Computers & Graphics 60 (2016) 9–22

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Technical Section

Symmetry aware embedding for shape correspondence$

Yusuke Yoshiyasu a,n, Eiichi Yoshida a, Leonidas Guibas b

a CNRS-AIST JRL (Joint Robotics Laboratory), UMI3218/RL, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba Central 1,1-1-1 Umezono, Tsukuba, Ibaraki 305-8560, Japanb Department of Computer Science, Stanford University, Stanford, CA 94305, USA

a r t i c l e i n f o

Article history:Received 11 December 2015Received in revised form7 June 2016Accepted 4 July 2016Available online 30 July 2016

Keywords:Shape correspondenceEmbeddingSymmetry

x.doi.org/10.1016/j.cag.2016.07.00293/& 2016 Elsevier Ltd. All rights reserved.

article was recommended for publication byesponding author.ail addresses: yusuke-yoshiyasu@aist.go.jp (Y.cs.stanford.edu (L. Guibas).

a b s t r a c t

In this paper, we present symmetry-aware embedding for shape correspondence, which is robust againstsymmetric (left–right) flips and rotational (front–back) flips. Unlike previous embedding approaches thatembed surfaces into a high-dimensional space, our technique is based on a low dimensional (3D)embedding. Our method can solve left–right flips by finding a 3D rigid transformation between twoembedding surfaces without reflections. Using the global reflectional symmetry plane to align twosurfaces, we can further reduce the problem to that of finding the rotation that corrects the signs of thefront–back and up–down directions (there are four possible solutions). We exploit this simple problemformulation and alleviate the front–back flips, by explicitly comparing the front and back of embeddingsurfaces based on the global and local extrinsic shape characteristics. Consequently, reasonably accuratepoint-to-point correspondences can be established simply by performing the nearest neighbor search inour embedding space. Experimental results based on a shape correspondence benchmark showed thatour method produces stable matching results.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

In the last decade, the non-rigid shape correspondence field hasadvanced remarkably and many sophisticated techniques emerged[1–6]. Early approaches gradually deform a surface into another inthe original 3D space [2,3]. The methods based on graph matchingsolves an assignment problem by minimizing intrinsic distortionssuch as discrepancies in geodesic distances. Recent techniques[1,6–8] embed surfaces into a high-dimensional space using spectralembedding techniques, such as Isomap [9] and Laplacian eigenmaps[10], to transform an intrinsic shape matching problem into aneasier extrinsic one. Another class of techniques [5,11] use conformalmaps to evaluate intrinsic distortions.

Although the embedding approach requires the optimization ofa high-dimensional linear transformation only, it introduces a fewchallenges that are difficult to overcome. First, the embeddingapproaches encounter the problem called symmetry (left–right)flips because the reflectional symmetry cannot be distinguished byintrinsic shape characteristics. Second, finding a map (lineartransformation) between two sets of high-dimensional eigenbasis

H. Zhang

Yoshiyasu),

is not an easy problem especially when the surfaces start todeviate from isometry. Under the isometric assumption, a map canbe approximated as a rigid transformation which is close to adiagonal matrix. However, if the surfaces involves non-isomericdeformations, eigenbasis often exhibit ordering flips and multi-plicities, which means that we can hardly predict the structureand patterns of a map. One can imagine this optimization problembecomes harder as the dimension of eigenbasis gets higher.

On the other hand, the approaches based on conformal maps[5] are robust against left–right flips because conformal maps areorientation preserving. The more challenging problem is therotational (front–back) flip. As reported in Blended Intrinsic Maps(BIM) [5], it is difficult to judge front–back flips by evaluatingconformal energy which is not sensitive to the changes in extrinsicshape characteristics (curvatures).

In this paper, we propose a novel shape correspondence tech-nique based on a low-dimensional embedding—symmetry-awareembedding. With the use of a low dimensional embedding, we canmake the solution space of the shape correspondence problemmuch smaller than that of high-dimensional embedding techni-ques. Low-dimensional embedding has been widely employed inthe shape retrieval field. We extend one of the popular lowdimensional embedding techniques, the least squares multi-dimensional scalings (LS-MDS) algorithm [12]. This embedding isorientation-preserving and can avoid left–right flips. We devise a

Fig. 1. Shape correspondence by symmetry aware embedding. Our method embeds surfaces in such a way that they are aligned according to the global reflectionalsymmetry plane (a). Reasonably accurate point-to-point correspondences can be established simply by performing the nearest neighbor search in the embedding space. Notethat our method is robust against left–right and front–back flips (b).

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2210

deformation algorithm that can align and symmetrize thisembedding with respect to the reflectional symmetry plane evenin the presence of imperfect symmetries and missing parts.

Once we embed two surfaces with our symmetry-awareembedding, our task is to align them with each other up to localscales. Since our embedding is symmetrized, the correspondenceproblem is reduced to that of finding the best match from fourpossible solutions i.e., determining the signs of up–down and front–back directions. The front–back flip is thus explicitly tackled duringthis matching phase, by comparing the front and back of embed-ding surfaces based on the global and local extrinsic shape char-acteristics. After alignment, point-to-point correspondences can beestablished using the nearest neighbor search in the embeddingspace (Fig. 1). Experimental results showed that our technique ismore stable than the previous symmetry robust approaches [11,13],producing reasonably accurate correspondences.

The main contributions of this work are summarized asfollows:

� We propose a low-dimensional (3D) embedding approach toshape correspondence. Although accuracy of shape correspon-dence is not as good as that of high-dimensional techniques, ourlow-dimensional approach can produce correspondences morestably. Our method can thus provide a good initial correspon-dences for high-dimensional embedding approaches.

� We provide a solution to alleviate the front–back flip problem.We compare the front and the back of embedding surfacesbased on extrinsic shape characteristic, such as curvatures, todisambiguate the flip.

� We propose an intrinsic symmetry detection technique that isrobust to imperfect symmetries and missing parts. It detects aglobal reflectional symmetry plane in 2D embedding space.

