Pattern Recognition Letters 30 (2009) 1310–1320

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Pattern Recognition Letters

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

Affine-invariant contours recognition using an incremental hybridlearning approach q

A. Bandera a,*, R. Marfil a, E. Antúnez b

a Grupo ISIS, Dpto. Tecnologı́a Electrónica, Universidad de Málaga, Spainb PRIP, Vienna University of Technology, Austria

a r t i c l e i n f o

Article history:Received 31 August 2008Received in revised form 24 May 2009Available online 18 June 2009

Communicated by M. Kamel

Keywords:Planar shape recognitionAdaptive curvature functionIncremental analysis

0167-8655/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.patrec.2009.06.004

q This work has been partially supported by the SpInnovación (MICINN) and FEDER funds Project No. TINde Andalucía Project No. P07-TIC-03106, and by theGrants P18716-N13 and S9103-N13.

* Corresponding author. Tel.: +34 5 213 2845; fax:E-mail addresses: ajbandera@uma.es, bandera@dte

a b s t r a c t

In this paper, a planar shape recognition system is proposed. This proposal is based on a global incremen-tal scheme which combines two learning mechanisms: the Incremental Non-parametric DiscriminantAnalysis and the mode analysis method. At the feature selection stage, a novel adaptive curvature esti-mator for shape characterization is presented. This method describes the planar shape using an affine-invariant triangle-area representation obtained from its closed contour. Contrary to previous approaches,the triangle side lengths at each contour point are adapted to the local variations of the shape, removingnoise from the contour without missing relevant points. In order to reduce the dimensionality of theshape descriptor, an Incremental Non-parametric Discriminant Analysis is conducted to seek directionsfor efficient discrimination (incremental eigenspace learning). At the classification stage, the incrementalmode analysis is employed to classify feature vectors into a set of spherically-shaped groups (incrementalprototype learning). The classification is conducted based on the k-nearest neighbor approach whose pro-totypes are updated by the mode analysis method. This scheme enables a classifier to learn incremen-tally, on-line, and in one-pass. Experimental results show that the proposed shape recognition systemis well suited for shape indexing and retrieval.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Pattern recognition is an important task in computer visionwhich plays a crucial role in a large number of applications suchas image retrieval or object recognition. Particularly, the increasingnumber of applications relying on multimedia databases has moti-vated that object representation and classification will be the sub-ject of much research. Among other image features which are usedto achieve this goal, like colour or texture, shape is commonly con-sidered the most promising tool to represent and identify objects(Alajlan et al., 2007). Shape description refers to the process of pre-senting the shape in a suitable format for storage and matching(Marji and Siy, 2003). One of the most typically used shape descrip-tors is the shape contour, and different methods for representing ithave been proposed (Loncaric, 1998). Among them, this paper is fo-cused on those techniques which attempt to represent shapesusing the curvature of their outer boundaries.

ll rights reserved.

anish Ministerio de Ciencia e2008-06196 and by the JuntaAustrian Science Fond under

+34 5 213 1447..uma.es (A. Bandera).

Object recognition also constitutes one of the typical examplesof real-world applications where a complete set of training sam-ples cannot be usually provided in advance when building a classi-fier. For instance, there exists a recent interest in the mobilerobotics community for using object detection and recognition ap-proaches to provide natural landmarks for the sake of simulta-neous robot localization and environment mapping (Asmar et al.,2006). In this framework, objects are detected little by little andthe properties of the real scenario where they are acquired couldbe slightly changed as time passes. Therefore, the learning of a sys-tem must be also conducted sequentially in an on-line manner. Onthe other hand, it is desirable that the human supervisor only pro-vides training samples to the robot when it does not correctly clas-sify autonomously perceived patterns. On-line learning, alsotermed incremental learning, is primarily focused on processingthe data in a sequential way so that in the end the classifier is noworse than a hypothetical classifier trained on the batch data (Kun-cheva, 2004).

In this paper, we describe a global incremental scheme forshape-based object recognition which can perform without an apriori knowledge about the shape of the classes which compoundthe feature space where objects will be represented. The proposedapproach combines an incremental non-parametric discriminantanalysis (Raducanu and Vitrià, 2007) and an incremental, cen-troid-based version of the mode analysis classifier (Wishart,

