Pattern Recognition Letters 31 (2010) 1650–1657

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Variational shape matching for shape classification and retrieval

Kamal Nasreddine a,*, Abdesslam Benzinou a, Ronan Fablet b

a Ecole Nationale d’Ingénieurs de Brest, Laboratoire RESO, 29238 Brest, Franceb Telecom Bretagne, LabSTICC, 29238 Brest, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 February 2009Available online 19 May 2010Communicated by G. Borgefors

Keywords:Shape classificationShape retrievalContour matchingShape geodesicsMulti-scale analysisRobustness

0167-8655/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.patrec.2010.05.014

* Corresponding author. Fax: +33 298056689.E-mail addresses: nasreddine@enib.fr (K. Na

(A. Benzinou), ronan.fablet@telecom-bretagne.eu (R. F

In this paper we define a multi-scale distance between shapes based on geodesics in the shape space. Theproposed distance, robust to outliers, uses shape matching to compare shapes locally. The multi-scaleanalysis is introduced in order to address local and global variabilities. The resulting similarity measureis invariant to translation, rotation and scaling independently of constraints or landmarks, but constraintscan be added to the approach formulation when needed. An evaluation of the proposed approach isreported for shape classification and shape retrieval on the part B of the MPEG-7 shape database. The pro-posed approach is shown to significantly outperform previous works and reaches 89.05% of retrievalaccuracy and 98.86% of correct classification rate.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction and related work

This work addresses the definition of a robust distance betweenshapes based on shape geodesics. The proposed distance is appliedto shape classification and shape retrieval. Recently, computer vi-sion has extensively studied object recognition and known signifi-cant progress, but current techniques do not provide entirelysignificant solutions (Veltkamp and Hagedoorn, 2001; Daliri andTorre, 2008).

Regarding shape analysis and classification, similarity measuresmay be defined from information extracted from the whole area ofthe object (region-based techniques) (Kim and Kim, 2000), or fromsome features which describe only the object boundary (boundary-based techniques) (Costa and Cesar, 2001). The latter category mayalso comprise skeleton description (Lin and Kung, 1997; Sebastianand Kimia, 2005). Skeleton description of shapes has a lowersensitivity to articulation compared with boundary and region-descriptions, but it is with the cost of higher degree of computa-tional complexity due to tree or graph matching (Sebastian et al.,2003; Sebastian and Kimia, 2005). On the other hand, boundary-based object description is considered more important thanregion-description because an object’s shape is mainly discrimi-nated by the boundary. In most cases, the central part of objectcontributes little to shape recognition.

ll rights reserved.

sreddine), benzinou@enib.frablet).

The boundary-based approach described in this paper is estab-lished on a comparison between matched contours. Contourmatching has been already widely applied to object recognitionbased on shape boundary (Diplaros and Milios, 2002). Two majorclasses of techniques can be distinguished: those based on rigidtransformations, and those based on non-rigid deformations(Veltkamp and Hagedoorn, 2001). Methods of the first type searchoptimal parameters which align feature points assuming that thetransformation is composed of translation, rotation and scalingonly. They may lack accuracy. Methods based on elastic deforma-tions rely on the minimization of some appropriate matching crite-rion. They may present the drawback of asymmetric treatment ofthe two curves and in many cases lack of rotation and scalinginvariance (Veltkamp and Hagedoorn, 2001). Existing techniquestypically take advantage of constraints specific to the applicationsor use shape landmarks. These points are generally defined as min-imal or maximal shape curvature (Del Bimbo and Pala, 1999;Super, 2006), as zero curvature (Mokhtarian and Bober, 2003), ata distance from specific points (Zhang et al., 2003), on convex orconcave segments (Diplaros and Milios, 2002), or any other criteriasuitable to involved shapes.