The rest of the paper is organized as follows: Section 2 brieflysummarizes related work. Section 3 reviews and analyzes theprevious embedding approaches, Isomap and LS-MDS. Section 4describes the key idea and Section 5 shows the overview of ourframework. The technique for constructing our symmetry-awareembedding is presented in Section 6. The shape correspondencealgorithm based on our embedding is described in Section 7. Weshow experimental results in Section 8 and conclude in Section 9.

2. Related work

In the following, we will briefly review previous work on shapecorrespondence and symmetry detection. Please refer to well-organized surveys on shape correspondence [14] and symmetrydetection [15].

Deformation-driven framework: Early approaches to non-rigidshape correspondence adopt the deformation-driven strategy. Liet al. [3] and Huang et al. [2] proposed local techniques that

iteratively deform a template model using the as-rigid-as-possibledeformation algorithm. In contrast, Zhang et al. [16] took a globalapproach that selects the best matching result from among allpossible candidates of sparse correspondences, by non-rigidlydeforming the source model and evaluating the distortions ofthe deformed results.

Intrinsic graph matching: Instead of deforming surfaces, manytechniques seek for correspondences that minimize intrinsic dis-tortions of two shapes. This can be formulated as a graph matchingproblem, which is a quadratic assignment problem (QAP). Themain topic here is how to define the distortion measure such thatit can be minimized efficiently while producing accurate corre-spondences. Bronstein et al. [17] used the Gromov–Hausdorffdistance as their distortion measure in their generalized multi-dimensional scaling (GMDS) framework. Mobius voting by Lipmanand Funkhouser [4] uses conformal maps. Kim et al. [5] proposedBlended Intrinsic Maps (BIM) that blends multiple conformal mapsgenerated from sparse correspondences. Recently, a skeleton-based matching technique was proposed by Au et al. [18].

Embedding approach: Embedding approaches obtain two sets ofbasisfunctions for two shapes using such as Isomap [9] andLaplacian eigenmaps [10] and solve a linear assignment problemin the high-dimensional embedding space. Jain and Zhang [1]greedily matched eigenbasis from all possible combinations,ignoring multiplicity. Wuhrer et al. [19] used LS-MDS to improvethe scalability of the spectral embedding method [1], starting fromthe solution of [1] as an initial. Mateus et al. [7] proposed eigen-function signatures for matching eigenbasis, which is the histo-gram of the absolute values of eigenbasis. Ovsjanikov et al. [6]proposed functional maps that establishes correspondences offunctions, instead of point-to-point correspondences, whichimproves robustness of matching in practice. Pokrass et al. [8]extended functional maps and enforced sparsity and diagonalconstraints on a map as a strong isometric regularizer.

Symmetry detection: Symmetry has been studied extensivelyin computer vision and computer graphics for 2D images and 3Dobjects [15]. Podolak et al. [20] proposed a planar reflectionalsymmetry transform, a continuous measure of the reflectionalsymmetry. Mitra et al. [21] proposed a symmetry detectionalgorithm for discovering and extracting partial and approximatesymmetries of 3D geometric models, based on local descriptors.They then proposed a symmetrization technique [22] forenhancing symmetries of the model based on this symmetrydetection technique. Xu et al. [23] focused on partial intrinsicreflectional symmetries. Lipman et al. [24] analyze and representsymmetries in a point set based on their symmetry factoreddistance and embedding. Intrinsic symmetry detection has alsobeen studied in the geometry field. Here, a self map that maps asurface to itself is detected. For example, previous approachessolve this problem based on GMDS [25], Laplacian eigenbasis [26]and Mobius Voting [27].

Fig. 2. Difficulties related to previous embedding approaches. (a) High-dimen-sional embedding exhibits sign flips, ordering switchings and also multiplicitieswhen two shapes are not isometric. For such an eigenbasis pair, a mappingbetween them reveal highly complicated patterns which is difficult to predictbeforehand. (b) The low-dimensional embedding shapes produced by Isomap (thefirst to third eigenbasis) and LS-MDS are not symmetric and twisted. Theseembedding surfaces are difficult to align with PCA and symmetry alignmenttechniques. (c) Our method embeds surfaces in such a way that they are alignedaccording to the reflectional symmetry plane. We only need to select the bestembedding shape from among four candidates. There is no ordering flips andmultiplicities.

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Symmetry-robust approaches: Recently, several correspondencetechniques that are robust against symmetry flips are introduced.Liu et al. [11] used a symmetry axis to establish correspondences,which first matches 1D curves along axes and propagates thematches to the rest of the surface. Ovsjanikov et al. [28] decom-pose basisfunctions into symmetric and asymmetric ones and thenperform matching using them. Yoshiyasu et al. [29] proposed asymmetry-aware local descriptor using local depth images. Zhanget al. [13] proposed a symmetry-robust global descriptor using thegradients of harmonic fields and surface normals, which producessparse correspondences. Sahillioglu and Yemez [30] proposed amethod that selects the matching result without symmetric flipsfrom multiple optimal solutions.

3. Background

Here, we briefly review embedding approaches that are rele-vant to our technique, i.e., Isomap and LS-MDS. We also analyzethe two approaches and describe their challenges when used forshape correspondence. We denote a set of points of the originaldata in an N-dimensional space as an n� N matrix, x¼ ½x1…xn�T .In this paper, we assume N¼3.