A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320 1311

1969) as two learning schemes. As other recognition approaches,our proposal can be split into two main stages: feature selectionand classification. In the first stage, the proposed system employsa curvature-based approach. Curvature-based algorithms charac-terize the contour by computing its curvature at each point. Inour case, curvature is estimated using an adaptive scheme whichallows to correctly describe the contour details present at differentnatural scales. To reduce the dimensionality of this shape descrip-tor, a non-parametric discriminant analysis (NDA) is conducted.NDA is an eigenspace-based method, which is focused on seekingdirections for efficient discrimination. Since it does not haveembedded assumptions about the structure of the feature space,they can analyze arbitrarily structured feature spaces (Comaniciuand Meer, 2002). In the proposed approach, the incremental NDA(INDA) will be used for eigenspace learning (incremental featureselection). On the other hand, the classification stage combinesthe incremental mode analysis with a k-nearest neighbor algo-rithm. The incremental mode analysis is a fast one-pass, dis-tance-based clustering algorithm (Bandera et al., 1999), whichincludes the input feature vector in the most suitable existing clus-ter, creating a new one if necessary. As other approaches, e.g. theEvolving Clustering Method (ECM) (Kasabov, 2002), it is capableof achieving this goal with no supervision nor previous training.However, contrary to other clustering algorithms, it cannot opti-mize a classification criterion because it does not memorize infor-mation about all samples which were supplied. In our on-lineclassifier, this algorithm is used to cluster the input patterns intoa reduced set of groups and to determine the cluster centers ofthese groups (prototypes). The classification is finally conductedbased on the k-nearest neighbor algorithm.

Fig. 1 shows an overview of the proposed system. Briefly, theon-line process of classification and learning is conducted sequen-tially as follows. First, we assume that N training samples are givenin advance to form an initial eigenspace model. Then, the incre-mental mode analysis is used to group the transformed trainingsamples into a set of clusters, which will be only represented bytheir cluster prototypes. When a query input is presented to thesystem, the classification is carried out. The classification of thefeature vector obtained from the projection of the input shape intothe eigenspace is conducted based on a k-nearest neighbor algo-rithm whose prototypes are provided by the incremental modeanalysis. If the training sample is misclassified, this input and itsclass label are applied to the INDA to update the current eigen-space model. The updated eigenspace model is utilized for trans-forming the query input into a feature vector, and this vector aswell as the updated prototypes are used to train the classifier usingthe incremental mode analysis. Thus, the proposed on-line classifi-cation stage is able to simultaneously perform the feature selectionand the classifier learning in one-pass, being training samples pre-sented only once to learn. Finally, it must be noted that one condi-tion is imposed by the proposed approach: the incremental modeanalysis must always perform an over-classification of the param-

Fig. 1. Overview of the proposed contou

eter space, i.e. the number of spherically-shaped groups must begreater than the real one.

This paper is organized as follows. Section 2 describes previouswork related to the representation and classification stages of theproposed shape recognition system. Sections 3 and 4 present thetwo stages of the proposed method. The experimental resultsrevealing the efficacy of the method are described in Section 5.The paper concludes along with discussions and future work inSection 6.

2. Related work

2.1. Curvature-based shape contour descriptors

The literature on planar shape representation is relatively huge(Loncaric, 1998). However, in this section, we only focus on meth-ods that are based on curvature because of their close relation withour work. Besides, these descriptors are very popular. Thus, a cur-vature-based approach, the curvature scale space, has been used inthe MPEG-7 standard (Torres et al., 2007).

By definition a curvature function encodes the boundary of ashape in terms of their local curvature or orientation. LetcðtÞ ¼ ðxðtÞ; yðtÞÞ be a parametric plane curve. Its curvature func-tion jðtÞ can be calculated as (Mokhtarian and Mackworth, 1986;Fontoura and Marcondes, 2001)

jðtÞ ¼_xðtÞ€yðtÞ � €xðtÞ _yðtÞð _xðtÞ2 þ _yðtÞ2Þ3=2 ð1Þ

This equation implies that estimating the curvature involves thefirst and second order directional derivatives of the plane curvecoordinates, ð _x; _yÞ and ð€x; €yÞ respectively. This is a problem in thecase of computational analysis where the plane curve is representedin a digital form (Fontoura and Marcondes, 2001). In order to solvethis problem, two different approaches are typically employed:those that approximate the plane curve coordinates (interpolation-based curvature estimators), and those that estimate the curve orien-tation at each contour point with respect to a reference direction(angle-based curvature estimators). However, other methods to esti-mate the curvature can be found in the literature. Thus, Alajlan et al.(2007) have proposed a shape retrieval approach based on the tri-angle-area representation (TAR), where the curvature at each con-tour point is measured using the area of the triangle defined bythis point and two equally-separated contour neighbors.