Shape analysis from geodesics in shape space has emerged as apowerful tool to develop geometrically invariant shape comparisonmethods (Younes, 2000). Using shape geodesics, we can state thecontour matching as a variational non-rigid formulation ensuringa symmetric treatment of curves. The resulting similarity measureis invariant to translation, rotation and scaling independently onconstraints or landmarks, but constraints can be added to the

1 The regularization term is considered in order to obtain a smooth transformationfunction.

K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657 1651

approach formulation when available. This paper is an extension ofthe work presented in (Younes, 2000) to the task of shape classifi-cation and the task of shape retrieval.

The following is a summary list of the contributions of ourwork:

– Geodesics in shape space have been introduced to developefficient shape warping methods (Younes, 2000). Recently,we have exploited the corresponding similarity measure todefine a new distance for shape classification and applied itto marine biological archives (Nasreddine et al., 2009b;Nasreddine et al., 2009a). This distance takes advantage oflocal shape features while ensuring invariance to geometrictransformations (e.g. translation, rotation and scaling). To dealwith local and global variabilities, we derive here a newmulti-scale approach proposed for shape classification andshape retrieval.

– We establish the gain of the proposed method over state-of-artmethods for shape classification and shape retrieval. The test iscarried out on a complex shape database, the part B of theMPEG-7 Core Experiment CE-Shape-1 data set (Jeannin andBober, 1999). This database is the largest and the most widelytested among available test shape databases (Daliri and Torre,2008).

The subsequent is organized as follows. In Section 2 is detailedthe proposed framework for shape matching in the shape space,from where a robust similarity measure between two shapes is ta-ken. We discuss in Section 3 the benefit of the proposed similaritymeasure on shape matching performances. Sections 4 and 5 de-rive a multi-scale distance proposed for shape classification andshape retrieval. In Section 6 we evaluate the proposed distancefor shape classification and shape retrieval for part B of theMPEG-7 shape database and we compare results to other state-of-art schemes.

2. Proposed contour matching

In this paper a boundary-based approach is considered. Thecomparison between shapes is based on a similarity measure usingshape geodesics. The proposed similarity measure is applied toshape classification and retrieval. A multi-scale analysis is per-formed to take into account both local and global differences inthe shapes.

2.1. Shape geodesics

There are various ways to solve for shape matching problem,and many similarity measures have been proposed in the case ofplanar shapes (Veltkamp, 2001). Shape geodesics have emergedas a powerful tool to develop geometrically invariant shape com-parison methods (Younes, 2000). Shapes are considered as pointson an infinite-dimensional Riemannian manifold and distances be-tween shapes as minimal length geodesic paths. Retrieving thegeodesic path between any two closed shapes resorts to a match-ing issue with respect to the considered metric. Let us consider twoshapes C and eC locally characterized by the angle between the tan-gent to the curve and the horizontal axis (h and ~h respectively). Fol-lowing (Younes, 2000), the matching issue is stated as theminimization of a shape similarity measure given by:

SMC;eCð/Þ ¼ 2arc cosZ

s2½0;1�

ffiffiffiffiffiffiffiffiffiffiffi/sðsÞ

pcos

hðsÞ � ~hð/ðsÞÞ2

����������ds; ð1Þ

where s refers to the normalized curvilinear abscissa defined on[0,1], / is a mapping function that maps the curvilinear abscissa

on C to the curvilinear abscissa on eC and /s ¼ d/ds . The similarity

measure considered here includes a measure of the differencebetween the two orientations h and ~h, cos hðsÞ�~hð/ðsÞÞ

2

� �, and a term

that penalizes the torsion and stretching along the curve, ðffiffiffiffiffiffiffiffiffiffiffi/sðsÞ

pÞ.

Curve parametrization via angle function h(s) naturally leads toa representation which complies with the expected invarianceproperties (translation and scaling). A translation of the curvehas no effect on h, and an homothety has no effect on the normal-ized parameter s. Thus curves modulo translation and homothetywill be represented by the same angle function h(s). A rotation ofangle c transforms the function h(s) into the function h(s) + c mod-ulo 2p. For rotation invariance, the minimization of SMC;eCð/Þ overall choices for the origins of the curve parameterizations isconsidered.