Isomap: Isomap [9] is an extension of the classical MDS [31].The classical MDS minimizes the error measure called the strain

that is the discrepancies in the squared Euclid distances betweenthe original data and the embedding. Instead, Isomap uses thesquared geodesic distances. Let Φ¼ ½Φ1…Φn�T be a set of pointsin an M-dimensional embedding space. Let us define the geodesicdistance between point i and j and denote it as gij. Then, the strainis defined as:

EstrainðΦÞ ¼Xio j

wijðJΦi�Φj J2�g2ijÞ2 ð1Þ

where wij is the weight that determines the relative importanceof each distance to the above error measure. The inequality io jindicates the set of the point pairs connecting all of the datapoints. Setting wij to 1, Eq. (1) can be rewritten in the matrixform as:

EstrainðΦÞ ¼ ‖JðD2�Δ2ÞJ‖2Fwhere D and Δ are the distance matrices containing the Eucliddistances of embedding and the geodesic distances of theoriginal data, respectively. Note that Δ is symmetrized by Δij ¼Δji ¼ ðgijþgjiÞ=2 , which is a common procedure. Here J is thecentering matrix whose entry is:

Ji;j ¼1�1=n if i¼ j

�1=n if ia j

(

Using the property of the squared distance matrix, the strain canbe written as:

EstrainðΦÞ ¼ ‖ΦΦT �K‖2F

where K¼ �12JΔ2J. The new point configuration Φ is obtained

by the solution of the eigenvalue decomposition of K:

Φ¼ VΛ12

where Λ is the diagonal matrix containing the M top eigenva-lues, λ1…λM , and V is the corresponding eigenvectors.

Least squares MDS (LS-MDS): Instead of embedding a shape intoa high-dimensional space, LS-MDS [12] transforms the data in theoriginal space. LS-MDS is an extension of the metric MDS [32]where the distance matrix is constructed from geodesic distances.The metric MDS uses an alternative more interpretable errormeasure than the strain, which is called the stress:

EstressðxÞ ¼Xio j

wijðJxi�xj J�gijÞ2 ð2Þ

The stress in Eq. (2) is minimized using the technique callediterative majorization (aka. SMACOF) proposed by De Leeuw et al.[32]. Again, setting wij ¼ 1, the new point configuration ~x isobtained by:

~x ¼ 1nBðxÞx ð3Þ

where

Bi;j ¼gij=Jxi�xj J if ia j�P

kBi;k if i¼ j

8<:

Eq. (3) is iteratively solved until convergence. It is proven that thestress will monotonically decrease after each iteration withSMACOF.

Analysis: Isomap uses spectral decomposition which does notuse a geometric meaningful error measure. It therefore cannotcapture the original intrinsic properties sufficiently with low-dimensional basis only and typically produces the embeddingwith shrinkage (Fig. 2(b)). Thus, recent correspondence techniquesusually use a high-dimensional embedding. However, matchingtwo shapes using a high-dimensional embedding is not straight-forward because of sign flips, ordering switchings and multiplicity

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2212

as illustrated in Fig. 2(a); in particular, when the two models arenot isometric, the map reveals a complicated pattern. In addition,there is the symmetric flip problem i.e., we cannot figure outwhich is the left and right from the embedding itself. In contrast,LS-MDS is more geometric meaningful in that it better preservesintrinsic properties (Fig. 2(b)). Further, this technique isorientation-preserving. Thus it can avoid symmetric (left–right)flips, if such embedding surfaces can be used in shape corre-spondence. Nevertheless, LS-MDS tends to produce asymmetricand twisted embedding surfaces on which PCA (variance-based)and symmetry alignment [20] techniques do not always workcorrectly. Consequently, it is not straightforward to use theembedding constructed by the above techniques in shapecorrespondence.

4. Key idea

To overcome the challenges of the embedding approachesdescribed in the previous section, we propose a method for con-structing a symmetrized low-dimensional embedding that can beused for shape correspondences. We refer to our embedding assymmetry-aware embedding. The main target of our correspon-dence technique is the surfaces with a global reflectional sym-metry. In particular, we are most interested in matching theshapes with thin front–back structures like humanoid models,which are prone to front–back flips.

The key to solving symmetric (left–right) flips is to use a 3Dembedding that is orientation-preserving. We extend LS-MDSsuch that the embedding is aligned according to the globalreflectional symmetry plane and restrict the map of two embed-ding surfaces to rigid transformations (no reflections).

The key to solving rotational (left–right) flips is to explicitlycompare the extrinsic shape characteristics of the front and theback. In order to find the map of two embedding surfaces, we usethe global reflectional symmetry plane to align them. This reducesthe problem to that of finding the rotation that matches the signsof the front–back and up–down directions (Fig. 2(c)). The advan-tage of this simple formulation is that it can compare the front andthe back of two surfaces based on global and local extrinsic shapecharacteristics. Once the signs are matched, reasonably accurate

Fig. 3. Overview

point-to-point correspondences can be established by performingthe nearest neighbor search in the embedding space.

5. Overview

Overview of our algorithm is illustrated in Fig. 3. We divide ouralgorithm into the embedding stage and correspondence stage.

5.1. Embedding stage

The embedding stage consists of an initialization step and threedeformation steps. Note that we do not directly detect thereflectional symmetry plane of the 3D embedding surface gener-ated by LS-MDS, which is asymmetric and twisted, because sym-metry detection on such a surface tends to fail. Instead, we con-struct an initial 2D embedding using Isomap, perform symmetrydetection in 2D and deform the initial embedding to a 3D surface.The first deformation step extends the extremities of the model(Step 1: Extremity extension) in 2D. Since Step1 proceeds in 2Dspace, we can avoid generating twisted shapes in 3D space. Thesecond deformation step recovers the volume of an embedding(Step 2: Volume recovery). Finally, the third step symmetrizes theembedding (Step 3: Symmetrization).

5.2. Correspondence stage

Once we obtain the embedding surfaces, we match the signs ofthe front–back and up–down axes based on the geometric simi-larities computed from global and local shape characteristics.Finally, point-point correspondences are obtained by performingnearest neighbor search in the embedding space.

6. Embedding stage

In the embedding stage, we first construct an initial 2Dembedding based on Isomap and then deform it into 3D embed-ding. We take this two step approach because 3D embeddingsurfaces from LS-MDS are not only asymmetric but also twisted.This prevents us from detecting an accurate symmetry plane andperforming symmetry alignment in 3D. We found that reducing

of algorithm.

Fig. 4. Definition of embedding axes.

