ, Ω ξ ) ρ N (Ω λ, Ω ξ ) = 0. Furthermore, we provie a theoretical upper boun for these errors that shows how fast they ecrease in the worst case leaing to a principle strategy for choosing N. Section III-E ens the section by focusing on the invariance of WESD an nwesd to global scale (relative size) ifferences between objects. Specifically, we iscuss how an approximate scale invariance can be attaine for WESD an nwesd by following the same strategy propose in [3]. A. Analysis of the Heat-Trace an Derivation of WESD We erive WESD by analysing the mathematical link between the spectrum of an object an its geometry. This link is given by the heat-trace efine in Equation (1). Let us consier the heat-trace as a function of both t an the spectrum, Z(t, λ 1, λ 2,...). The main question we answer is how much the Z(t, ) function changes when we change the n th moe of the spectrum from λ n to ξ n. Consiering the polynomial expansion equivalent to Z(t, ) given in Equation (1), one can see that the change in the value Z(t, ) is irectly relate to the changes in the coefficients a s/2 an so to changes in the integrals over the local invariants. By analysing the influence of the change in the n th moe on Z(t, ), we actually analyse the influence of this change on the integrals over local geometric invariants. Following this line of thought, we

5 5 quantify the influence of the change from λ n to ξ n on Z(t, ) in terms of λ n an ξ n. This can be one by efining n Z 0 Z(t,..., λ n 1, λ n, λ n+1,... ) Z(t,..., λ n 1, ξ n, λ n+1,... ) t, which is simply the L 1 -norm of the ifference between the functions that is linke to the ifference between λ n an ξ n. Replacing Z(t, ) with its efinition leas to n Z = 0 e λnt e ξnt t (7) Without loss of generality let us assume ξ n λ n. Then e λnt e ξnt for t > 0. We can then evaluate the integral in Equation (7) as n Z = 0 e λnt e ξnt t = λ n ξ n λ n ξ n. n Z captures the influence of the ifference at the nth moe on Z(t, ). Now, aggregating these influences across all moes leas to the efinition of WESD [ ] 1/p [ ( ) ] p 1/p ρ(ω λ, Ω ξ ) = ( n Z) p λn ξ n =. λ n ξ n Surprisingly, the formulation of WESD, which results from the analysis presente above, also has very beneficial properties that makes it theoretically soun an useful in practical applications. These properties will be analyse in the following. Before elving into this analysis though let us make two remarks. The first relates ρ SD (, ) (Equation (2)) to the analysis of the heat-trace presente above. Remark 1. Let us efine n,m Z m 0 t m Z (t,..., λ n 1, λ n, λ n+1,...) m t m Z (t,..., λ n 1, ξ n, λ n+1,...) t. One can see that n,0 Z = n Z. Evaluating this integral yiels n,m Z = λ m 1 n ξn m 1. By setting m = 2 ρsd (, ) can be erive as follows [ ] ( ) 1/2 2 ρ SD (Ω λ, Ω ξ ) =. n=0 n,2 Z This relationship not only relates WESD to ρ SD (, ) but also provies the link between ρ SD (, ) an the heat-trace. One notices that the functional ifference efinition use in this remark iffers from the previous one use to erive WESD, see Equation 7. This is because ρ SD cannot be erive from the L1-istance efinition use previously but can be erive from the less ieal efinition use in this remark. The existence of an alternative erivation of ρ SD that woul start from an appropriate functional ifference is an open question. The secon remark notes the link between WESD an Global Point Signatures (GPS), a local shape escriptor, presente in [5]. Remark 2. GPS, as presente in [5], is efine for each point in an object { Ω λ as } the infinite series GPS Ωλ (x) {Φ λ,n (x)} f n (x), where x Ω λ an f n (x) is λ 1/2 n the n th eigenfunction. GPS has a connection to WESD arising from the following element-wise integrals [ ] 2 Φ 2 λ,n(x)x = λ 1/2 n f n (x) x = λ 1 n, Ω λ Ω λ where the equality arises from the fact that eigenfunctions form an orthonormal basis in Ω λ [11], i.e. Ω λ f n (x)f m (x)x = δ(n m) with δ( ) being the Dirac s elta. Consiering this integral, WESD can also be regare as a istance between GPS of two objects as { [ ] p } 1 p Φ 2 λ,n(x)x Φ 2 ξ,n(x)x = ρ(ω λ, Ω ξ ). Ω λ Ω ξ This link also provies an alternative view on the normalisation factor λ 1/2 n use in GPS. In [5] author justifies this normalisation factor by noting that for an object the Green s function can be written as an inner prouct in the GPS omain, see Section 4 in [5]. This is later use to argue the geometric meaning of GPS as authors point out the use of Green s function in ifferent shape processing tasks. Our link between GPS an WESD provies an alternative view on the normalisation factor as it connects this local signature to the heat-trace Z(t). B. Existence of the Pseuometric WESD WESD is efine as the limit of an infinite series as given in Equation (3). For such a istance to be a proper one, actually a pseuometric in this case, the limit of the infinite series shoul exist for any two spectra. In the case of WESD, this is not evient because it is efine over the entire spectra an each spectrum is a ivergent sequence. The first corollary presente below proves that when p > /2 WESD inee satisfies this conition, i.e. the infinite series converges. The corollary further provies an upper boun for this limit, which is use to construct nwesd. We woul like to note that for the ease of presentation, the proofs for all the following corollaries an lemmas are given in Appenix B in the supplemental material. Corollary 1. Let Ω λ R an Ω ξ R be any two close omains with piecewise smooth bounaries an {λ} an {ξ} be their Laplace spectrum. Then the weighte spectral istance [ ( ) ] p 1/p λn ξ n ρ(ω λ, Ω ξ ) = λ n ξ n converges for p > 2. Furthermore, { [ ρ(ω λ, Ω ξ ) < C + K ζ ( ) 2p 1 ( 1 2 ) 2p ]} 1 p, (8)

6 6 where ζ( ) is the Riemann zeta function an the coefficients C an K are given as C ( ) 2 p + 2 4π 2 B ˆV 1 ( ) i 1 i µ + 4 i=1,2 [ + 2 ( ) 2 K 4π 2 B ˆV 1 ] p µ ˆV max(v (Ω λ ), V (Ω ξ )), µ max(λ 1, ξ 1 ), where V ( ) enotes the volume (or area in 2D) of an object. The Inequality (8) states that WESD has a shape-epenent upper boun. We thus can map the WESD to the [0, 1) interval through normalising it with this upper boun. The nwesd score, given in Equation 4 is constructe base on this strategy. Since its existence is establishe next we prove that WESD is a pseuometric, i.e. satisfies the other criteria to be a pseuometric, such as the triangle inequality. Corollary 2. ρ(ω λ, Ω ξ ) is a pseuometric for 2. We note that WESD is not a metric because the spectrum is invariant to isometries, which is a esirable property for shape analysis. However, in aition to this, the spectrum is also invariant to isospectral non-congruent shapes. This is not esirable but oes not cause problems in practice as iscusse in Section II an also confirme in our experiments. C. On the multi-scale aspect of WESD The previous section highlighte the role of p on the convergence properties