Interpolation-based curvature estimators interpolate the planecurve coordinates and then, they differentiate the interpolationcurves. Thus, Mokhtarian and Mackworth (1986) propose to con-volve x½t� and y½t� with a one-dimensional Gaussian filter definedby

hðt;wÞ ¼ 1ffiffiffiffiffiffiffi2pp

we�0:5�ðt=wÞ2 ð2Þ

r-based shape recognition system.

1312 A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320

being w the filter width. This process removes the plane curve noiseby filtering the curve descriptor at a fixed cut frequency (single scalemethod). However, features appear at different natural scales andsingle scale methods only detect the features unaffected by the fil-tering process. Thus, single scale interpolation-based methods ob-tain a curvature function which strongly depends on the value ofw (Orrite et al., 1996): if w is large, contour details like cornerscan be missed, and if it is low, noise is not removed. To avoid thisproblem, multiscale methods have been proposed. These ap-proaches typically use iterative feature detection for different cutfrequencies. Thus, a popular solution is the curvature scale space(CSS) (Mokhtarian and Mackworth, 1986). This algorithm filtersthe curve descriptor with a Gaussian kernel and imparts smoothingat different levels of scale (the scale being proportional to the widthof the kernel). From the resulting curve descriptor, features associ-ated to the original shape are identified (Mokhtarian and Mack-worth, 1986). Using a similar approach, Adamek and O’Connorhave proposed a multiscale representation that makes use of bothconcavities and convexities of contour points (Adamek and O’Con-nor, 2004). The multiscale convexity–concavity (MCC) representa-tion obtains different scales by smoothing the boundary withdifferent Gaussian kernels. The relative displacement of a contourpoint with respect to its position in the preceding scale level is usedto measure its curvature. Instead of these iterative approaches, an-other solution is to adapt the cut frequency of the filter at eachcurve point as a function of the local properties of the shape aroundit, e.g. using a Gaussian filter of variable bandwidth (Ansari andDelp, 1991).

Curvature can be also estimated using angles between vectorswhich are defined as a function of the curve coordinates (angle-based curvature estimators). A contour curvature jðtÞ can be de-fined as the variation of the curve slope WðtÞ with respect to t, thatis, the inverse of the curvature radius, qðtÞ:

jðtÞ ¼ @WðtÞ@t

¼ 1qðtÞ ð3Þ

In order to extract jðtÞ from a digital contour, Sarkar (1993) pro-poses to use as slope the difference between consecutive chaincodes in a 8 neighborhood. This slope allows changes of 45�. Pavli-dis (1980) extracts the k-slope of a given contour from its chaincode. In this case, the slope at each point of the contour is equalto that of the line connecting that point with its k neighbor ahead.Since k can be bigger than 1, discretization noise is efficiently re-moved from the function. In the same way, the curve filtering andcurvature estimation are mixed in Agam and Dinstein (1997),where the curvature at a given point is defined as the difference be-tween the slopes of the curve segments on the right and left side ofthe point. The size of both curve segments is fixed. Arica and Vural(2003) define the curvature at each contour point as a random var-iable that draws its value from the angles between equally distantneighbor points at that point. Then, few order moments are com-puted for this random variable at each contour point.

It can be noted that, in the angle-based algorithms describedabove, the value of k determines the cut frequency of the curve fil-tering in a similar way that the Gaussian filter bandwidth does. So,these algorithms are single scale methods in which only featuresunaffected by the filtering process may be detected. Beus and Tiu(1987) propose a multiscale angle-based approach which modifiesthe Freeman’s algorithm Freeman (1978) by averaging the resultsobtained for several values of k and upper-bounding the lengthof segment presenting no discontinuities. However, this approachis slow and, in any case, it must choose the cut frequencies for eachiteration (Bandera et al., 2000). Since in the case of interpolation-based methods, another solution is to adapt the cut frequency ofthe filter at each curve point as a function of the local properties

of the shape around it (Teh and Chin, 1989). A k-slope algorithmwhich estimates the curvature using a k value which is adaptivelychanged according to the local information of the boundary is pro-posed by Bandera et al. (2000). Similar approaches have been sub-sequently proposed (Reche et al., 2002; Urdiales et al., 2002; Marjiand Siy, 2004). The approach proposed in this paper is based on thesame concept.

