Scale-invariant heat kernel signatures for non-rigid shape recognition

Michael M. BronsteinDepartment of Computer Science, Technion – Israel Institute of Technology

Haifa 32000, Israelmbron@cs.technion.ac.il

Iasonas KokkinosDepartment of Applied Mathematics, Ecole Centrale de Paris

Grande Voie des Vignes, 92295 Chatenay-Malabry, Franceiasonas.kokkinos@ecp.fr

Abstract

One of the biggest challenges in non-rigid shape re-trieval and comparison is the design of a shape descrip-tor that would maintain invariance under a wide class oftransformations the shape can undergo. Recently, heat ker-nel signature was introduced as an intrinsic local shape de-scriptor based on diffusion scale-space analysis. In this pa-per, we develop a scale-invariant version of the heat ker-nel descriptor. Our construction is based on a logarith-mically sampled scale-space in which shape scaling cor-responds, up to a multiplicative constant, to a transla-tion. This translation is undone using the magnitude ofthe Fourier transform. The proposed scale-invariant lo-cal descriptors can be used in the bag-of-features frame-work for shape retrieval in the presence of transformationssuch as isometric deformations, missing data, topologicalnoise, and global and local scaling. We get significant per-formance improvement over state-of-the-art algorithms onrecently established non-rigid shape retrieval benchmarks.

1. Introduction

Today, only a small fraction of Internet repositories ofvisual and geometric data is tagged and accessible throughsimple text search. Fast growth of these repositories makescontent-based retrieval one of the next grand challenges insearch and organization of such information. Particularlydifficult is the problem of shape retrieval, as geometricshapes manifest a vast variability due to different scale, ori-entation, non-rigid deformations, missing data, and also ap-pear in a variety of different formats and representations.

In principle, the common denominator of shape retrievalapproaches is the creation of a shape descriptor or signa-

ture which captures the unique properties of the shape thatdistinguish it from shapes belonging to other classes on theone hand, and is invariant to a certain class of transforma-tions a shape can undergo on the other [40, 39]. In rigidshape analysis, different types of invariance were addressed.Rotation and translation invariance can be achieved usingvolume and area descriptors [45], spherical harmonics [16],geometric moments et al. [38], and distribution of pair-wiseEuclidean distances [29].

Dealing with non-rigid shapes requires compensatingfor the degrees of freedom resulting from deformations.Elad and Kimmel [12] and follow-up works [26, 8] pro-posed modeling shapes as metric spaces with intrinsic (e.g.geodesic) distances, which are invariant to inelastic defor-mations. Ling and Jacobs [22] and Bronstein et al. [7] usedthis framework with a metric defined by internal distancesin 2D shapes. Reuter et al. [33, 32] used the Laplacianspectra as intrinsic shape descriptors.

A particular type of intrinsic geometry is generated byheat diffusion processes on the shape. Coifman and Lafon[11] popularized the notion of diffusion geometry, which isclosely related to scale-space methods in image processing[36]. Rustamov [34] was one of the first to use such dis-tances in shape analysis, applying the method of Osada etal. [29] to commute time distances (this method is similarto the recent work of Mahmoudi and Sapiro [24] who useddiffusion distances instead). In [5], shapes were analyzed asmetric spaces equipped with diffusion metrics.

In image analysis, bottom-up approaches have becomepopular, notably due to the works of Zisserman et al.[35, 10] and Schmid et al. [27]. Using these approaches,an image is described as a collection of local features (“vi-sual words”) from a given vocabulary, resulting in a repre-sentation referred to as a bag of features. In shape analy-sis, such approaches have been introduced more recently by

1

Ovsjanikov et al. [6] and Toldo et al. [42] (see [28, 20] forearlier similar ideas).

The bag of features paradigm relies heavily on the choiceof the local feature descriptor that is used to create the vi-sual words. In image analysis and 2D shape retrieval, typ-ical features are blobs [25] and corners [27], and the de-fault choice for a local descriptor is the scale-invariant fea-ture transform (SIFT) [23] or one of its varieties [2, 41].Scale-invariant local descriptors can be constructed in twoways. First way is to use scale-space analysis of the imageto locally estimate the scale [21, 23]. Descriptors are thenextracted from appropriately scaled image patches. Sec-ond way is to use a combination of logarithmic samplingwith Fourier analysis to compensate for the scaling effects[17] (such an approach is also commonly used to computea global image rotation and scaling in the context of regis-tration [9, 46]).