2.2. Robust variational formulation

Given two shapes C and eC respectively encoded by h(s) and ~hðsÞ,the matching problem comes to the registration of two 1D signals(Nasreddine et al., 2009b; Nasreddine et al., 2009a). The registra-tion consists in retrieving the transformation that best matchespoints of similar characteristics. Formally, it resorts to determiningthe transformation function /(s) such that hðsÞ ¼ ~hð/ðsÞÞ. Here, thisissue is stated as the minimization of an energy EC;eCð/Þ involving adata-driven term, EC;eC

D , that evaluates the similarity between thereference and aligned signals and a regularization term,1 ER.

EC;eCð/Þ ¼ ð1� aÞEC;~CD ð/Þ þ aERð/Þ; ð2Þ

ERð/ðsÞÞ ¼Z

s2½0;1�j/sðsÞj

2 ds; ð3Þ

where a is a variable that controls the regularity. From time causal-ity, the minimization of EC;eCð/Þ has to be carried out under the con-straint /s > 0.

To ensure more robustness against outliers, we have introduceda robust criterion as a modification of the similarity measure is-sued from shape geodesics (Nasreddine et al., 2009b). Using a ro-bust estimator q, the shape registration issue resorts then tominimizing:

EC;eCð/Þ ¼ ð1� aÞEC;eCD ð/Þ þ a

Zs2½0;1�

/sðsÞj j2 ds;

EC;eCD ð/Þ ¼ arc cos

Zs2½0;1�

ffiffiffiffiffiffiffiffiffiffiffi/sðsÞ

pcos

qðrðsÞÞ2

��������ds; ð4Þ

where rðsÞ ¼ hðsÞ � ~hð/ðsÞÞ. Several forms of the robust estimator qwere proposed (Black and Rangarajan, 1996). We will use the Lecl-erc estimator given by:

qðrÞ ¼ 1� expð�r2=ð2r2ÞÞ; ð5Þ

with r is the standard deviation of data errors r.

2.3. Numerical implementation

To solve for the minimization of EC;eCð/Þ, two methods are con-sidered: dynamic programming and an incremental scheme.

A dynamic programming algorithm is applied as follows. Givena discretisation step and the discretized vectors hðsiÞi¼1::N and~hð~sjÞj¼1::M , the algorithm considers in the plane ½s1; sN � � ½~s1;~sM � thegrid G which contains the points p = (x,y) such that either x = si

and y 2 ½~s1;~sM �, or y ¼ ~sj and x 2 ½s1; sN�. We fetch a continuousand increasing matching function that is linear on each portion

Fig. 1. Test on synthetic shapes. We have applied a known transformation (1(b)) on the shape of 1(a) to get the shape 1(c). s and ~s are the normalized curvilinear abscissas onthe curves.

Fig. 2. Results of shape matching on synthetic contours depicted in Fig. 1 using the dynamic programming for different values of a 2 ½0;1�.

1652 K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657

K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657 1653

that does not cut the grid. The value of the energy EC;eCð/Þ iscalculated at each point of the grid depending on the values at pre-vious points, and the minimum is chosen. This procedure is iter-ated over all choices for the origins of the curves. This algorithmis more detailed in (Trouvé and Younes, 2000).

As an alternative, we have proposed an incremental iterativeminimization (Nasreddine et al., 2009b), which is shown to be

Fig. 3. Test on synthetic shapes. Occluded shape obtained from the shape Fig. 1(c).