of WESD an therefore on its existence. We now emonstrate that p also provies WESD a multi-scale characteristic. The sensitivity of WESD to the shape ifferences at finer scales epens on the value of p. Specifically, we show that the higher the value p the less sensitive WESD is to finer scale etails an its sensitivity increases as p gets lower. The multi-scale aspect of WESD arises from the relationship between the Laplace operators an heat iffusion processes [31]. We first present an intuitive summary of this relationship, which is about the multi-scale aspect of Z(t) an t in particular. For a more mathematical treatment we refer the reaer to [6]. As state in [6] an [14], t can be interprete as the time variable in a heat iffusion process within an object. A useful visual analogy to consier here is the Laplacian smoothing of a surface where t woul correspon to the amount of smoothing. Similar to the surface smoothing, as t increases, the local geometric etails of an object, such as sharp riges or steep valleys, lose further their influence on the Z(t) value. As a result Z(t) becomes somewhat insensitive to these local geometric etails, i.e. shape etails at finer scales. From an alternative view, the value of Z(t) loses its information content with regars to local geometric etails. This effect intuitively summarizes the multi-scale characteristic of the heat-trace with respect to t. Having explaine the multi-scale aspect of Z(t), we now analyse how this aspect is reflecte upon the eigenvalues. To o so let us efine the influence ratio D(n, t) e λnt Z(t). This ratio captures the influence of the n th moe on the heat-trace. In other wors, the higher the ratio, the higher the influence of λ n on the value of Z(t) at that specific t. The following lemma compares the influence ratios of ifferent moes an how this comparison epens on t. Lemma 1. Let Ω λ R represent an object with piecewise smooth bounary an D(l, t) e λ l t Z(t) be the corresponing influence ratio of moe l at t. Then for any two spectral inices m > n > 0 D(n, t) > D(m, t), t > 0 an particularly for two t values such that t 1 > t 2 D(m, t 1 ) D(n, t 1 ) < D(m, t 2) D(n, t 2 ). The first inequality of the lemma inicates that the lower moes in the spectrum have more influence on the value of Z(t) than the higher moes. The secon inequality shows that the influence of the higher moes become more prominent as t ecreases. Consiering that for lower t values Z(t) is more informative with regars to shape etails at finer scales, Lemma 1 suggests that the higher moes are more important for finer scales than for coarser scales. We illustrate this observation on a synthetic example shown in Figure 2 with the pair (a) + (b) being an example showing coarser shape ifferences an the pair (a) + (c) showing finer ifferences. The plots given in Figure 2 () an (e) show the corresponing spectral ifferences observe at moes between 1 an 150. Between (a) an (b) the shape ifferences are at the coarse level. Accoring to Lemma 1 these ifferences shoul show up at the very first moes. On the other han, between (a) an (c) the ifferences are at a finer scale an furthermore the objects are very similar at the coarse level. Lemma 1 states that these ifferences therefore, shoul show up at higher moes an the ifferences at the lower moes shoul be low. Satisfying these expectations, the ifferences at the first few moes shown in plot () have relatively large values compare to the ones in plot (e). Furthermore, the amplitue of the ifferences at higher moes are generally larger in plot (e) than in plot(), especially after 100. In orer now to connect these finings to WESD an p let us present the following corollary, which stuies the influence of p on the components insie the infinite sum efining the istance. Corollary 3. Let Ω λ an Ω ξ be two objects with piecewise smooth bounaries. Then for any two scalars with with p > /2, q > /2, p q an for all n with λ n ξ n > 0 there exists a M > n so that m M ( ) p λm ξ m λ mξ m ( λn ξ n ) p λ nξ n ( ) q λm ξ m λ mξ m ( ) q λn ξ n λ nξ n Thus, the relative contributions of the higher spectral moes on ρ(ω λ, Ω ξ ) with respect to the contributions of the lower moes epen on the value of p. Specifically, the higher spectral moes become more influential as p ecreases. Combining this fining with the result of Lemma 1, we follow that as p

7 7 () between (a) an (b) (a) (b) (c) (e) between (a) an (c) Fig. 2: Multi-scale characteristics of ifferent spectral moes:(a), (b) an (c) show three synthetic shapes. In () we plot the absolute ifferences between the corresponing moes of (a) an (b) with respect to the spectral inex. In (e) we plot the same ifference for the shapes in (a) an (c). The shape ifference between (a) an (b), which is at a coarser level, is alreay capture at the lower spectral moes. The ifference between (a) an (c) results in lower ifferences in lower spectral moes because these objects are more similar at a coarser level. At the higher spectral moes though, the ifference between (a) an (c) becomes more prominent since these objects iffer more substantially at the finer scales. The plots in () an (e) emonstrate that the higher moes for a given object are more important for finer scale shape etails. increases WESD gives less importance to ifferences at higher spectral moes an therefore becomes less sensitive to the shape ifferences at finer scales. This provies WESD with a multi-scale aspect with respect to p an also provies us the intuition for choosing a proper value for p. D. Finite Approximations of WESD an nwesd One of the important practical questions regaring spectral istances is the number of moes to be inclue in the calculation of the istance. The computation of eigenvalues an eigenfunctions can be expensive an inaccurate especially for the higher moes. Therefore, spectral istances require the user to set a finite number of moes to be use. This parameter is often referre to as the signature size. Having efine the istance over the entire sequence, we refer to it as the truncation parameter. This actually provies a ifferent perspective on the number of moes use to compute the istance. In previous works, such as [3], [5], [20], the value of this parameter, viewe as the signature size, is often set arbitrarily an its effect on the istances have not been carefully analyse. Here, viewing it as a truncation parameter, we stuy its influence. Specifically, we formulate the ifference between using the entire spectra to only using a finite number of moes as an approximation/truncation error. So we analyse how this error changes with respect to the truncation parameter. We specifically show in the next corollary that the errors in approximating WESD an nwesd by the first N moes converges to zero as N increases. Furthermore, we provie an upper boun for both errors as a function of N. Corollary 4. Let ρ N (Ω λ, Ω ξ ) be the