Fig. 5. Comparison of the first eigenvalues. Isomap rarely produces eigenbasiscorresponding to repeated eigenvalues for the first several dimensions as opposedto Laplacian eigenbasis, because Isomap eigenbasis are ordered from the basis thatmost captures global intrinsic structure of the original model. Typically, the firstfew eigenvalues (the variances of the first few axes of an embedding surface) aredifferent, if the object is not a sphere.

1 When the model is close to sphere or cylinder (e.g., four legged animal withshort extremities), we experienced the cases with repeated eigenvalues. However, wefound this is attributable to the farthest point algorithm which does not distributesampling points sufficiently to capture the global reflectional symmetry and can besolved by producing multiple initial shapes with different sampling results.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–22 13

one dimension to 2D space and performing symmetry detection inthere works well.

Definition of embedding axes: Fig. 4 illustrates the definition ofthe embedding axes. We define X, Y and Z-axis such that theycorresponds to the direction of the largest variance (up–down),reflectional symmetry (left–right) and thickness (front–back) ofthe embedding, respectively. The reflectional symmetry plane islocated in the middle of Y-axis. Note that there is no semanticmeaning to the signs of these axes; the sign (þ) in the front–backaxis can be “front” or “back”, depending on the reference modelwe pick.

6.1. Generating initial 2D embedding

As our method is an iterative deformation-based technique, it isimportant to provide a good initial shape to construct a well-formed 3D embedding. We demand the reflectional symmetry axisof the initial 2D embedding to roughly align with Y-axis. To gen-erate such a 2D initial embedding, we select the two of the leadingIsomap eigenbasis, arrange them into X and Y-coordinates andperform symmetry detection in 2D to detect the reflectionalsymmetry plane.

To select the two of the eigenfunctions, we follow the globalintrinsic symmetric detection technique based on Laplace eigen-basis [26]. This technique exploits the fact that the Laplaceeigenbasis is orthogonal basis and under isometric assumptiontheir non-repeated basis produce reflectional symmetries only,which simplifies the problem. Note that the eigenfunctions cor-responding to the repeated eigenvalues can introduce rotationalsymmetries in the embedding space, which is difficult to detect.

Here, we provide a theoretical analysis on Isomap eigenbasis toemploy them in symmetry detection. Since pairwise distancematrix K is a real symmetric matrix, Φ is an orthogonal basis ofthe M-dimensional eigenfunction space of eigenvalue λi. Because areflectional symmetry transformation T is isometric mapping,~Φ i ¼ Φð1Þ

i ○T…ΦðMÞi ○T

h iis also an orthogonal basis of the corre-

sponding eigenfunction space of the eigenvalue λi [26,33]. Then,there exists an M-dimensional orthonormal matrix ~Φ i ¼ΦiCwhere CTC¼ CCT ¼ I, (I is an M-dimensional identity matrix). Notein case i is non-repeating, Cii ¼ 71. Consequently, Isomap eigen-basis corresponding to non-repeating eigenvalues can be dividedinto positive and negative:

� ΦðKÞi ○T ¼ΦðKÞ

i : we call such ΦðKÞi positive

� ΦðKÞi ○T ¼ �ΦðKÞ

i : we call such ΦKÞi negative

Our task is therefore to arrange a positive basis in X-coordinateand negative basis in Y-coordinate. To efficiently accomplish this,we select the two basis from among the several leading Isomapeigenbasis. As can be seen from Fig. 5, Isomap rarely produceseigenbasis corresponding to repeated eigenvalues for the firstseveral dimensions, which means that those eigenbasis are notrelated to rotational symmetries.1 We also observed that, for themost objects with a notable global reflectional symmetry, wherethe (geodesic) distance between the left most and the right mostpoints is long enough, at least one negative basis exists in theleading three Isomap eigenbasis. It is therefore highly likely thatwe can find a positive and negative eigenbasis from among theseveral leading eigenbasis. We create six candidates of 2Dembedding from the top four eigenbasis (1st–2nd, 1st–3rd, 1st–4th, 2nd–1st, 3rd–1st and 4th–1st) as in Fig. 6(a). To handle thecase where the first eigenbasis is negative, we also evaluated threeadditional cases where the first basis is arranged into Y-axis.

To evaluate the six candidates, we compute a symmetry scorethat is similar to the symmetry distance [20] which measures thedeviation from the closest symmetric shape. Specifically, we gen-erate a binary image (interior is 1 and exterior is 0) for eachcandidate and its mirrored version. We compute the overlappingarea as a measure of how much the candidate is similar to theclosest symmetric shape; we count the number of pixels in theoverlapping regions (white regions in Fig. 6 (a)) and divide it bythe number of all pixels within the 2D projection.

Accounting for imperfect symmetries: For the surfaces withimperfect symmetries (one leg is longer than the other) or missingparts, which deviates from self isometry, the nonrepeated eigen-function is neither positive nor negative. In this case C becomes ageneral orthonormal matrix, not a diagonal matrix. We found thatsuch imperfect symmetries introduce small perturbations in C,which introduces a tilt of the 2D embedding from the reflectionalsymmetry axis. To account for such a tilt, instead of assuming purereflections, we apply rotations to the 2D embedding so that it canbe aligned with the reflectional symmetry plane. We apply rota-tions by 3° from �45° to 45° to find the most symmetric shapes.For the human model example shown in Fig. 6, the 1st and 3rdbasis with 3° tilted are chosen. Notice how this technique detects

Fig. 6. Initial embedding construction. We create six candidates of 2D embeddingfrom the top four eigenbasis (1st–2nd, 1st–3rd, 1st–4th, 2nd–1st, 3rd–1st and 4th–1st). Because the embeddings are not always properly aligned with the axes as in(b) left, we rotate them by 3° from �45° to 45° to find the most symmetric shapes.This process improves robustness against asymmetry like Armadillo with missingarm and leg ((b) right). The symmetric score (provided at the bottom of eachimage) is measured by counting the pixels of overlapping regions (white), which isthen normalized by the pixels in all projected areas (gray þ white). The 2Dembedding with the highest score is selected (emphasized in the red dashed line).(For interpretation of the references to color in this figure caption, the reader isreferred to the web version of this paper.)