Finally, it must be noted that shapes can change drastically asthe point of view changes due to perspective transformation. Whenthe object is far from the camera, this viewpoint change can beapproximated by an affine transformation (Avrithis et al., 2001).If the shape recognition system must deal with shapes correspond-ing to different affine transformations, it will be interesting thatthe shape descriptor remains constant under arbitrary affine trans-formations (Obdrzálek and Matas, 2006). It must be noted that thedifferential geometry provides an affine curvature function, whoseanalytic formulation is based on high order numerical derivatives(Chaker et al., 2003). The problem is the same as the one statedat the beginning of this Section, and previously mentioned interpo-lation-based curvature estimators can be extended to estimate thisaffine curvature (e.g., the affine curvature scale space Mokhtarianand Abbasi (2001)). In any case, affine curvature does not usuallyprovide good results when compared to other techniques (see Cha-ker et al., 2007). Other solutions to deal with affine transformationsis to embed the invariance in the matching or recognition pro-cesses or to normalize the input shape. The goal of image normal-ization is to reduce different views of the same shape to itscanonical size and principal orientation (Shen and Ip, 1997). Apartfrom the affine transformation parameters, to which the normali-zation is invariant, no other information is discarded; the normal-ization process consists in fact of an affine transformation and theshape of the original curve remains unchanged. This will be theadopted solution to deal with affine transformations in this paper.

2.2. Incremental learning

The aim of incremental learning is to process the input data in asequential way: each submitted data point is classified and then, itand its label are added to the training set, updating the classifier toaccommodate the new training point. It is usually assumed a staticenvironment, where forgetting of the learned knowledge is notconsidered. The adaptation to a changing environment may comeas an effortless bonus (Kuncheva, 2004), as long as some forgettingmechanism is put into place.

In pattern recognition and data mining, input vectors often havea large size. Therefore, it is interesting to select the informative in-put variables before the classification is conducted. This impliesthat when building a classification system, we should considertwo types of incremental learning: the incremental feature selec-tion and the incremental learning of the classifier. In order to ob-tain a dimensionality reduction of the original data, eigenspace-based methods are the most common techniques. The aim of theseapproaches is mainly focused on obtaining either an efficient datarepresentation or an effective data discrimination (Raducanu andVitrià, 2007). Thus, Linear Discriminant Analysis (LDA), also knownas Fisher Discriminant Analysis (FDA), seeks directions for efficientdiscrimination, while Principal Component Analysis (PCA) seeksdirections efficient for representation. Although the typical imple-mentation of these two approaches assumes that a complete data-base for training is given in advance and learning is conducted inone batch, they have been modified to perform incremental learn-ing. Thus, Hall and Martin (1998) proposed an Incremental PCA(IPCA) which is based on the updating of the covariance matrixthrough a residue estimating procedure. This work was subse-quently extended to allow a chunk of new samples to be learnedin a single step. This incremental approach was based on the

A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320 1313

merging and splitting of eigenspace models (Hall et al., 2000). Tospeed up the IPCA computation, a fast approximation algorithmwhich computes the principal components of a sequence of sam-ples without estimating the covariance matrix has been proposed(Weng et al., 2003). Based on these previous works, Pang et al.(2004) described an incremental classifier for face membershipauthentication where both feature selection and classification aremodeled using one-pass incremental learning method. In this ap-proach, IPCA is used for feature selection, and the k-nearest neigh-bor with prototypes updated using evolving clustering method(ECM) (Kasabov, 2002) is used as an incremental classifier. In asubsequent work, Pang et al. (2005) have also proposed an Incre-mental Linear Discriminant Analysis (ILDA) for the classificationof data streams.

Non-parametric approaches for eigenspace learning have beenalso proposed. Contrary to the previously mentioned parametricapproaches, these approaches do not assume that the samplespresent a normal distribution. Therefore, they are more effectivewhen dealing with general data distributions. Besides, they captureproperly the structural information between class boundaries(Raducanu and Vitrià, 2007). Bressan and Vitrià (2003) proposeda non-parametric form to estimate the within-class scatter matrix,normalizing the distance between each point and their nearestneighbors instead of assuming a Gaussian distribution. A similarstrategy was employed by Raducanu and Vitrià (2007) to developthe Incremental Non-parametric Discriminant Analysis (INDA).The present work extends this previous work to include an incre-mental classifier, a k-nearest neighbor with prototypes updatedusing the incremental mode analysis, which is simultaneously car-ried out with the incremental eigenspace learning.