In shape analysis, on the other hand, there is no com-monly agreed upon feature descriptor similar to SIFT. Innon-rigid shape retrieval applications, an ideal feature de-scriptor should be first of all intrinsic and thus deformation-invariant. Second, it should cope with missing parts, andalso be insensitive to topological noise and connectivitychanges. Third, it should work across different shape repre-sentations and formats (e.g. point clouds and meshes) andbe insensitive to sampling. Finally, the descriptor shouldbe scale-invariant. The last two properties are especiallyimportant when dealing with shapes coming from Internetrepositories such as Google 3D Warehouse, where shapesappear in a variety of representations and with arbitraryscales.

Different approaches such as contour and edge features[30, 18], spin images [14], local patches [28, 42], confor-mal factor [4], differential operators [44], and local volumeproperties [13] were used as feature descriptors in shape re-trieval literature. Unfortunately, none of them satisfy all ofthe above desired properties (for example, volumetric andpatch-based methods are not intrinsic, and conformal factoris sensitive to topology).

Recently, a local feature descriptor based on multiscaleheat kernels was proposed [37]. This descriptor satisfies allof the above properties except for scale invariance. Scaleinvariance poses an additional challenge, for a few reasons.Compared to images, shapes typically contain less featuresthat would be roughly analogous to blobs or corners, andthere is no clear generalization of such structures to 3Dsurfaces. Feature detection based on intrinsic scale-spaceanalysis such as [37] would find a few reliable points (usu-ally with high curvature), at which scale estimation can bedone. In flat regions, no scale estimation is possible. Forthis reason, Ovsjanikov et al. [6] avoided feature detectionand used a dense feature descriptor computed at every pointof the shape in combination with statistical weighting to re-

duce the influence of trivial points.Contribution. In this paper, we develop a scale-invariantversion of the heat kernel signature by combining this de-scriptor with the recent approach of [17] to scale invariancein images. Our construction is based on a logarithmicallysampled scale-space in which shape scaling corresponds,up to a multiplicative constant, to a translation. This trans-lation is then undone using the magnitude of the Fouriertransform. Since our descriptor does not rely on local scaleestimation, it is computable at every point including flat re-gions, and can be thus used in the shape retrieval frameworkof [6], as well as for other applications such as dense corre-spondence between shapes.

2. BackgroundIn the following discussion, we model shapes as Rie-

mannian manifolds (possibly with boundary) and use theheat conduction properties as shape descriptors. Heat prop-agation on non-Euclidean domains is governed by the heatdiffusion equation,

(∆X +

∂

∂t

)u = 0, (1)

where, ∆X denotes the positive semi-definite Laplace-Beltrami operator, a Riemannian equivalent of the theLaplacian. The solution u(x, t) of the heat equation with theinitial conditions u(x, 0) = u0(x) (and respective boundaryconditions if X has a boundary) describes the amount ofheat on the surface at point x in time t. The solution of (1)with point heat distribution u0(x) = δ(x−z) as initial con-ditions is called the heat kernel and denoted by KX,t(x, z).

On compact manifolds, the heat kernel can be presentedas [15]

KX,t(x, z) =∞∑

i=0

e−λitφi(x)φi(z). (2)

where λ0, λ1, ... ≥ 0 are eigenvalues and φ0, φ1, ... are thecorresponding eigenfunctions of the Laplace-Beltrami op-erator, satisfying ∆Xφi = λiφi.Heat kernel signatures. Sun et al. [37] proposed using theheat kernel signature (HKS)

h(x, t) = KX,t(x, x) =∞∑

i=0

e−λitφ2i (x) (3)

as local shape descriptors. The HKS is intrinsic and thusisometry-invariant (two isometric shapes have equal HKS),multi-scale and thus capture both local features and globalshape structure, and also informative: under mild condi-tions, if two shapes have equal heat kernel signatures, theyare isometric [37].

10 20 30 400

100

200

300

400

500

600

Figure 1. Construction of a bag of features shape descriptor. Left:dense HKS local descriptor (shown three components as RGB col-ors); middle: local descriptor quantized in a geometric vocabularyof size 48 (each color represents a geometric word); right: bag offeatures counting the frequency of appearance of each geometricword.