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2214

the reflectional symmetry plane of Armadillo with missing extre-mities (Fig. 6(b)).

6.2. Generating 3D symmetric embedding

In this section, we describe our basic deformation frameworkto construct symmetry-aware embedding: a global-local mini-mization approach to the metric MDS and the subspace MDSalgorithm. We then present the optimization technique and theenergy terms we minimize for constructing a symmetry-awareembedding from the 2D initial embedding.

6.2.1. Basic deformation algorithmGlobal-local approach to MDS: The key observation for trans-

forming LS-MDS into a useful tool for shape correspondence is tointerpret the stress in Eq. (3) as a spring model that is widely usedin cloth simulation [34]. In fact, Eq. (3) is equivalent to the springmodel except that the connectivity is global and the rest lengthsare geodesic distances. Following Liu et al. [35] who recentlypointed out the equivalence between the implicit Euler methodand the global-local optimization approach [36,37], we can furtherarrive at the following equation:

Estress ¼Xio j

wijðxi�xj�gijdijÞ2

where dij is the edge direction dij ¼ ðxi�xjÞ=Jxi�xj J . Roughlyspeaking, this can be thought of as iteratively deforming a surfacesuch that its geodesic paths become straight lines. Note that thiscan be rewritten in the matrix form as:

Estress ¼ ‖Mx�Gd‖2F ð4Þwhere M and G is an edge incidence matrix and a diagonal matrixcontaining geodesic distances, respectively. Here, d is a matrixcontaining edge directions. Eq. (4) is iteratively minimized by theglobal-local optimization technique that alternates two steps: thefirst step optimizes x by fixing d and the second step obtains edgedirections d from the current embedding x.

The above change in the formulation, where we obtain anembedding by iteratively solving least squares optimization, pro-vides us convenient ways to enforce useful constraints such as thesymmetrization constraint.

Subspace multidimensional scaling: To improve efficiency androbustness of deformation, one promising strategy is to use asubspace optimizer e.g., those based on the harmonic coordinatesubspace [38] and PCA subspace [39]. Here, we introduce a sub-space optimizer based on the eigenbasis constructed by Isomap.This is achieved by representing the position x as a linear com-bination of Φ, i.e.:

x¼Φw ð5Þwhere w is a M � 3 matrix, w¼ ½wx;wy;wz� that contains coeffi-cients. With Eq. (5), the problem of constructing a low dimen-sional embedding is transformed into the optimization of w. Wecan derive the subspace MDS energy ESMDS from Estress (Eq. (3)) as:

ESMDSðwÞ ¼ ‖MΦw�Gd‖2F¼ ‖Mw�Gd‖2F ð6Þ

Since the eigenvalues of the Isomap eigenbasis decay rapidly, weneed to optimize the first ten basis only, in practice. In fact, thespectral embedding technique [1] produces reasonable corre-spondence results with the first six basis. Thus, we can expect thatthe size of w is much smaller than that of x, which results in alarge amount of speed up when compared to the original problemthat optimizes x.

6.2.2. 3D embedding optimizationWe gradually deform the initial 2D embedding into a 3D

symmetric embedding by minimizing the energy presented in thefollowing.

Energy terms: To generate a symmetric embedding that isreadily usable in non-rigid shape matching, we minimize Eq. (6)with two constraints, which becomes a combination of the threeenergy terms as follows:

EðwÞ ¼wSMDSESMDS

þwsymEsym

þwnormalEnormal ð7Þ

Fig. 7. Evolution of embedding.

Fig. 8. Matching signs based on the global geometric error. The error is measured with the average of Euclid distances between the extreme points of the reference and thetarget. The rightmost is the correct alignment with the lowest error.

Fig. 9. Diffused mean curvature distribution.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–22 15

where Esym enhances symmetries of an embedding and Enormal

makes the directions of the surface normals outward and keepsthe volume of embedding.

Symmetrization term: To enhance symmetries of the embeddingshape, we incorporate an energy that attracts the points of theembedding toward the closest symmetric shape. To do so, wefollow the symmetrization algorithm proposed by Mitra et al. [22]and calculate the position of the closest symmetric shape as fol-lows. Let p and q be a pair of points that is paired by symmetrydetection. The goal of symmetrization is to displace the pointssuch that the resulting shape gets close to perfect symmetry byminimally altering its shape. Given a reflectional transformation T,the target position is computed by: p0 ¼ ðpþT �1ðqÞÞ=2 andq0 ¼ ðTðpÞþqÞ=2. In our case, because the initial embedding isalready closely aligned with the reflectional axis, it is reasonable toexpect that the reflectional transformation is an inversion of theY-coordinates. Let ~x i be the current embedding position at vertex i.Then, ~x i is mapped by a reflectional transformation to yi ¼ T ~x i,where T¼ diagð½1; �1;1�Þ. Consequently, the point on the closestsymmetric shape corresponding to ~x i is calculated as:

x0i ¼

~x iþyidxðiÞ2

where idxðiÞ indicates the index of the nearest point in y from xi.We can thus define the symmetrization energy as follows:

Esym ¼ ‖x�x0‖2F ¼ ‖Φw�x0 J2F

Surface normal term: To preserve the volume of the embedding,we minimize the surface normal term similar to the balloonenergy used in Snakes [40] as follows:

Enormal ¼ ‖Φw�ð ~xþβnÞ‖2F

where n is a matrix containing vertex normals of the embeddingsurface and β is a scaling factor. Minimization of this energyrecovers the volume. More importantly, it makes surface normalsoutward because we inflate a 2D embedding into a 3D embeddingby pushing the vertices toward outward normal directions.