3. Affine-invariant shape representation approach

Several authors have pointed out that a shape recognition sys-tem will be useful if it is able to allow explicit invariance undergeometric operations of translation, rotation and scaling (Avrithiset al., 2001; Tabbone et al., 2006; Rahtu et al., 2006). In this paper,we consider the recognition of shapes which are affected by affinetransformations. The affine transformation includes rotation, scal-ing, skewing, and translation. It preserves parallel lines and equi-spaced points along a line, and, in some cases, it can also be usedto approximate the perspective transformation. A two-dimensionalaffine transformation possesses six degrees of freedom. Therefore,to determine an affine transformation, six independent constraintshave to be found on the shape. The vector of first order algebraicmoments or centre of gravity of a shape gives two constraints,i.e. resolves translation, and the matrix of second central algebraicmoments or covariance matrix provides three constraints (Obdr-zálek and Matas, 2006). Together, these two shape features fixthe affine transformation up to an unknown rotation, which couldbe determined from local extrema of curvature or from contourpoints of extremal distance to the centre of gravity. In this paper,we employ the algorithm described by Heikkilä (2004), whichuse third algebraic moments to determine the unknown rotation.

In our approach, the input shape will be firstly normalized bythe covariance matrix, allowing for affine-invariant measurementof curvature (Obdrzálek and Matas, 2006). Then, the curvaturefunction associated to the outer contour of the shape will be esti-mated using a triangle-area representation (TAR) (Alajlan et al.,2007). Contrary to previous TAR-based approaches, in our proposalthe triangle side lengths at each contour point are adapted to thelocal variations of the shape, removing noise from the contourwithout missing relevant points. Thus, this representation isinvariant to affine transformation, but it also exhibits robustnessagainst noise. Next subsections describe the processes to normalize

the input shape and to delimit the region-of-support of eachboundary point, and the algorithm to obtain the triangle-area rep-resentation of the shape.

3.1. Shape normalization

Let fxi; yigni¼1 be the Cartesian coordinates of the set of boundary

points of the input shape. The process to compute the centre ofgravity and the covariance matrix use the first order algebraic mo-ments and the second order central algebraic moments, which aredefined by Obdrzálek and Matas (2006)

lpq ¼Xn

i¼1

Z xi

xi�1

Z yi�1þðyi�yi�1Þx�xi�1xi�xi�1

0xpyqdydx ð4Þ

l0pq ¼Xn

i¼1

Z xi

xi�1

Z yi�1þðyi�yi�1Þx�xi�1xi�xi�1

0ðx� l10Þ

pðy� l01Þqdydx ð5Þ

Then, the centre of gravity of the shape C is l ¼ ðl10;l01ÞT , and the

covariance matrix is calculated as

R ¼l020 l011

l011 l002

� �ð6Þ

Once the covariance matrix is computed, the shape is normalized sothat the covariance matrix of the transformed shape equals to theidentity matrix (Obdrzálek and Matas, 2006). This is achieved bytransforming every shape pixel by the inverse of the covariance ma-trix. Assuming local planarity of the input shape, this geometricnormalization, together with the position of the centre of gravityof the shape, provides a rotation-variant transformed shape.

To determine the unknown rotation, the algorithm proposed byHeikkilä (2004) is employed. Third moments of the normalizedshape form a complex number, whose angle changes covariantlywith rotation. This angle is estimated as

a ¼ tan�1 l021 þ l003

l030 þ l012

� �ð7Þ

Fig. 2b show the normalized versions of the shapes at Fig. 2a. In thisexample, input shapes are associated to a non-planar object. Hence,normalized shapes are not very similar. However, it must be notedthat the aim of this normalization process is to provide invarianceunder geometric operations of translation, rotation and scaling. Thisissue is successfully achieved.

3.2. Adaptive estimation of the region-of-support

From the pioneering paper of Teh and Chin (1989), manyresearchers have argued that the estimation of the curvature reliesprimarily on the precise calculation of the region-of-support asso-ciated to each curve point. Thus, in the Teh and Chin’s proposal, theregion-of-support for each point is determined based on the chordlength and the perpendicular distance to the chord. The region-of-support is extended from each boundary point outwards until cer-tain conditions are violated (Teh and Chin, 1989). In this paper, wefollow the scheme proposed by Bandera et al. (2000), which pro-poses to monitor the normalized difference between the arc andchord lengths. The region-of-support continues growing from eachside of the point of interest as long as this ratio remains below afixed threshold, which is set to provide a more robust curve repre-sentation in presence of noise.

To specify the region-of-support associated to the point i of ashape boundary, the algorithm must determine the maximumlength of boundary presenting no significant discontinuities onthe right and left sides of the point i, tf ½i� and tb½i�, respectively(see Fig. 3). The tf ½i� value is estimated by comparing the chordlength from point i to its tf ½i�th neighbor (ki; iþ tf ½i�k2) with the

Fig. 2. Shape normalization: (a) input shapes; and (b) shapes normalized to have an identity covariance matrix and the same orientation.