Fig. 4. Results of shape matching using the iterative scheme for different values of a 2Fig. 1(a). s and ~s are the normalized curvilinear abscissas on the curves.

computationally more efficient than the dynamic technique inthe case of registration without landmarks (see Section 3 for com-parison). At iteration k, given current estimate /k we solve for an

incremental update: /kþ1 ¼ /k þ d/k such that d/k ¼ argmind/EC;~C

ð/k þ d/Þ. The initialization of the algorithm is given by the iden-tity function taken in turn for all choices for the origins of thecurves. For each of these initializations, the algorithm iteratestwo steps:

1. the computation of the robust weights xki issued from the line-

arization of the Leclerc estimator as xki ¼ 2

r2 expð�r2ðsiÞr2 Þ (Black

and Rangarajan, 1996),2. the estimation of d/k ¼ fd/kðsiÞg as successive solutions of

the linearized minimization d/k ¼ argmind/

PiE

ki . The key

approximation of this linearization is: ~hð/kþ1Þ ¼ ~hð/k þ d/kÞ �~hð/kÞ þ ~hsð/kÞ � d/k. For a = 0, the equation we obtain does nothave a unique solution. The resulting d/kðsiÞ for a – 0 is givenby:

�0;1�. We register here the occluded shape of Fig. 3 with respect to the reference

2 Defi

1654 K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657

d/kðsiÞ ¼NðsiÞDðsiÞ

; ð6Þ

gðsiÞ ¼ ð1� aÞsinxk

i rðsiÞ2

� �½~hð/kðsiÞÞ � ~hð/kðsi�1ÞÞ�;

NðsiÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/kðsiþ1Þ � /kðsi�1Þ

qgðsiÞ cos

xki rðsiÞ2

� �þ 2a 2/kðsiÞ � /kðsi�1Þ � /kðsiþ1Þ

h�d/kðsi�1Þ � d/k�1ðsiþ1Þ

i;

DðsiÞ ¼12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/kðsiþ1Þ � /kðsi�1Þ

qg2ðsiÞ � 4a:

Fig. 5. Results of shape matching. Aligned shapes by the robust and non robustalgorithms; the reference shape is given in Fig. 1(a) and the shape to be aligned inFig. 3.

Table 1Optima MSEh obtained by the robust and the non robust algorithms with the gain dueto the robust solution for initializations of / at points which are far from the correctsolution from different angles (correct solution corresponds to angle = 0). Thisexperiment is carried out on synthetic shapes given in Fig. 1.

Angle (�) MSENonRobusth MSERobust

h Gain ¼ MSENonRobusth �MSERobust

h

MSENonRobusth

� 100 (%)

35 0.293 0.087 70.3045 8.66 0.089 98.9790 0.296 0.085 71.28135 1.78 0.086 95.17

3. Shape matching performances

To study the influence of the robust criterion and of the regular-ization term, we evaluate here the matching process for syntheticcontours (one contour is obtained by applying a known transfor-mation to the other one). Some examples of these synthetic shapesare given in Fig. 1 with a representation of the used transformationfunction /.

In Fig. 2 we report the mean square error MSEh ¼ Eðjh� ~hð/Þj2Þobtained for different values of a 2 ½0;1�. This result is issued fromthe dynamic programming algorithm. For high values of a, theregularity term is favored over the similarity measure and thealignment results in high MSEh values. For small values of a, therobust algorithm ensures solutions with smaller errors (MSEh ¼0:085) corresponding to MSE/ ¼ Eðj/applied � /estimatedj

2Þ � 0:001.The gain2 due to the robust solution is represented in Fig. 2; this gainis optimum for a = 0 and reaches 90%. The aligned shapes given inFig. 2c and d show the superiority of the robust solution. The consis-tency of this result has been verified by testing many transformationfunctions with different shapes.

Using the incremental iterative scheme, the minimization leadsto the same optimum as the dynamic programming except fora = 0. For the iterative scheme the regularity term is necessary, ashould have a nonzero value to lead to a unique solution. Experi-mentally, a value of a in the range [0.1, 0.2] is optimal.