Ovsjanikov et al. [6] used the HKS to construct globalshape descriptors following the bag of features paradigmused in image retrieval applications [35, 10]. First, the HKSdescriptor is computed at every point of the shape (Figure 1,left). Next, using vector quantization, for each point on theshape, the HKS is replaced by the index of the most similarentry in a geometric vocabulary consisting of representativeheat kernel signatures or “geometric words” (Figure 1, mid-dle). The vocabulary is constructed offline by performingclustering in the HKS space. Finally, the distribution of ge-ometric words on the shape is computed, resulting in a bagof features representation (Figure 1, right).

Sensitivity to scale. A notable disadvantage of the heatkernel signatures is their sensitivity to scale. Given a shapeX and its scaled version X ′ = βX , the new eigenvaluesand eigenfunctions will satisfy λ′ = β2λ and φ′ = βφ. Wetherefore have the following equation:

h′(x, t) =∞∑

i=0

e−λiβ2tφ2

i (x)β2 = β2h(x, β2t), (4)

relating the signature h′ at time t for X ′ with the signatureh at time β2t for X .

In some cases, the scaling effect can be undone usingsome global pre-normalization of the shape. Possible waysare normalizing the bounding box of the shape or its covari-ance (geometric moments), normalize the intrinsic diameterof the shape, i.e. the longest geodesic distance, or normalizethe Laplace-Beltrami eigenvalues. The first approach willwork only in rigid shapes, as non-rigid deformations changethe bounding box. The second and the third approaches areinsensitive to deformations, but would fail if the shape hasmissing parts. In the following, we describe an approachfor local normalization of the heat kernel signature, whichdoes not suffer from this problem.

3. Scale-invariant heat kernel signatures

In order to achieve scale invariance, we need to removethe dependence of h from the scale factor β. This is possi-ble through the following series of transformations appliedto h. First, at each shape point x we sample the heat signa-ture logarithmically in time (t = ατ ) and form the discretefunction

hτ = h(x, ατ ). (5)

Based on Eq. 4, scaling the shape by β will result in a time-shift by s = 2 logα β and amplitude-scaling by β2 (Fig-ure 2, left):

h′τ = β2hτ+s. (6)

Second, we remove the multiplicative constant β2 bytaking the logarithm of h, and then the discrete derivativew.r.t. to τ (Figure 2, middle). The first step turns the mul-tiplicative factor into an additive constant, 2 log β, whichthen vanishes in differentiation:

h′τ = hτ+s, (7)

(here, hτ = log hτ+1 − log hτ ).Finally, taking the discrete-time Fourier transform of hτ

turns this shift in time into a complex phase;

H ′(ω) = H(ω)e2πωs, (8)

where H and H ′ denote the Fourier transform of h and h′,respectively, and ω ∈ [0, 2π]. The phase is in turn elimi-nated by taking the Fourier transform modulus (FTM):

|H ′(ω)| = |H(ω)|. (9)

We thus have constructed the scale-invariant quantity|H(ω)| (denoted as SI-HKS and shown in Figure 2, right)from the HKS at each point x, without performing scale se-lection. This allows us to compute descriptors at any pointof our shape, where scale selection based on maxima de-tection could be impossible. Moreover, most of the signalinformation is contained in the low-frequency componentsof the FT, so we can build a compact descriptor by sampling|H(ω)| at a small number of low frequencies.

One caveat of our approach could be that scaling theshape and then resampling the function hτ makes the sam-ples at the boundaries of the range of τ change. This canhave dramatic effects if the signal information is concen-trated at the boundaries of the scale-space. Fortunately, theHKS is typically smooth at low- and high- scales and there-fore its derivative is equal to zero for a broad range of τs atthe beginning and end of h.

0 50 100 150 200 250 300 350−15

−10

−5

0

τ0 50 100 150 200 250 300 350

−0.04

−0.03

−0.02

−0.01

0

τ0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

k

Figure 2. Construction of the scale-invariant heat kernel signature. Left: heat kernel signatures h (red) and h′ (blue) computed at acorresponding point on a shape and its version scaled by the factor of 11, plotted on a logarithmic scale. h and h′ differ by scale and shiftin τ . Middle: hτ and h′τ , where the multiplicative constant is undone and the change in scale corresponds to a shift in τ only. Right: first10 frequencies of |H(ω)| and |H(ω)| used as scale-invariant HKS; the two descriptors computed at the two different scales are virtuallyidentical.