Fig. 3. Calculation of the maximum length of contour presenting no significant discontinuity on the right side of point iðtf ½i�Þ: (a) Contour of the object and point i; (b)different values for

Pli ¼ laði; iþ tf ½i�Þ and dE ¼ ki; iþ tf ½i�k2; and (c) estimation of the correct tf ½i� (in this case, tf ½i� ¼ 4).

1314 A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320

arc length, i.e. the length of the boundary between points i andiþ tf ½i� which is defined as

laði; iþ tf ½i�Þ ¼Xtf ½i��1

k¼i

kk; kþ 1k2 ð8Þ

The arc and chord lengths tend to be equal in absence of severe cur-vature changes, even if trajectories are noisy. Otherwise, the lengthof the chord is quite shorter than the length of the arc. The tf ½i� valueis the largest value that satisfies

laði; iþ tf ½i�Þ � ki; iþ tf ½i�k2 < Ut ð9Þ

where Ut is a threshold which must be empirically set. The tb½i� va-lue is also set according to the described scheme, but using i� tb½i�instead of iþ tf ½i�.

3.3. Curvature-based shape description

Many researchers have used the area of the triangle, formed bythe outer boundary points, as the basis for shape representations(Shen et al., 2000; Alajlan et al., 2007). The proposed shape recog-nition system employs a curvature estimator to characterize theshape contour which is based on this triangle-area representation(TAR). Given a shape and, once our proposal have determined the

local region-of-support associated to every point of its contour,the process to extract the associated TAR consists of the followingsteps:

(1) Calculation of the local vectors ~fi and ~bi associated to eachpoint i. These vectors present the variation in the x and y axisbetween points i and iþ tf ½i�, and between i and i� tb½i�. Ifðxi; yiÞ are the Cartesian coordinates of the point i, the localvectors associated to i are defined as

~fi ¼ ðxiþtf ½i� � xi; yiþtf ½i� � yiÞ ¼ ðfxi; fyiÞ

~bi ¼ ðxi�tb ½i� � xi; yi�tb ½i� � yiÞ ¼ ðbxi; byiÞ

ð10Þ

(2) Calculation of the TAR associated to each contour point. Thesigned area of the triangle at contour point i is given by Alaj-lan et al. (2007):

ji ¼12

bxibyi

10 0 1fxi

fyi1

�������������� ð11Þ

When the contour is traversed in counter clockwise direction, posi-tive, negative and zero values of TAR mean convex, concave andstraight-line points, respectively.

Fig. 4. (a) Shape #1; (b) moment-based normalized shape #1; (c) adaptive TARassociated to (b); (d) shape #2; (e) moment-based normalized shape #2; and (f)adaptive TAR associated to (e). Shape #2 has been obtained by transforming shape#1.

A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320 1315

Fig. 4 shows two shapes. The shape at Fig. 4d has been obtained

by transforming the shape at Fig. 4a

A ¼ 0:1 1:3

0:7 0:3

� �; t ¼

�20:00:0

� �!. Fig. 4c and f present the two adaptive TAR associated

to the moment-based normalized shapes (Fig. 4a and e). It can benoted that the representations are practically equal although theypresent slightly different contour lengths.

The advantage of measuring the curvature in an adaptive waycan be appreciated in Fig. 5. Fig. 5a shows the corners detectedby thresholding the adaptive TAR associated to the object contour(threshold value equal to 0.6, and triangle side lengths rangingfrom 3 to 15). It can be noted that all corners are correctly detected.On the contrary, Fig. 5b and c show the corners obtained by thesame process when two constant triangle side length values areused. It can be appreciated that when a low value is used (t ¼ 3),the TAR is too noisy and false corners detection occurs. On the con-trary, if a high value is used (t ¼ 15), the representation is exces-sively filtered, and some corners are lost.

3.4. Incremental Non-parametric Discriminant Analysis

Let us assume that N training curvature functions fxgi belongingto M classes Ci have been presented. Let us assume that each class

Fig. 5. (a) Corners detected by thresholding the TAR obtained using an adaptivetriangle side length t; (b) corners detected by thresholding the TAR obtained using afixed t ¼ 3; and (c) corners detected by thresholding the TAR obtained using a fixedt ¼ 15.