truncate approximation of ρ(ω λ, Ω ξ ) base on the first N moes an ρ N (Ω λ, Ω ξ ) of ρ(ω λ, Ω ξ ). Then p > /2 an lim N ρ ρn = 0 lim ρ N ρn = 0. Furthermore, for a given N 3 the truncation errors ρ ρ N an ρ ρ N can be boune by ρ ρ N < { C + K [ ζ ( ) 2p 1 { [ N ( 1 C + K n C + K ρ ρ N < 1 [ C + K ζ ( 2p n=3 [ N ( 1 2 ) 2p ]} 1 p n=3 ( 1 n ) 2p ]} 1 p (9) ] ) 2p ) 1 ( 1 2)2p 1 p ] (10) The above corollary has important practical implications. First of all, the sensitivities of ρ N an ρ N with respect to N ecreases as N increases. For any application relying on the shape istances, such as constructing low imensional embeings, this reuce sensitivity is particularly important as it provies stability with respect to N both for the istance an for the application using the istance. We note that the opposite is true for ρ N SD, which is one of the main isavantages of this istance. In aition, Corollary 4 can guie the choice for the number of moes N an the norm type p. Specifically, the error upper bouns given in Equations 9 an 10 provie the worst case errors for a given pair of shapes without the nee to compute the eigenvalues. So for instance, once a number of moes are compute then base on the istance value obtaine so far an the worst case error compute using the upper bouns, one can ecie whether to compute more moes or not. Furthermore. these upper bouns are shape-specific as they epen on C an K. One can go one step further an efine a shape-inepenent resiual ratio for N 3 an p > /2 as R(N, p) 1 N n=3 ( 1 n)2p ζ ( ) ( 2p 1 1 ) 2p 2 1 p. (11) that satisfies R(N, p) > ρ ρ N, for which the proof is given in Proposition 1 in Appenix B. R(N, p) can be use to select the parameters N an p as it quantifies the quality of the approximation for a given pair of (N, p) in terms of the error upper bouns. In Figure 3, we plot R(N, p) versus N for ifferent settings of p an = 2, 3. Besies the obvious point that the error upper boun ecreases for increasing N we also notice that

8 8 Fig. 3: Choosing N: The figures plot the resiual ratio R(N, p) versus N for ifferent p values in 2D (left) an in 3D (right). As expecte the error upper boun rops with increasing N. The rate of ecrease also becomes faster with increasing p. This inverse relation suggests the trae-off between N an the sensitivity of WESD to finer scale shape ifferences since WESD becomes less sensitive as p increases, see Section III-C. i) the behavior in 2D an 3D are similar an ii) the rate of ecrease of the error upper boun is much faster for higher p. Consiering the multi-scale aspect of WESD capture in p, this behavior is interesting. It emonstrates that the choice of p an N are correlate an suggests a trae-off between the rate of ecrease of the truncation error an the sensitivity of WESD to shape ifferences at finer scales. In theory, the choice of these parameters epens on the application an the expecte shape ifferences. If one expects coarse scale ifferences then choosing a large p an small N might be sufficient. However, if one is intereste in finer scale ifferences then a small p value will be require, which in turn will require a large N value to have a ecent approximation. The important aspect of R(N, p) is that it is universal, i.e. it oes not epen on the objects. So it can be use in any type of application to choose the parameter pair N, p an to have a rough estimate of the computational costs for computing the istance WESD. We note once again, the specific values shoul be chosen base on the application an the shapes at han. E. Invariance to global scale ifferences We en this section stuying the impact of global scale ifferences on WESD an how invariance to such ifferences can be attaine. We woul like to note that the notion of global scale in this section refers to the relative size of an object, which is not to be confuse with the notion of multiscale iscusse in Section III-C. The spectrum of an object epens on the object s size, i.e. a global scale change alters all the eigenvalues by a constant multiplicative factor [11]. As a result, the global scale ifference between two objects contributes to the spectral shape istance WESD. In some applications this contribution might not be esirable, for instance in an object recognition task, where objects in the same category have varying sizes. Therefore, it is a useful property of a shape istance to allow invariance to global scale ifferences. Reuter et al. [3] propose ifferent approximations for normalising the effects of scale ifferences on the spectrum. In particular, the authors use two ifferent normalisations in their experiments in [7], [27], [29]. Both normalisations irectly act on the eigenvalues. The first one normalises the eigenvalues with respect to the volume (area in 2D or surface area for Riemannian manifols) an is given as λ n λ n V 2/ Ω λ. The secon one normalises the eigenvalue with respect to the first eigenvalue in the sequence as λ n λ n /λ 1. Both of these strategies can be use when computing istances with WESD. Furthermore, since these strategies o not alter the mathematical characteristics of the entire spectrum the theoretical properties of WESD an nwesd hol either way. For our experiments we aopt the first strategy, volume normalisation, using the volume as efine in Eucliean geometry. When using the volume normalise eigenvalues, the only change that applies to the technical etails presente so far is ˆV in Equation 8 becomes ˆV = 1. The rest applies irectly without any moification. We woul also like to note that estimating the global scale ifference between two arbitrary objects is not always a wellpose problem. It is especially har when the objects are of ifferent category, e.g. an octopus an a submarine. Furthermore, the scale normalisation is application epenent an it might not be esirable for all applications. In Section IV-C2 we present such an example where we analyse the temporal change of the left ventricle shape uring a heart cycle. In this case, the volume change is essential for analysing the heart of the same patient so that scale invariance is not appropriate. IV. EXPERIMENTS This section presents a variety of experiments on synthetic an real ata highlighting the strengths an weaknesses of WESD an nwesd. We start by briefly explaining the etails of the numerical implementation of WESD use in the experiments presente here. Then in Section IV-B, the propose istances are applie to synthetically generate objects emonstrating that (i) Orering objects with respect to their shapes using nwesd results in a visually coherent series (Section IV-B1), (ii) WESD is useful for constructing low imensional embeings, in particular it yiels stable embeings with respect to the signature size N, (Section IV-B2) an (iii) WESD is a suitable istance for shape retrieval, which is shown through experiments on the SHREC ataset [27] (Section IV-B3). Lastly, in Section IV-C WESD is applie to real objects extracte from 3D meical images. We focus on two examples from a wie variety of applications WESD an nwesd can be beneficial to: population stuies of brain structures an analysis of 4D cariac images. A. Implementation Details There are two ifferent aspects in the implementation of WESD: the numerical computation of the Laplace spectra an the parameter settings. First, any numerical metho tailore towars computing the eigenvalues of the Laplace operator can be use to compute WESD. Examples of such metho are liste in [3], [32]. Our specific implementation represents objects simply as binary images with the foregroun efining