4. Numerical computationNumerical computation of the HKS and the SI-HKS is

done using formula (2), in which a finite number of terms istaken and the continuous eigenfunctions and eigenvalues ofthe Laplace-Beltrami operator are replaced by the discretecounterparts. The discretization of the Laplace-Beltramioperator depends on the representation on the shape. Forshapes represented as point clouds, the Laplace-Beltramioperator can be approximated using [3]. For triangularmeshes, one of the most common discretizations is thecotangent weight scheme [31], defined for any function fon the mesh vertices as

(∆Xf)i =1ai

∑

j

wij(fi − fj), (10)

where wij = cot αij + cot βij for j in the 1-ring neigh-borhood of vertex i and zero otherwise (αij and βij are thetwo angles opposite to the edge between vertices i and jin the two triangles sharing the edge), and ai are normaliza-tion coefficients proportional to the area of triangles sharingthe vertex xi. This discretization preserves many impor-tant properties of the continuous Laplace-Beltrami operator,such as positive semi-definiteness, symmetry, and locality,and in addition it is numerically consistent [43]. In matrixnotation, Eq. (10) can be written as

∆Xf = A−1Lf, (11)

where A = diag(ai) and L = diag(∑

l 6=i wil

)− (wij).

The eigenvalues and eigenfunctions of the Laplace-Beltrami operator discretized according to 11 are com-puted by solving the generalized eigendecomposition prob-lem [19]

AΦ = ΛLΦ, (12)

where Λ is the (k+1)×(k+1) diagonal matrix of the small-est eigenvalues λ0, ..., λk, and Φ is an N × (k + 1) matrixof corresponding eigenfunctions φ0, ..., φk such that φil isthe value of the lth eigenfunction at the point xi. Anotherway of approximating Laplace-Beltrami eigenfunctions on

triangular meshes is using finite element methods (FEM)[32].

The discrete heat kernel signature is approximated by

h(xl, τ) ≈k∑

l=0

e−λlατ

φ2il = Ψe−TΛ, (13)

where T = diag(ατ ) and Ψ = (φ2il). Since the heat kernel

depends only on the eigenfunctions and eigenvalues of theLaplace-Beltrami operator, at least in theory, one can com-pare shapes in different representations (e.g., point cloudsto meshes). This property of heat kernel signatures is es-pecially appealing in Internet shape retrieval applications,where the variety of shape representations and formats isenormous.

5. ResultsWe used the ShapeGoogle database [6], consisting of

1061 shapes with simulated transformations. As of today,this is the largest non-rigid shape retrieval benchmark avail-able. The database contained shapes from 469 differentclasses. For thirteen shape classes, the following transfor-mations were simulated: 208 isometry, 208 global scale(varying approximately between 0.7 and 1.35), 128 localscale (local “swelling” of the shape), and 48 partiality+scale(missing parts in shapes with different global scaling). Ex-amples of transformations are shown in Figure 4.

Heat kernel signatures (HKS) and the proposed scale-invariant heat kernel signatures (SI-HKS), respectively,were used as local shape descriptors. For the discrete com-putation of the heat kernels, we used the cotangent weightapproximation of the Laplace-Beltrami operator and k =200. For HKS, we used the parameters as in [6] (six scales1024, 1351, 1783, 2353, 3104 and 4096), which were exper-imentally found to give optimal performance on the Shape-Google database. In order to construct the SI-HKS, we useda logarithmic scale-space with base α = 2 and τ rangingfrom 1 to 25 with increments of 1/16. After applying thelogarithm, derivative, and Fourier transform, the first 6 dis-crete lowest frequencies were used as the local descriptor.

Original

Scaled

Figure 3. Comparison of HKS (left) and the proposed scale-invariant HKS (right). First and third rows: three components of HKS andSI-HKS, represented as RGB color and shown for shapes differing by global (first row) and local (third row) transformations. Second andfourth rows: HKS and SI-HKS at three point on the head (blue), hand (green), and foot (red) of the human shape.