Ci is composed by mi samples, Ci ¼ fxi1; x

i2; . . . ; xi

mig, where �xi is the

mean vector of class Ci. Non-parametric discriminant analysis(NDA) seeks a transformation U over the set of training samplesin such a way that the ratio of the between-class matrix Sb andthe within-class matrix Sw is maximized. In the non-parametricscheme, these matrices are defined as Raducanu and Vitrià (2007)

Sb ¼XM

i¼1

XM

j¼1;j–i

Xmi

t¼1

wtCi ;Cj

dðxit;lCj

ðxitÞÞ ð12Þ

Sw ¼XM

i¼1

Xj2Ci

dðxij; �x

iÞ ð13Þ

where dðu; vÞ is a function which measures the distance betweentwo curvature functions u and v. This distance must take into ac-count the different boundary lengths of the shapes to compare. Inour case, the dynamic time warping (DTW) has been employed tofind the best alignment between two shape representations. Thismethod is able to accommodate elastic matching, and therefore, ithas been suggested as a flexible distance metric. lCj

ðxitÞ is the local

k-nearest neighbor mean, defined by

lCjðxi

tÞ ¼1k

Xk

p¼1

NNpðxit ;CjÞ ð14Þ

being NNpðxit;CjÞ the pth nearest neighbor from xi

t to the class Cj.The term wt

Ci ;Cjis a weighting function which is calculated as

wtCi ;Cj¼

minðdðxit ;lCi

ðxitÞÞ; dðxi

t ;lCjðxi

tÞÞÞdðxi

t ;lCiðxi

tÞÞ þ dðxit ;lCj

ðxitÞÞ

ð15Þ

It can be noted that the weight wtCi ;Cj

takes a value close to 0.5 onclass boundaries and drop to zero as the sample moves away, to-wards the centre of its class.

Assuming that the scatter matrices have been computed from atleast two classes, let us consider that a new training sample y ispresented to the system. Two situations can be distinguished(Raducanu and Vitrià, 2007):

� The input sample y belongs to the class CL.The between-classscatter matrix is updated as

S0b ¼ Sb � Sinb ðCLÞ þ Sin

b ðCL0 Þ þ Soutb ðyÞ ð16Þ

where CL0 ¼ CLS

y, and Sinb ðCLÞ and Sout

b ðyÞ are defined by

Sinb ðCLÞ ¼

XM

i¼1;i–L

Xmi

t¼1

wtCi ;CL

dðxit;lCL

ðxitÞÞ ð17Þ

Soutb ðyÞ ¼

XM

i¼1;i–L

dðy;lCiðyÞÞ ð18Þ

On the other hand, the update equation of the within-class scat-ter matrix is

S0w ¼XM

j¼1;j–L

SwðCjÞ þ SwðCLÞ þmL

mL þ 1dðy; �xLÞ ð19Þ

� The input sample y belongs to a new class CMþ1.In this case, thebetween-class scatter matrix is updated using the followingequation

S0b ¼ Sb þ Soutb ðCMþ1Þ þ Sin

b ðCMþ1Þ ð20Þwhere Sout

b ðCMþ1Þ and Sinb ðCMþ1Þ are defined as

Soutb ðCMþ1Þ ¼

XM

i¼1

dðy;lCiðyÞÞ ð21Þ

Sinb ðCMþ1Þ ¼

XM

i¼1

Xmi

t¼1

wtCi ;CMþ1

dðxit ;lCMþ1

ðxitÞÞ ð22Þ

The within-class scatter matrix remains unchanged (S0w ¼ Sw).

1316 A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320

Once the scatter matrices are obtained, the transformation matrixU can be computed by conducting an eigenvalue decomposition ofthe matrix D ¼ S�1

w Sb,

DU ¼ UK ð23Þ

The columns of the transformation matrix correspond to the dis-criminant eigenvectors. In our approach, eigenvectors with smalleigenvalue are discarded to compress a high dimensional data(i.e., the curvature function) to a low-dimensional feature with anenhanced discrimination. For instance, Fig. 6 shows the feature vec-tors associated to the curvature functions depicted in Fig. 4. In thiscase, a set of 17 eigenvectors has been chosen to ensure that theprojection of the data onto this reduced set covers at least 90% ofthe data’s spread,

PKi¼1ki=

Piki > 0:9. It must be noted that the num-

ber of eigenvectors to maintain a constant value of coverage of thedata’s spread could change during the incremental eigenspacelearning.

4. Classification stage

Mode analysis is a derivative of single linkage clustering whichsearches for natural sub-groupings of the data by estimating dis-joint density surfaces in the sample distribution (Wishart, 1969).Although this algorithm is suitable for working on-line, it requiresthe storage of every processed sample because it uses this informa-tion for further classifications. Therefore, the process is computa-tionally expensive when the number of input samples is high(the computing time is Oðn3Þ and the storage requirement isOð1=2n2Þ). In order to reduce its computational cost, this algorithmcan be modified to perform incremental recognition. The incre-mental mode analysis conducts one of the following actionsaccording to its input:

� if the Euclidean distance between the prototype of every groupf�xignp

i¼1 and the input sample x is greater than a threshold Ui, anew group np þ 1 is created and the input sample becomes itsprototype �xnpþ1 ¼ x, or

� if the Euclidean distance between the prototype of one or sev-eral groups and the input sample is lower than Ui, the samplex is included in the closest group k. The new group prototype�x0k and covariance matrix R0k can be obtained from

�x0k ¼ 1nkþ1 ðnk�xk þ xÞ

R0k ¼nk

nkþ1 ðx� �x0kÞðx� �x0kÞT ð24Þ

nk being the number of samples in group k before x is presented.