9 9 Ω. Using the Cartesian gri of the image, it iscretizes Ω through finite ifference scheme (see also Chapter 2 of [32]). This step yiels a sparse matrix of which we compute the eigenvalues via Arnoli s metho presente in [33] an implemente in MATLAB R. We choose this specific implementation as 1) it is simple 2) it oes not introuce any aitional parameters an 3) when working with images it avois any extra preprocessing steps, such as surface extraction or mesh construction. With regars to the secon implementation aspect, we set the parameters N an p empirically. Base on Section III-D, we set p = 1.5 or p = 2.0 in 2D an p = 2.0 in 3D. These values result in a relatively fast iminishing upper boun of the truncation error with respect to N (see Figure 3) while being sensitive to shape ifferences at finer scales. In both 2D an 3D, we chose N = 200 for the number of moes as the truncation error seeme to vanish at that point. Furthermore, in aition to the theoretical consierations on the effects of N an p on WESD given in Section III-D, in Sections IV-B2 an IV-B3 we experimentally stuy the effects of these parameters on applications using WESD, specifically on constructing low imensional embeings an shape retrieval. B. Synthetic Data We conuct three experiments: first two are on 2D objects an the last one is on 3D objects. For all of the experiments, we use the scale invariant versions of the spectra obtaine by normalising the eigenvalues with the object s volume as escribe in Section III-E. As a result the istances WESD an nwesd become almost invariant to global scale ifferences. 1) Orering of Shapes: For the first experiment we create two synthetic atasets. Each ataset consists of a reference object an ranom eformations of this reference. These eforme versions are generate by transforming the reference via ranom eformations of varying magnitue an amount of nonlinearity. As a result the atasets contain objects that are very similar to the reference ones an objects that are substantially ifferent. Figures 4(a) an (b) show some examples from these atasets where the binary images to the very left show the reference objects. In the first ataset, the reference object is a isc. There are a total of 500 ranom eformations of this reference isc. The first 400 are generate via non-linear eformations while the last 100 are isometric transformations. In the secon ataset, the reference is a slightly more complicate object (see Figure 4(b)) an in total there are 400 ranom transformations of this reference. The first 300 are generate by non-linear eformations an the last 100 prouce via isometric transformations. All objects are iscretize as binary maps with a size of pixels. The numerical computations are performe on these image gris as iscusse earlier. We compute the nwesd scores (ρ N with p = 1.5 an N = 200) between the reference an the eforme objects in each ataset. Base on these scores, we then orere the eforme objects accoring to their similarity in shape to the reference. Figures 4 (c) an () show examples of the resulting orerings. We notice that the orerings are visually meaningful (a) (b) (c) () Fig. 4: Shape-base orering of objects: We generate two artificial atasets each consisting of a reference object an its ranom eformations. Samples from the atasets are shown in (a) an (b). The binary images to the very left show the reference objects for each ataset. We then orere all the eforme objects with respect to the nwesd scores between the object an the reference. The graphs in (c) an () plot these orerings. Base on visual inspection the orering is quite reasonable., i.e. the further the eforme objects visually eviate from the references, the higher their nwesd score is. Furthermore, all the objects generate via isometric transformations yiele scores close to zero as a result of the invariance of the propose scores to this type of transformation. 2) Low Dimensional Embeings: In the secon experiment we focus on creating low imensional embeings. We compare the embeings constructe by WESD with the ones constructe using ρ N SD (Equation (2)), the istance propose in [3]. We o so base on the TOSCA ataset (toolbox for surface comparison an analysis), [34], [35]. This ataset contains binary segmentations of 5 human, 5 centaurs an 5 horses as shown in Figure 5(a). We compute the pairwise affinity matrices between objects via ρ N SD an WESD (ρn with p = 2.0). We then apply the ISOMAP algorithm [36] to these matrices, which maps the 15 objects to a 2D plane base on the pairwise shape istances. We repeat this experiment for affinity matrices compute using ifferent number of spectral moes, i.e. N = 50, 100, 200, to emonstrate the effect of the signature size (truncation parameter) on both istances. The plots in Figures 5(b),() an (f) present the resulting 2D embeings of the ataset using ρ N SD. The embeings are substantially ifferent for ifferent N. This variation arises ue to high sensitivity of ρ N SD towars the signature size N. Moreover, the embeings obtaine using higher N are less satisfactory in terms of separating the three ifferent

10 10 (a) (b) ρ N SD, N=50 (c) ρ N, N=50 () ρ N SD, N=100 (e) ρ N, N=100 (f) ρ N SD, N=200 (g) ρ N, N=200 Fig. 5: Low imensional embeings: (a) The 15 objects use in this experiment. The graphs plot the 2D embeings of the objects base on the affinity matrices constructe by ρ N SD an WESD (ρ N ). Each row presents the results base on ifferent N: 50, 100 an 200 from top to bottom respectively. The structures of the 2D embeing base on ρ N SD are quite ifferent for ifferent N. WESD however, prouces embeings that are similar. This emonstrates the stability of the embeing with respect to N when WESD is use. object classes. This is actually expecte since the spectral moes with higher inices ominate the value of ρ N SD even though they are not informative with regars to the overall geometry an thus, negatively impact the outcome. The plots in Figures 5(c), (e) an (g) present the embeings obtaine using WESD. The embeings obtaine at ifferent N are very similar. This shows that the construction of the low imensional embeing is stable with respect to N when WESD is use. This is a irect consequence of the convergent behavior of WESD iscusse