Shape descriptors were constructed using bags of geo-metric words proposed in [6]. For HKS and SI-HKS, a ge-ometric vocabulary of size 48 was built using clustering inthe signature space (six-dimensional in both cases). TheHKS and SI-HKS at each point of the shape were replacedby the closest geometric word from the vocabulary usingsoft vector quantization. We used the approximate nearestneighbor algorithm [1] as implemented in the ANN tool-box.1 The distribution of geometric words (48-dimensionalbag of features) was used as the shape descriptor. L1 dis-tance was used to compare the bags of features.

For comparison, we show the results of the ShapeDNAapproach [33], describing shapes by the vector of the firsteigenvalues of the Laplace-Beltrami operator. We used first15 eigenvalues to construct the ShapeDNA descriptors (thisparameter was empirically selected to achieve optimal per-formance on the ShapeGoogle database). Eigenvalues werecomputed using the same cotangent weight discretization.L2 distance was used to compare the ShapeDNA descrip-tors.

Shape retrieval performance was quantified using theprecision-recall (PR) curve (Figure 5), plotting the tradeoffbetween precision (ratio of the number of relevant shapesretrieved and the total number of shapes retrieved) and re-

1Code available from http://www.cs.umd.edu/˜mount/ANN

call (ratio of the number of relevant shapes retrieved and thetotal number of existing relevant shapes that could be ide-ally retrieved). We used the mean average precision (mAP)as a single number to quantify the retrieval quality (averageprecision is computed as the area below the precision-recallcurve for each query, and the mAP is the average of AP overall queries).

Table 1 shows the performance of shape retrieval usingbags of features built of HKS and SI-HKS. Our approachshows a dramatic improvement in the presence of varyingscale (99.5% mAP compared to 61.32% with HKS) andalso better performance for local scaling transformations(92.60% mAP compared to 85.83%). HKS-based bags offeatures produce negligibly (by 0.01%) worse results thatSI-HKS on the class of isometric deformations. ShapeDNAshows similar nearly perfect performance on the class ofisometries, but performs very poorly on scale and local scaletransformations (36.72% and 72.17% mAP, respectively)

Figure 6 shows examples of first five matches retrievedusing HKS and SI-HKS. With HKS, a scaled down centauris confused with a dog (row a) and a scaled up horse is con-fused with an elephant (row c); while SI-HKS produces cor-rect matches (rows b and d). In the presence of local scaling,because of the local nature of the descriptor, it remains un-changed far from the deformed parts. The SI-HKS showsbetter robustness to such scaling compared to HKS (e.g., in

Figure 4. Example of transformations used in our shape retrievalexperiment (left to right): null, scale down, scale up, two examplesof local scale, partiality+scale.

Table 1. Shape retrieval performance (mAP in percents) usingHKS and SI-HKS based bags of features and ShapeDNA [33].Best result is shown in bold.

Transformation Queries HKS SI-HKS ShapeDNAIsometry 208 99.96% 99.97% 99.52%Scale 208 61.32% 99.95% 36.72%Local scale 128 85.83% 92.60% 72.17%Partiality+scale 48 54.67% 89.95% 27.42%All 1061 85.30% 97.25% 74.47%

Figure 6 (g) local scaling transformations make HKS con-fuse between male and female shapes).

6. Conclusions

We presented an extension of the heat kernel signatureallowing to deal with global and local scaling transforma-tions. The use of Fourier transform magnitude to extract ascale-invariant quantity out of the heat kernel signature isadvantageous over attempts to perform scale localization,which works only at prominent feature points. Our ap-proach allows to create a dense scale-invariant feature de-scriptor defined at every point of the shape. Besides invari-ance to global scaling, the scale-invariant HKS shows betterresilience to local scaling transformations. Such transfor-mations can arise, for example, due to locally-elastic defor-mations that stretch or shrink the shape surface. In futurework, we intend to explore the proposed method in the con-text of part-based approaches in which the local descriptoris confined to a part of a shape and not computed acrossparts, and a separate descriptor is computed for each part.This way, inelastic deformations could be addressed.

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������������Figure 6. Shape retrieval results. Left: queries, right: first matches using HKS (a,c,e,g) and SI-HKS (b,d,f,h).