In Fig. 3, we report another test for a synthetic shape obtainedby applying an occlusion on the shape given in Fig. 1c. The resultsof its matching to the reference shape given in Fig. 1a are reportedin Figs. 4 and 5. We see that the robust algorithm is more robustagainst the occlusion, it is still able to align the curves and to re-trieve the applied transformation with minor errors. The transfor-mation estimated by the non robust algorithm (Fig. 4b) is incontrast far from the real one (Fig. 1b).

The relevance of the robust solution is even more visible whenwe analyze the evolution of the incremental algorithm through theinitializations in turn for all choices for the origins of the curves.We report in Table 1 matching results for initialization far fromthe correct solution, we notice that with the robust criterionMSEh decreases through iterations to attain the optimum. In con-trast MSEh values remain greater when the non robust criterion isused and only a local minimum is reached. These experimentsshow that this criterion is robust to the initialization of the choiceof the origins of the curves. Hence, only one arbitrary initializationmay be considered in practice.

Regarding computational complexity, the incremental methodis also more efficient when shape matching with no landmarks isaddressed. The dynamic programming needs a relatively longertime. For example, for the synthetic contours considered in Fig. 1,this time reaches 9.7 times that required by the robust iterativescheme.

ned as: MSENonRobusth �MSERobust

h

MSENonRobusth

� 100.

4. Distance-based shape classification

In this section, we exploit shape geodesics for shape classifica-tion. The alignment cost used in Eq. (4) is taken as the similaritybetween any two shapes. On the basis of a general algebraic andvariational framework, (Younes, 2000) has proved that the con-structed cost function meets all the conditions necessary for a truedistance between planar curves.

Formally, the distance between two shapes S1 and S2 is definedas:

dðS1; S2Þ ¼ ES1 ;S2D ð/�Þ where /� ¼ argmin

/ES1 ;S2 ð/Þ: ð7Þ

In this work, a multi-scale characterization is issued from thecombination of shape matching costs at different scales. Here,the scale is defined as the resolution of shape sampling, as in (Attal-la and Siy, 2005).

In order to exploit local and global variabilities, the distanceused for shape comparison is a combination of distances measuredat different scales. Formally, the distance between shapes S1 and S2

is defined as follows:

dðS1; S2Þ ¼1N

XN

k¼1

dkðS1; S2Þ; ð8Þ

where dk is the distance defined in Eq. (7) between the same shapesat the kth scale and N the number of considered scales.

Assuming we are provided with a set of categorized shapes,(Sl,Cl), where Sl is the shape of the lth sample in the databaseand Cl its class, the classification of a new shape S may be issuedfrom a nearest neighbor criterion.

Fig. 6. Examples of shapes that are visually dissimilar from other samples of their own class.

Fig. 7. Examples of pair of shapes issued from different classes but highly similar.

K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657 1655

5. Distance-based shape retrieval

In addition to shape classification performance, we also addressshape retrieval (Del Bimbo and Pala, 1999). A retrieval problem con-sists in determining which shapes in the considered database arethe most similar to a query shape. The classification accuracy of ashape descriptor does not necessarily give a relevant guess of theretrieval efficiency (Kunttu et al., 2006). As for classification, thedistance used for shape retrieval is the distance defined in Eq. (8).

6. Comparison to other schemes

To compare the proposed approach to the state-of-the-art shaperecognition approaches, we proceed to an evaluation of shape clas-sification and retrieval performances on the part B of the MPEG-7shape database (Jeannin and Bober, 1999). This database is com-posed of a large number of different types of shapes: 70 classesof shapes with 20 examples of each class, for a total of 1400 shapes.The classes include natural and artificial objects. The shape recog-nition on this database is not simple because elements present out-liers so that some samples are visually dissimilar from othermembers of their own class (Fig. 6). Furthermore, there are shapesthat are highly similar to examples of other classes (Fig. 7).

We do not discuss edge detection here; it is an obvious step inimage analysis. The dataset of shape outlines are issued from anautomated extraction of the outlines using the Matlab image pro-cessing toolbox.3

With a view to being invariant to flip transformation, the opti-mal matching between two shapes results from Eq. (4) wherematching costs are computed between the first shape and the sec-ond one flipped or not.