in Sections III-B an III-D. As illustrate by this experiment, this property has very important practical implications. The experimental analysis presente above requires computing a high number of eigenvalues in avance. Besies this option, through Corollary 4, WESD provies us the opportunity to perform the same analysis without the nee to compute eigenvalues in avance. Once N moes are compute, one can use Equation 9 to analyse the stability of the embeing with respect to N an ultimately use this analysis to ecie whether N is enough. In the following we perform such an analysis for this experiment. At three ifferent N = (100, 150, 200), we compute the error upper bouns for each element of the affinity matrix using Equation 9. Let us refer to these bouns as E N (Ω 1, Ω 2 ), so we can write ρ(, ) [ρ N (, ), ρ N (, ) + E N (, )). For the analysis we assume ρ(, ) can lie anywhere in this range, i.e. uniformly istribute. By ranomly sampling these ranges for each element, we generate 5000 ifferent affinity matrices an buil low-imensional embeings for each matrix. In Figure 6 we plot the embeings obtaine with ρ N, with markers, along with the minimum an the maximum lowimensional coorinates each object attaine uring ranom sampling - inicate with a rectangle aroun each point. These rectangles show the maximum error on the low-imensional embeing that we might be introucing by truncating the computation of WESD at N. Observing these graphs: i) one can provie a guarantee that aing more moes after N = 200 will not change the embeing much, ii) at N = 150 one coul have stoppe computing more moes because the embeing cannot change substantially an iii) at N = 100 the embeing theoretically can change so more moes might be necessary. An important point to note is that our experimental finings suggest that the theoretical upper bouns given in Corollary 4 are very conservative. In this experiment, for instance, the embeing oes not change much between using N = 100 an N = 200, i.e. the maximum absolute coorinate change vector for all the objects is ( , ). 3) Shape-base Retrieval of 3D Objects: In this last experiment with synthetic ata, we focus on the application of shapebase object retrieval, i.e. given a test object ientifying other similar objects within a ataset using shape information. Similarity in this context can be efine in various ways but the efinition use here is semantic similarity, meaning that objects that are of the same semantic category (e.g. human boies, aeroplanes, etc) are similar an objects of ifferent categories are not. Shapes of similar objects have similar traits an properties. Shape istances use for retrieval purposes shoul be able to capture these traits yieling the lowest values between similar object pairs. Here, WESD s value for shapebase retrieval is evaluate using the publicly available ataset SHREC presente in [27] 3. SHREC ataset consists of 600 3D non-rigi objects from 30 ifferent categories, i.e. 20 objects per category. Ob- 3 Available at

11 11 NN FT ST E DCG WESD (ρ N ) p=3.15, N= WESD (ρ N ) p=2.0, N= ρ N SD N=12, norm ρ N SD N=12, norma ρ N SD N=15, norm MDS-CM-BOF SD-GDM-meshSIFT (a) (b) p = 3.15 an N = 100 (c) p = 3.15 () N = 100 Fig. 7: Shape-base Object Retrieval Results on SHREC Dataset. a) Retrieval scores obtaine by WESD for two ifferent sets of N an p values along with the scores obtaine by the istance propose in [3] (values taken from [27].) b) Precision-Recall curves obtaine for shape retrieval via WESD for the entire ataset. c) Effect of the signature size N on the retrieval scores obtaine by WESD for a fixe p = ) Effect of the norm type p on the same scores for a fixe N = 100. Fig. 6: The theoretical analysis of the effects of truncating the computation of WESD at N moes on the low-imensional embeings: Points (markers) inicate the low-imensional embeings obtaine by using ρ N (, ). The rectangle aroun a point enotes the theoretical maximum extent that point might move to if infinite number of moes were use to compute the istance, i.e. if ρ(, ) were use. These bouns are compute using using Equation 9 without the nee to compute more eigenvalues than N. jects from the same category iffer with substantial nonlinear eformations, which makes retrieval in this ataset challenging. To evaluate the retrieval accuracy of WESD, first each object was converte from its original watertight surface mesh iscretization to a 3D binary image using the Iso2mesh software package 4. Then pairwise shape istances across the entire ataset were compute using WESD an the affinity matrix was constructe, where each entry is a pairwise istance. This affinity matrix was then evaluate using the software provie with the ataset 3. The evaluation consists of a variety of retrieval accuracy scores such as Nearest Neighbor (NN), First-Tier (FT), Secon-Tier (ST), E-Measure (E), Discounte Cumulative Gain (DCG) an Precision-Recall curve. The first two rows of the table in 4 Figure 7(a) list these scores obtaine using WESD for two ifferent settings of the p an N values. Aitionally, the next three rows of the same table show the results obtaine using ρ N SD (Equation 2, [3]), as liste in [27]5, using two ifferent types of scale normalisation (norm1: normalising with respect to the first eigenvalue, norma: area normalisation, see Section III-E for further etails). These accuracy scores show that WESD an ρ N SD perform very similar in retrieval from the SHREC ataset. The plot given in Figure 7(b) shows the precision-recall curve of WESD (p = 3.15 an N = 100) for the entire ataset. The curve is very similar to the best curve obtaine using ρ N SD shown in [27]. Once again, this confirms that both istances perform similarly. For a complete comparison, the table given in Figure 7(a) also reports the results of the two methos that achieve the highest retrieval scores in the SHREC challenge: MDS- CM-BOF ( [37]) an SD-GDM-meshSIFT ( [38], [39]). Both of these methos construct multi-step pipelines that provie powerful tools for retrieval. However, one has to note that these methos also have ae complexities in contrast to WESD an ρ SD, such as: i) the nee for a training ataset, ii) the corresponence requirements an the resulting nee for remeshing, iii) computation of local key-points for feature computation, iv) high computational cost of geoesic istance matrix, an iv) local feature or point matching to compute a global istance using local features. Nevertheless, their high accuracies emonstrate the avantage of combining ifferent funamental components to achieve powerful retrieval tools. Lastly, the graphs shown in Figure7(c) an () provie an analysis of the retrieval results with respect to the parameters N an p. Graphs in Figure 7(c) plot the change of ifferent 5 We note that for these latter results a slightly ifferent notation is use here than in [27] to conform to the overall notation of this article.