Optimization: The energy Eq. (7) is minimized with the alter-nating optimization method where the first step optimizes w bysolving the normal equation:

w¼ ðATAÞ�1ATb ð8Þ

Here the system matrix A and the right hand side b is defined asfollows:

A¼wSMDS �Mwsym �Φwnormal �Φ

0B@

1CA;b¼

wSMDS � Gdwsym � x0

wnormal � ð ~xþβnÞ

0B@

1CA

Fig. 10. (a) Intrinsic symmetry detection based on our embedding. (b) Result on amodel with imperfect symmetries. The left arm and leg are longer than the right. Inthis case, Laplacian eigenbasis are neither positive nor negative.

Fig. 11. Qualitative comparison with previous techniques. BIM cannot always dis-tinguish between the front and the back based on their conformal distortionmeasure. Our technique solves the front–back flips based on the sign matchingtechnique that uses global and local extrinsic characteristics. The symmetry-axismatching technique fails on the Armadillo dataset because the left arm of onemodel is missing and the symmetry axis is identified at the wrong region. On theother hand, our technique is robust against such incompleteness. The methodbased on Laplacian eigenbasis suffers from symmetry flips, which implies that thehigh-dimensional spectral embedding approaches [1,6,8] cannot also avoid thisproblem. In contrast, ours can distinguish the left and the right.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2216

Again, the second step computes edge directions d. The embed-ding positions are then obtained by Eq. (5). We use a Choleskysolver provided with CHOLMOD library [41] to solve Eq. (8).

6.3. Algorithm summary, implementation issues and parameters

We compute geodesic distances using the heat diffusion basedtechnique proposed by Crane et al [42]. To obtain Isomapembedding Φ, we perform the Landmark Isomap method with150 sample points obtained using the farthest point samplingtechnique and perform interpolation using the biharmonic inter-polation. We used M¼10 in all steps.

We generate an initial embedding with the method describedin Section 6.1. Although the initial embedding is approximatelysymmetric, it is sometimes prone to self-intersections dependingon the eigenbasis chosen. We observe that the self-intersectionsoccur at around extreme points and therefore we can eliminateoverlapping regions by applying mesh fairing to these regions.Specifically, we use the harmonic embedding technique [43] withposition constraints provided at the points within the 10% ofbounding rectangle diagonal from the centroid. This generates a2D disk like embedding which does not have overlaps ofextremities.

While there are three energy terms in our deformation fra-mework, we set wsym proportional to wnormal. Thus, parametertuning is relatively simple as we need to consider wnormal only. Theparameters are set as follows:

Step 1. Extremity extension: We only incorporated the SMDSterm without the surface normal term so that Step1 proceeds in2D space and can avoid producing twisted shapes in 3D space. We

therefore set the parameters as: wSMDS ¼ 1, wsym ¼ 0 andwnormal ¼ 0. This step typically takes 20 iterations.

Step 2. Volume recovery: We set the parameters as: wSMDS ¼ 1,wsym ¼ 0, wnormal ¼ ϵ and β¼ 0:1. Here we set ϵ in the range ofϵ¼ ½10�100�. We terminate this process when the embeddingsurface does not deform any more, i.e., the average norm of thedisplacements between the embeddings in the previous and cur-rent iterations is below the threshold. This step typically takes 40iterations.

Step 3. Symmetrization: We set the parameters as: wSMDS ¼ 1,wsym ¼ 0:5� ϵ and wnormal ¼ 0. By setting wnormal ¼ 0, we candeflate the embedding surface that has been inflated too much inStep2. This step typically takes 40 iterations.

Fig. 7 shows the evolution of embedding surfaces.

Fig. 12. Comparisons with 2D embedding and 3D LS-MDS embedding. Two-dimensional embedding does not provide global geometry characteristics of thefront and the back. Thus, it cannot solve front back flips. Although, LS-MDS surfacesare aligned according to the reflectional symmetry plane, they are twisted and notsymmetric. Thus, the left arm of Victoria is matched to the right arm of David andthe right hand of the Victoria is matched to the head of David.

Fig. 13. Benchmark result on SCAPE dataset without symmetry maps allowed.

Fig. 14. Benchmark result on Human dataset without symmetry maps allowed.

Fig. 15. Benchmark result on Armadillo (SHREC) dataset without symmetry mapsallowed.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–22 17

7. Correspondence stage

The embedding we obtained so far is symmetrized with respectto the reflectional symmetry axis, but the signs of embeddedsurfaces are not consistent to each other. Fortunately, the sign ofthe Y-coordinate will be determined automatically by restrictingthe transformation to a rigid one, which means that there are onlyfour possible solutions (the signs of the X and Z-coordinates).Point-to-point correspondences can then be obtained using thenearest neighbor searches. We match the signs in two step:matching based on global and local geometric matching that usesextrinsic shape characteristics. Note that we use the global shapefeatures of the embedding surface and the local features of theoriginal surface. If the global matching step outputs an ambiguousresult such as the case where the variance of the Z-coordinates aresmall, like Humanoid, we perform matching based on local

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2218

geometric features. In the following, we refer to the first model asthe reference and the second model as the target.

Matching by global geometric features: To match the signs ofembedding axes, we first use the method based on global geo-metric features. This is achieved by extracting extreme points ofthe embedding where we detect local maxima of average geodesicdistance (AGD). We then find correspondences of these extremepoints between the reference and the target. The target model istested for four orientations where we rigidly transform the targetmodel by applying four different rotations, i.e., diag([1,1,1]), diag([�1,�1,1]), diag([1,�1,�1]) and diag([�1,1,�1]) as shown inFig. 8. The matching error is calculated simply by averaging theEuclid distances between the correspondence pairs of extremepoints. We select the rotation with the smallest global geometricerror (e.g., Fig. 8, rightmost).

Matching by local geometric features: For the case where theglobal geometric measure does not show significant differences(here we set the threshold as ðe1�e2Þ=e1o0:3 where e1 is thesmallest global geometric error and e2 is the second smallestglobal geometric error), such as the case of matching the Z-axis ofhumanoid models, we perform geometric matching based on localgeometric details.