Fig. 8. The correct classification rate (in %) on the MPEG-7 shape database versusthe values of a (a is the coefficient that controls the regularity of the solution).3 Website: http://www.mathworks.com/products/image/.

Shape representation is given by points equally sampled alongthe boundary. Shape sampling at different scales with 32, 48, 64and 192 points is considered.

Classification rates are issued from the leaving one out methodwhere each shape in turn is left out of the training set and usedas a query image. Retrieval accuracy is measured by the so-calledBull’s eye test (Jeannin and Bober, 1999): for every image in thedatabase, the top 40 most similar shapes are retrieved. At most20 of the 40 retrieved shapes are correct hits. The retrieval accu-racy is measured as the ratio of the number of correct hits of allimages to the highest possible number of hits which is 20 � 1400.

Table 2Recognition accuracy measured as nearest neighbor classification rate and retrieval accuracy measured by the bull’s eye test on the MPEG-7 shape database.

Aspect Method Retrieval accuracy Classification rate

Single-scale approachesGlobal schemes Skeleton DAG (Lin and Kung, 1997) 60% NA

Multilayer eigenvectors(Super, 2006) 70.33% NAElliptic FD (Nixon and Aguado, 2007) NA 82%Zernike moments (Kim and Kim, 2000) 70.22% 90%

Local schemes

Matching based Shape context (Belongie et al., 2002) 76.51% NAParts correspondence (Latecki, 2002) 76.45% NACurve edit distance (Sebastian et al., 2003) 78.17% NAInner-distance shape context (IDSC) (Ling and Jacobs, 2007) 85.40% NARacer (Super, 2003) 79.09% 96.8%Normalized squared distance (Super, 2003) 79.36% 96.9%Fixed correspondence (Super, 2006) 80.78% 97%Fixed correspondence + Chance probability functions (Super, 2006) 83.04% 97.2%Fixed correspondence + aggregated-pose chance probability functions (Super, 2006) 84% 97.4%Proposed scheme (64 points) 85.7% 95.05%

Multi-scale approachesGlobal schemes Multi-scale Fourier Descriptors 2D (Direkoglu and Nixon, 2008) NA 95.5%Local schemesOther criteria Wavelet (Chuang and Kuo, 1996) 67.76% NA

Curvature Scale Space (Mokhtarian, 1996) 75.44% NAOptimized CSS (Mokhtarian and Bober, 2003) 81.12% NA

Matching based Shape tree (Felzenszwalb and Schwartz, 2007) 87.7% NAHierarchical procruste matching (McNeill and Vijayakumar, 2006) 86.35% 95.71%String of symbols (Daliri and Torre, 2008) 85.92% 98.57%Proposed scheme 89.05% 98.86%

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As mentioned in Section 3, the best shape matching in term ofmean square error is obtained for a = 0.1. The results of shape clas-sification carried out on this database do not change significantly(±0.01%) by taking a in the range [0.05,0.2]. Note that the valueof a intervenes in the process of convergence of the shape match-ing and not in the expression of the distance of Eq. (8). In Fig. 8 wereport the variation of the correct shape classification rate with re-spect to a.

The proposed approach based on shape geodesics has beencompared to state-of-the-art schemes for part B of the MPEG-7dataset as reported in Table 2. Methods are categorized accordingto single-scale versus multi-scale and local versus global ap-proaches. By global, we refer here to methods such that the shapedescriptors hold information from all points along the shape (e.g.,Fourier methods, Zernike moments) in contrast to techniquesexploiting local shape features such as matching-based or wave-let-based schemes.