12 12 retrieval scores with respect to the number of moes use N, i.e. signature size, keeping p fixe at Graphs in Figure 7() plot the changes with respect to the norm type p keeping N fixe at 100. These graphs show that as N increases the scores seem to increase slowly an then converge. On the other han, p has a stronger effect on the results than N, particularly on FT, ST an E scores. However, the changes in the scores with respect to changes in N or p are rather small especially compare to the relatively larger fluctuation of the FT score of ρ N SD with respect to the two sample N values provie in the table in Figure 7(a). The experiment presente above showe that the retrieval power of WESD is similar to that of the istance ρ N SD propose by Reuter et al. [3]. The sounness an theoretical properties of WESD o not come at the expense of lower retrieval power. On the contrary, WESD is able to leverage the escriptive power of the spectra while its properties guarantee that it oes not suffer from similar rawbacks as other istances, such as sensitivity to signature size. C. Real Data The experiments on real ata are conucte on segmentations of 3D structures obtaine from magnetic resonance images (MRI). First, we apply WESD to subcortical brain structures. The experiment emonstrates WESD s capabilities to ifferentiate categories of objects even in the presence of high intra-class variability. In the secon experiment, we focus on temporal analysis of cariac images. We apply nwesd to elineations of the bloo pool of the left ventricle obtaine from 3D + time cariac MRI. The experiment shows that the shape issimilarity measurements between time points correlates with the ynamic processes of the beating heart. 1) Clustering Sub-Cortical Structures: Meical research frequently relies on morphometric stuies analysing anatomical shapes from meical images [40]. In this experiment we construct a low imensional embeing of subcortical structures extracte from Magnetic Resonance Image (MRI) scans of ifferent iniviuals base on WESD as well as shape-dna base istance, ρ N SD, as propose in [3]. For this experiment, we use the publicly available LPBA40 ataset [41] 6. The ataset contains manual segmentations of various subcortical structures from MRI brain scans of 40 healthy subjects. Figures 8(a), (b) an (c) show some examples from these structures. The are two main ifficulties associate with such atasets. First, the structures have very large intraclass (inter-subject) variability, i.e. the shape of an anatomical structure is often very ifferent across subjects. Secon, the segmentations were obtaine by manually elineating the 3D objects on successive 2D slices. This creates inconsistencies between segmentations in two successive slices. Such inconsistencies in the en manifest themselves as local artefacts on the object. The protrusion that can be seen on the top of the secon hippocampus in Figure 8(a) is an example of such an artefact. These artefacts can influence shape istances negatively. We select six structures for each patient: left/right cauate nucleus, left/right putamen an left/right hippocampus, 6 website: resulting in 240 structures in total. We then create pairwise affinity matrices of the 240 structures first using ρ N SD with N = 200, as propose in [3], an then WESD (ρ N with p = 2 an N = 200). Finally, we use the ISOMAP algorithm [36] to construct 2D embeings of the structures. Figures 8() an (e) show the resulting embeings. We observe that the embeing obtaine via WESD well clusters the ata with respect to the anatomical structures. The separation of the clusters for the SD case, however, is more ambiguous, especially between putamen an hippocampus. The embeings presente above were obtaine by irectly using the manual segmentations without any preprocessing. A natural question is how o these embeings change if the effects of various artefacts are reuce say via surface smoothing. To answer this question, we smooth the surface of the anatomical 3D moels an recompute the embeings, which are shown in Figures 8(f) an (g). The embeing obtaine with ρ N SD, although to a lesser extent, still suffers from similar ambiguity as in Figure 8(). The new embeing base on WESD on the other han, compare to Figure 8(e), even more clearly separates ifferent anatomical structures. However, we also note that this type of preprocessing can also prouce unesirable artefacts such as altering the topology of the anatomical object. This is the case for one cauate in Figures 8(f) an (g), which ens up as an outlier that is clearly separate from the other ata points. Consiering this, the fact that WESD is able to prouce visually pleasing embeings without the nee of preprocessing is an avantage. 