To do so, we compare the curvatures of the reference modeland the target model. We first compute mean curvatures anddiffuse slightly to remove high-frequency components. Becausethe prominent differences in local geometry mostly appear aroundthe symmetry axis, we also attenuate curvatures as their positionsget far from the symmetry axis. We therefore compute the mod-ified curvature (Fig. 9) as:

ci ¼ absðcmi Þ � 1� absðxyi Þ

maxðabsðxyÞÞ

!

where cmi is the diffused mean curvature and xyi is the Y-coordinate

of the embedding at point i.To determine the sign of Z-coordinates, we measure the total

curvature of the front and back facing triangles for the referenceand the target model. To identify the front and back facing trian-gles, we calculate the sign indicator function, which segments the

Fig. 16. Impact of the initial shape on the embedding result. By naively deformingthe initial embedding with overlapping region, it is highly likely that the final resultis distorted by twist. This can be resolved by a harmonic mapping with the con-straints provided at around centroid to eliminate the overlap of the extremities(top). For the model close to cylinder like Pig, the first four eigenbasis sometimescontain repeated eigenvalues, depending on the farthest point sampling used forlandmark Isomap. Initial embedding is therefore constructed without a globalreflectional symmetry. Starting from this initial, the final embedding is producedwith distortions (bottom middle). This problem can be solved by generating mul-tiple 2D embedding candidates from different sampling points and select the mostsymmetric one (bottom right).

embedding into front and back facing regions, from the surfacenormals of the embedding as:

szi ¼ sgnðni � ½0; 0; 1�T Þwhere ni is the surface normal of the embedding at vertex i.Finally, we compute the sign of Z-coordinate as:

Sz ¼ sgnXnszi � ci

!

If Sz is negative, then we flip the Y- and Z-coordinates.Point-to-point correspondence: We perform nearest neighbor

searches in the embedding space to establish point-to-point cor-respondences. We could use the Hungarian algorithm to forcecorrespondences to become one-to-one mappings. Furthermore,high-dimensional embedding techniques such as functional maps[6] or non-rigid regimentation techniques [1] would improve theaccuracy of matching. However, our primary focus here is to showthe ability of our method to produce correspondences stably byalleviating left–right and front–back flips. Thus we limit ourselvesto use the simplest NN search.

8. Results

Embedding results: Figs. 1 and 7 show the examples ofembedded surfaces constructed by our method. In Fig. 7, we showhow the embedding evolves with our technique. Our embeddingresult is symmetric and aligned with respect to its reflectionalsymmetry axis, which makes the subsequent shape correspon-dence process easy.

Intrinsic symmetry detection: Intrinsic symmetry detection is aprocess of finding a self map of a surface. Because our embeddingis already symmetrized, intrinsic symmetry detection can beachieved by inverting the Y-coordinates and performing thenearest neighbor search from the original embedded surface to the

Fig. 17. Impact of energy terms on embedding results and sensitivity toparameters.

Table 1Timings for embedding (in seconds).

Model #Vert Pre Init Embed Total

Human 15,154 7.98 0.87 16.48 25.33Arma. 25,431 12.78 1.28 22.40 36.46

‘#Vert’ indicates the number of vertices. ‘Pre’ indicates the timing for preprocesssuch as computation of geodesic distances. ‘Init’ is the timing for constructing aninitial 2D embedding. ‘Embed’ is the timing for constricting a 3D embedding.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–22 19

flipped version. In Fig. 10(a), we show the results of intrinsicsymmetry detection. Our method is able to detect intrinsic sym-metry of a mesh with missing parts like Armadillo. It is also pos-sible to detect the reflectional symmetry of the model withimperfect symmetries where the left arm and leg are longer thanthe right (Fig. 10(b)).

Shape correspondences: In Fig. 20, we show examples of corre-spondence results obtained using our embedding. As can be seenfrom Fig. 20, our technique robustly handles left–right, front–backflips and non-isometric deformations. The method is applicable toa relatively wide range of datasets, such as humanoids and four-legged animals, and handles imperfect symmetries. Our techniquecan also match man-made objects like chairs and planes.

In Fig. 11, we qualitatively compare our technique with BIM [5],the symmetry-axis (SymAxis) matching technique [11] and themethod that optimizes a map between two sets of Laplacianeigenbasis where we used similar strategy to [8] and enforced adiagonal constraint on a map to achieve isometry. BIM cannotalways distinguish between the front and the back based on theirconformal distortion measure. Our technique alleviates this pro-blem based on the sign matching technique that uses global andlocal extrinsic shape characteristics. SymAxis fails on the Arma-dillo dataset because the left arm of one model is missing and thesymmetry axis is identified at the wrong region. On the otherhand, our technique is robust against such incompleteness. Themethod based on Laplacian eigenbasis suffers from symmetry

Fig. 18. Failure cases. Top: Wrongly matching different extremities. Because ouralgorithm uses a low dimensional embedding, it does not provide sufficientinformation to match many extremities correctly. Middle: Failed symmetrydetection. Since our method relies on the symmetry detection, it fails when thesymmetry detection result fails. Bottom: Our matching method fails on the modelwithout surface details to distinguish between the front and back, like theteddy bear.

flips, which implies that the high-dimensional spectral embeddingapproaches [1,6,8] cannot also avoid this problem. In contrast, ourscan distinguish between the left and the right.

In Fig. 12, we compared our method with 2D and 3D LS-MDSembedding. Two-dimensional embedding does not provide globalgeometry characteristics of the front and the back. Thus, it cannotsolve front–back flips. Although LS-MDS surfaces are aligned withsymmetry alignment [20], they are twisted and not symmetric.Thus, the left arm of Victoria is matched to the right arm of Davidand the right hand of the Victoria is matched to the head of David.In contrast, our method produces reasonable correspondences,which is made possible by the deformation technique that con-structs a 3D symmetry-aware embedding from 2D embedding andthe alignment technique that uses both global and local extrinsicshape characteristics.