The proposed multi-scale approach outperforms reportedschemes with a correct classification rate of 98.86% correspondingto a gain in term of correct classification rate between 0.3% and17%. Regarding the bull’s eye, a score of 89.05% is reached. This isgreater by 1.35% than the best result reported previously. The high-est scores of previous works are those of methods based on shapematching and/or with hierarchical analysis (shape tree, hierarchi-cal procruste matching, string of symbols, IDSC, fixed correspon-dance with chance probability functions); this fact justifies thechoices operated to develop the proposed approach which relieson shape matching coupled with a multi-scale analysis.

From the results reported in Table 2, one may analyze the per-formances of the different categories of techniques. Performancescomparison between the single-scale and the multi-scale ap-proaches shows clearly that multi-scale analysis is very relevant.The single-scale approches reach an average rate of correct classi-fication of 94.04% and an average retrieval rate of 77.62% to becompared respectively to 97.16% and 81.91% for the multi-scaleapproaches. The performances of the method presented in this pa-per are improved by 3.81% in correct classification rate and by

3.35% in retrieval score when considering a multi-scale analysis in-stead of its single-scale form. The gain both in classification and re-trieval performances clearly state the relevance of the multi-scaleapproach for shape analysis.

Global methods are greatly outperformed by local schemes: forinstance for a single-scale analysis, 86% versus 96.73% and 66.85%versus 80.85% for the mean correct classification and retrieval ratesrespectively for the global techniques and local ones. The later canbe argued to provide more flexibility to exploit local shape differ-ences. As expected, a similar conclusion holds when comparingmulti-scale global and local schemes. It may also be noted thatmatching-based schemes also depict greater performances thanother local approaches (e.g., for multi-scale ones, 97.1% and87.26% versus 95.5% and 74.77% for the mean correct classificationrate and mean retrieval rate respectively).

Compared to the other matching-based approaches, the gain re-ported for our approach may be associated with two main features.Before all, these results stress the relevance of the chosen shapesimilarity measure encoding geometric invariance to translation,rotation and scaling. The second important property, often not ful-filled by matching-based schemes, is the symmetry of the similar-ity measure, i.e. the measure of the similarity between shape 1 andshape 2 is the same than between shape 2 and shape 1. This prop-erty is guaranteed by the fact that the matching is stated as a min-imal path issue in the shape space. Regarding our multi-scalestrategy, we proceed similarly to Daliri and Torre (2008), the mul-ti-scale similarity measure is a mean over several scales. In previ-ous works (McNeill and Vijayakumar, 2006; Felzenszwalb andSchwartz, 2007), the multi-scale analysis comes up through theshape matching process where the shape matching at a given res-olution depends on all matchings performed at lower resolutions.

We further analyze the proposed multi-scale matching-basedscheme for object classes depicted in Fig. 9 for which a lowerretrieval accuracy is reported. These shapes within these classesare highly similar, the local curvature differs in a small numberof points only. Experimentally we notice that the use of the robustcriterion leads to consider these data points as outliers. For

Fig. 9. Examples of shapes from different classes with high similar curvature.

K. Nasreddine et al. / Pattern Recognition Letters 31 (2010) 1650–1657 1657

example, if we focus on the nearest 20 neighbors of the samples ofthe class spoon, more than 50% are elements of the classes watch,pencil, key and bottle; if we use the similarity measure withoutthe robust weights, 95% of the nearest 20 neighbors are of the sameclass, spoon. Using robust weights, the average retreival accuracy ispenalized due to the low accuracies obtained for these 6 classes,but overall it remains greater than without the use of the robustweights.

Future work will explore the combination of the proposed ap-proach to kernel-based statistical-learning. Recently, in (Yanget al., 2008) authors propose to combine classical metrics to learn-ing through graph transduction. It has been shown that this ap-proach yields significant improvements on retrieval accuracies.For example, the retrieval rate using the IDSC (Ling and Jacobs,2007) is improved by 5.6% when combined to the learning graphtransduction. We will focus on the combination of machine learn-ing techniques such as random forest and SVMs to the proposedmulti-scale matching-based similarity measure.

Acknowledgement

The authors thank Jean Le Bihan for fruitful discussions.

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