2) Analysing Heart Function in 4D MRI: Fourimensional imaging of patient anatomy is gaining interest in the meical community. The temporal analysis of anatomical structures is use to extract the characteristics of relate ynamic processes, which often inicate certain pathologies [42]. Furthermore, in the recent work [43] authors show that shape information, in aition to volumetric measurements, improve the accuracy of pathology relate classification tasks in such ynamical analyses. In this section, we apply nwesd to the shapes of the hearts extracte from four imensional cariac images of five ifferent patients. The scan of each patient captures a full cycle of one heartbeat as a series of 20 3D images. Each image shows the left ventricle (LV) at a specific point in the cycle, from which we manually segment the corresponing bloo pool. Our reference is the bloo pool extracte from the first frame (iastole). We compute the nwesd scores between this reference an all other shapes extracte from the series of images. Here, we o not normalise the eigenvalues with respect to the global scale since size change is an important aspect of the heartbeat ynamics. The graph given in Figure 9 shows the results of these measurements over time across the five patients. The figure also shows some exemplary images an shapes. We observe that the symmetry of the heartbeat along the systolic (as the bloo pumps out of the LV pool) an the iastolic phases (as the bloo fills in the pool) is well capture with the nwesd score. Furthermore, the ensystolic phase (the time point with the largest istance w.r.t. the reference) is at ifferent time points for ifferent patients, which is to be expecte since the ifferent patient scans are not synchronize in time. In summary, WESD well captures the

13 13 (a) Four Hippocampi () ρn SD - no preprocessing (b) Four Cauate Nuclei (e) WESD - no preprocessing (f) ρn SD - surface smoothing (c) Four Putamen (g) WESD - surface smoothing Fig. 8: 2D embeing of subcortical structures: 240 structures (80 cauate nucleus, 80 putamen an 80 hippocampus) are extracte from MR scans of 40 ifferent iniviuals. (a),(b) an (c) show some example structures from this ataset. Note the high intra-class variability an the artefacts ue to finite resolution an manual segmentations. () an (e) plot 2D embeings of these 240 structures obtaine base on the affinity matrices compute via ρn SD an WESD respectively. These embeings are compute without any preprocessing applie to the structures. The embeing obtaine with WESD istinctly clusters the objects with respect to the anatomical structures. The embeing in () however, shows some ambiguities in the separation. Graphs in (f) an (g) plot the similar embeings obtaine after smoothing the surfaces of the structures to remove artefacts. The embeing obtaine by ρn SD, although better than (), still suffer from similar problems. The embeing base on WESD on the other han, now even between better separates the groups. presente an prove the properties of WESD relate to its existence, computability an multi-scale aspect. The presente experiments showe that the theoretical properties of WESD have many practical avantages over previous works. These experiments further highlighte that WESD is beneficial for various applications. R EFERENCES Fig. 9: Analysing 3D + time (4D) cariac images: First column shows corresponing 2D slices of a 4D MRI ataset at time points t = {0, 6, 12}. The secon column, 3D shapes extracte at each of the time points. For five patients, we compute the nwesd shape issimilarity score of the LV bloo pool at each time point with respect to its shape at t = 0. The graph plots these scores. We note that the propose shape istance is able to capture the ynamic process of the LV shape changes an furthermore, the symmetry between the two phases of an heart beat: iastole an systole. ynamics of the beating heart, which is to be expecte given the continuous link between the ifferences in eigenvalues an the ifference in shape (see Section II). V. C ONCLUSION This article propose WESD, a new spectral shape istance efine over the eigenvalues of the Laplace operator. WESD is a theoretically soun shape metric that is erive from the heat-trace. The theoretical analysis given in this article [1] D. Zhang an G. Lu, Review of shape representation an escription techniques, Pattern Recognition, vol. 37, no. 1, pp. 1 19, [2] N. Iyer, S. Jayanti, K. Lou, Y. Kalyanaraman, an K. Ramani, Threeimensional shape searching: state-of-the-art review an future trens, Computer-Aie Design, vol. 37, pp , [3] M. Reuter, F.-E. Wolter, an N. Peinecke, Laplace-Beltrami spectra as Shape-DNA of surfaces an solis, Computer-Aie Design, vol. 38, pp , [4] B. Le vy, Laplace-beltrami eigenfunctions towars an algorithm that unerstans geometry, in Shape Moeling an Applications, [5] R. M. Rustamov, Laplace-beltrami eigenfunctions for eformation invariant shape representations, in Eurographics Symposium on Geometry Processing, [6] J. Sun, M. Ovsjanikov, an L. Guibas, A concise an provably informative multi-scale signature base on heat iffusion, in Eurographics Symposium on Geometry Processing, [7] M. Reuter, F.-E. Wolter, M. Shenton, an M. Niethammer, Laplacebeltrami eigenvalues an topological features of eigenfunctions for statistical shape analysis, Computer-Aie Design, vol. 41, pp , [8] M. M. Bronstein an A. M. Bronstein, Shape recognition with spectral istances, IEEE Transactions on Pattern Analysis an Machine Intelligence, vol. 33, no. 5, pp , [9] H. Weyl, Das asymptotische verteilungsgesetz er eigenwerte linearer partieller ifferentialgleichungen, Mathematische Annalen, pp , [10] M. Kac, Can one hear the shape of a rum? The American Mathematical Monthly, vol. 73, no. 7, pp. 1 23, [11] R. Courant an D. Hilbert, Metho of Mathematical Physics, vol I. Interscience Publishers, [12] V. Jain an H. Zhang, Robust 3 shape corresponence in the spectral omain, in International Conference Shape Moeling an Applications, 2006.