Benchmark test: We tested our technique on Surface Corre-spondence Benchmark [5], which include TOSCA [44], SCAPE [45]and SHREC datasets. The graphs shown in Figs. 13–15 are theBenchmark results on SCAPE, Human and Armadillo datasets,where we compare our method with Blended Intrinsic Maps (BIM)

Fig. 19. Imperfect symmetries and shape correspondence results.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–2220

[5], functional maps [6], spectral embedding [1], symmetry robustdescriptor [13] and symmetry-axis matching [11]. These graphsshow percentages of correspondences within certain error ranges.Note that we do not allow symmetric (left–right) maps for thisevaluation.

Our embedding approach with the nearest neighbor pointmatching is stable in that it produces reasonably accurate resultsfor most of the examples; the tails of the graphs reach almost100%. Our algorithm is slightly more successful on the SCAPEdataset than Human dataset. This is because, in the Human data-set, there are some cases where surface details of two surfacessignificantly differs from each other, e.g., one wears clothes with alot of wrinkles and one is naked, which violates the front–backmatching that is based on curvatures.

BIM produces more accurate correspondences than ours but itsometimes cannot avoid major failures such as confusions of thefront and the back. In fact, for SCAPE dataset, the rate of majorfailures (front–back confusions) was 7/71 for BIM, whereas it was3/71 for our method where all of the three cases were attributableto one particular pose. Functional maps is the most accuratemethod under the isometric assumption (Fig. 13). One weakness isthat it is somewhat difficult to generalize the method beyondisometry. The spectral embedding method can handle non-isometric deformations but is prone to symmetry flips. The

Fig. 20. Shape correspondence results. Notice that our met

symmetry-axis matching technique is stable and accurate for theSCAPE dataset where the symmetry axis can be reliably extractedand the shape of the symmetric axis curves do not change dras-tically among the dataset. However, the stability of correspon-dence algorithm degrades as non isometric deformations andimperfect symmetries are introduced in the dataset. The sym-metry robust descriptor can handle imperfect symmetries if theyare insignificant. Our technique is better than their technique inthis respect.

Effect of initial embedding: As our method is deformation-based,the quality of the initial embedding affects the final embeddingresult. In Fig. 16, we show how the initial embedding impacts theresult. For the humanoid models, there are often two negativeeigenbasis among the top four basis and one of them tends to haveoverlapping regions. By naively deforming the initial embeddingwith self-overlapping regions, it is highly likely that the final resultis distorted by twist. This can be resolved by a harmonic mappingwith constraints provided at around the centroid to eliminate theoverlap of the extremities.

For the four legged animal with short legs like Pig, the first foureigenbasis sometimes do not contain negative eigebasis, which isdependent on the result of the farthest point sampling used forlandmark Isomap. Initial embedding is therefore constructedwithout a global reflectional symmetry. Starting from this initial,

hod is not confused by left–right and front–back flips.

Y. Yoshiyasu et al. / Computers & Graphics 60 (2016) 9–22 21

the final embedding is distorted largely. We found that this pro-blem can be solved by generating multiple 2D embedding candi-dates from different sampling points and select the mostsymmetric one.

Energy terms: In Fig. 17, we evaluate the impact of each term onthe embedding result of the female model in Fig. 7. It is obviousthat the surface normal term is needed to recover volumes.Without the symmetry term, the embedding shape is moreasymmetric than the one obtained with it.

Sensitivity to parameters: To analyze the sensitivity of ouralgorithm to parameter values, we varied wnormal and checked howit affects the embedding result. We used the human, giraffe andarmadillo model in this test. Fig. 17 shows the RMS errors inembedding positions with respect to the result obtained usingwnormal ¼ 50, which is normalized by the bounding box diagonal.The RMS errors are below 1% in the range of wnormal ¼ ½10�100�,which means that our algorithm is fairly robust against the para-meter changes.

Timings: We implemented our algorithm with Matlab andCþþ on an Intel Core i7 3.4 GHz 64-bit workstation. The timingsof our method is shown in Table 1. To obtain the point-to-pointcorrespondences of two surfaces with around 15k vertices, it takesapproximately 60 s, which includes the embedding of two sur-faces, matching the signs of embedded surfaces and the nearestneighbor search.

Failure cases and limitations: Fig. 18 shows failure cases of ourtechnique. Our low dimensional embedding approach is not suf-ficient to match model with many extremities. The Ant models areoverall aligned correctly, but their extremities are matchedwrongly (Fig. 18, top). Since our method relies on symmetrydetection, it fails when the symmetry detection result is notaccurate enough (Fig. 18, middle). Our matching method fails onthe model without sufficient surface details to distinguishbetween the front and back, like the teddy bear (Fig. 18, bottom).

In Fig. 19, we have also evaluated how much our algorithm isrobust against imperfect symmetries. We tested the model withone leg missing, one leg and arm missing and half of the bodymissing. Our algorithm can handle the first case but beyond that isdifficult to handle. This experiment and the Armadillo results(Fig. 15) imply that our algorithm is not sensitive to imperfectsymmetries as long as its global reflectional symmetry structure issimilar to that of the original, e.g., two legs or arms of the humanmodel remain.

Our method is limited to match the models with globalreflectional symmetries and cannot handle other symmetry clas-ses like rotational symmetries and repetitive structures. This lim-itation could be overcome by using the symmetry factoredembedding method [24] that can discover various symmetrystructures automatically. Finally, our method cannot match topo-logically different objects.

9. Conclusions

We presented a shape correspondence algorithm based on alow-dimensional embedding that is aligned and symmetrizedwith respect to the reflectional symmetry axis. Our embeddingreduces the shape correspondence problem to that of selecting thebest map from four possible solutions (front–back � up–down).This formulation allows us to compare the front and back ofembedding surfaces based on extrinsic shape characteristics andto establish correspondences stably, by alleviating front-back flips.

Appendix A. Supplementary material

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.cag.2016.07.002.

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