14 14 [13] A. M. Bronstein, M. Bronstein, M. Mahmoui, R. Kimmel, an G. Sapiro, A Gromov-Hausorff framework with iffusion geometry for topologically-robust non-rigi shape matching, International Journal of Computer Vision, vol. 89, no. 2,3, pp , [14] F. Mémoli, A spectral notion of gromov-wasserstein istance an relate methos, Applie an Computational Harmonic Analysis, vol. 30, no. 3, pp , [15] M. Ovsjanikov, A. M. Bronstein, M. M. Bronstein, an L. J. Guibas, Shape google: a computer vision approach to invariant shape retrieval, in International Conference on Computer Vision Workshops, 2009, pp [16] M. Bronstein, Scale-invariant heat kernel signatures for non-rigi shape recognition, in Computer Vision an Pattern Recognition, [17] R. Lai, Y. Shi, K. Scheibel, S. Fears, R. Woos, A. Toga, an T. Chan, Metric-inuce optimal embeing for intrinsic 3 shape analysis, in Computer Vision an Pattern Recognition, [18] S. Gnutzmann, U. Smilansky, an N. Sonergaar, Resolving isospectral rums by counting noal omains, Journal of Physics A: Mathematical an General, vol. 38, no. 41, [19] S. Gnutzmann, P. Karageorge, an U. Smilansky, Can one count the shape of a rum? Physical Review Letters, vol. 97, no. 9, [20] R. Lai, Y. Shi, I. Dinov, T. Chan, an A. Toga, Laplace-beltrami noal counts: A new signature for 3 shape analysis, in International Symposium on Biomeical Imaging, [21] A. Pleijel, A stuy of certain green s functions with applications in the theory of vibrating membranes, Arkiv För Matematik, vol. 2, no. 6, pp , [22] H. McKean an I. Singer, Curvature an the Eigenvalues of the Laplacian, Journal of Differential Geometry, vol. 1, pp , [23] L. Smith, The asymptotics of the heat equation for a bounary value problem, Inventiones Mathematicae, vol. 63, pp , [24] M. Protter, Can one hear the shape of a rum? Revisite, SIAM Review, vol. 29, no. 2, pp , June [25] D. Vassilevich, Heat kernel expansion: user s manual, Physics Reports, vol. 388, no. 5-6, pp , [26] G. Jurman, R. Visintainer, an C. Furlanello, An introuction to spectral istances in networks (extene version), Preprint in ArXiv, Oct [27] Z. Lian, A. Goil, B. Bustos, M. Daoui, J. Hermans, S. Kawamura, Y. Kurita, G. Lavoué, H. V. Nguyen, R. Ohbuchi, Y. Ohkita, Y. Ohishi, F. Porikli, M. Reuter, I. Sipiran, D. Smeets, P. Suetens, H. Tabia, an D. Vanermeulen, SHREC 11 track: Shape retrieval on nonrigi 3D watertight meshes, in Proceeings of the Eurographics/ACM SIGGRAPH Symposium on 3D Object Retrieval, [28] C. Goron, D. Webb, an S. Wolpert, Isospectral plane omains an surfaces via riemannian orbifols, Inventiones Mathematicae, vol. 110, no. 1, pp. 1 22, [29] M. Niethammer, M. Reuter, F.-E. Wolter, S. Bouix, N. Peinecke, M.- S. Koo, an M. E. Shenton, Global meical shape analysis using the Laplace-Beltrami spectrum, in Meical Image Computing an Computer Assiste Intervention, [30] E. Whittaker an G. Watson, A course of moern analysis. Cambrige Mathematical Library, [31] L. Evans, Partial Differential Equations. American Mathematical Society, [32] W. F. Ames, Numerical Methos for Partial Differential Equations. Acaemic Press, [33] W. E. Arnoli, The principle of minimize iterations in the solution of the matrix eigenvalue problem, Quarterly of Applie Mathematics, vol. 9, no. 17, pp , [34] A. Bronstein, A. B. M.M. Bronstein, an R. Kimmel, Analysis of two-imensional non-rigi shapes, International Journal of Computer Vision, vol. 78, no. 1, pp , [35] A. Bronstein, M. Bronstein, an R. Kimmel, Numerical geometry of non-rigi shapes. Springer, [36] J. Tenenbaum, V. e Silva, an J. Langfor, A global geometric framework for nonlinear imensionality reuction, Science, vol. 290, no. 5500, pp , [37] Z. Lian, A. Goil, X. Sun, an H. Zhang, Non-rigi 3 shape retrieval using multiimensional scaling an bag-of-features, in Proceeings of ICIP, 2010, pp [38] C. Maes, T. Fabry, J. Keustermans, D. Smeets, P. Suetens, an D. Vanermuelen, Feature etection on 3D face surfaces for pose normalisation an recognition, in Proceeings of Biometrics: Theory, Applications an Systems, [39] D. Smeets, T. Fabry, J. Hermans, D. Vanermuelen, an P. Suetens, Isometric eformation moelling for object recognition, in Proceeings of Computer Analysis of Images an Patterns, 2009, pp [40] F. L. Bookstein, P. D. Sampson, A. P. Streissguth, an P. D. Connor, Geometric morphometrics of corpus callosum an subcortical structures in the fetal-alcohol-affecte brain, Teratology, vol. 64, pp. 4 32, Jul [41] D. Shattuck, M. Mirza, V. Aisetiyo, C. Hojatkashani, G. Salamon, K. Narr, R. Polrack, R. Biler, an A. Toga, Construction of a 3 probabilistic atlas of human cortical structures, NeuroImage, vol. 39, no. 3, pp , [42] T. Mansi, I. Voigt, B. Leonari, X. Pennec, S. Durrleman, M. Sermesant, H. Delingette, A. Taylor, Y. Boujemline, G. Pongiglione, an N. Ayache, A statistical moel for quantification an preiction of cariac remoeling: application to tetralogy of fallot, IEEE Transactions on Meical Imaging, vol. 30, no. 9, pp , [43] E. Bernaris, E. Konukoglu, Y. Ou, D. N. Metaxas, B. Desjarins, an K. M. Pohl, Temporal shape analysis via the spectral signature, in In Proceeings of Meical Image Computing an Computer Assiste Intervention, E ner Konukoglu was born in Istanbul, Turkey, in He receive a Ph.D. egree in computer science specializing in meical image analysis from Université e Nice an INRIA Sophia Antipolis, France. He is now a Research Fellow at Athinoula A. Martinos Center for Biomeical Imaging, Massachusetts General Hospital / Harvar Meical School. His research interests are meical image analysis, biophysical moels, machine learning an applications of partial ifferential equations. B en Glocker was born in Goettingen, Germany, in He receive a Ph.D. egree in meical computer science from Technische Universitaet Muenchen, Germany. He is currently a Postoctoral Researcher at Microsoft Research Cambrige, UK where he is a member of the Machine Learning an Perception group. His main research focus is on the application of machine learning techniques for meical image analysis. A ntonio Criminisi was born in 1972 in Italy. He receive a Degree in Electronics Engineering at the University of Palermo an obtaine a PhD in Computer Vision at the University of Oxfor. He is currently a Senior Researcher at Microsoft Research Cambrige, UK. His research interests are in the area of meical image analysis, object category recognition, image an vieo analysis an eiting, one-to-one teleconferencing, 3D reconstruction from single an multiple images with application to virtual reality, forensic science an history of art. K ilian M. Pohl receive a M.S. egree from the Department of Mathematics, University of Karlsruhe, an his PhD from the Computer Science an Artificial Intelligence Laboratory, Massachusetts Institute of Technology. Dr. Pohl is now an Assistant Professor at the Department of Raiology an hols a seconary appointment at the Bioengineering Grauate Group, University of Pennsylvania. His research focuses on creating algorithms for automatically quantifying an generalizing the information latent in meical images.