Fig. 6. (a–b) Feature vectors associated to the curvature functions depicted in Fig. 3.

It can be noted that the incremental mode analysis only needsto store the prototype of every group (centroid cluster analysis).Therefore, it is computationally cheap and does not need to storeall input samples. On the contrary, all input samples cannot beused to redefine the set of data groups (reference set) when anew sample arrives. This implies that the on-line estimated proto-type of a group could be sequentially changed from its initial value�xj to a excessively different final value �x0j. To avoid that, the proto-type update at Eq. (24) can be modified to introduce a new param-eter, the learning rate a, which controls the movement of theprototype. In our case, it is desirable that the incremental modeanalysis performs an over-classification of the input data. There-fore, it will generate small clusters which, in our experiments, willnot provoke excessive prototype’s movements.

In any case, the incremental mode analysis tends to force obser-vations into spherically-shaped data groups. This is not unusual:the same problem has been observed in more complex algorithmslike the hierarchical Ward’s method (Wishart, 1998). For instance,Fig. 7b and c illustrate the result obtained by classifying the set ofbidimensional samples present in Fig. 7a using the incrementalmode analysis with different thresholds. If Ui is equal to 132, themethod fails to separate the two natural data groups present inFig. 7a. When a low threshold value (Ui ¼ 28) is employed,although there are a large number of data groups and the groupshapes are all relatively small, the resulting data groups correctlymap the densest part of the feature space. It is interesting to notethat the choice of a distance (instead of the classical Euclidean dis-tance) plays an important role allowing to control this biased effect(Lomenie, 2004).

Once the new set of prototypes has been obtained, the k-nearestneighbor is employed to classify the input sample. Euclidean dis-tance can be used to measure the similarity between the input cur-vature function y and each stored prototype pj, which is also acurvature function. This distance dypj is computed as

dypj ¼ kUT y�UT pjk2 ð25Þ

where U is the transformation matrix provided by the INDA.

5. Experimental results

5.1. Shape description: A comparative study

In this Section, the performance of the proposed curvature esti-mator is demonstrated using two standard experiments on twodifferent data sets. The first data set is the well-known MPEG-7CE-shape-1 (Latecki et al., 2000). It contains 1400 silhouetteimages semantically classified into 70 classes. Each class has 20different shapes. Fig. 8 shows several shape images. The other dataset is the Kimia’s database (Sebastian et al., 2001) which consists of9 classes with 11 shapes in each cluster1 (see Fig. 9). Experimentswere performed on a Pentium IV 2.6 GHz PC. It must be noted that,in these experiments, we will only test the performance of the pro-posed curvature function for shape description.

In the first experiment, the retrieval performance of the pro-posed descriptor is evaluated using the MPEG-7 CE-shape-1 partB test. The total number of images in the database was used here.Each class was used as a query, and the number of similar images(which belong to the same group) was counted in the top 40matches (Bull’s Eye score). As it has been mentioned in previousworks (Latecki et al., 2000; Adamek and O’Connor, 2004; Alajlanet al., 2007), 100% retrieval is not possible, since some groups con-tain objects whose shape is significantly different so that is notpossible to classify them into the same group using only shape

1

http://www.lems.brown.edu/vision/researchAreas/SIID/.

Fig. 7. (a) Set of 125 bidimensional samples, (b) classification of the set (a) using the incremental mode analysis with Ui ¼ 132 (two data groups), and (c) classification of theset (a) using the incremental mode analysis with Ui ¼ 28 (20 data groups).

Fig. 8. Some example of shape images from the MPEG-7 CE-shape-1.

Fig. 9. The Kimia’s database Sebastian et al. (2001).

A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320 1317

Fig. 10. Results on Kimia’s data set of 99 shapes.

1318 A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320

information. In order to evaluate the performance of the proposedmethod, the results of several curvature-based approaches on thissame test has been compared. The chosen approaches are:

� The dynamic space warping (DSW) proposed by Alajlan et al.(2007), which characterizes the shape using a multi-scale trian-gle-area representation (MTAR).

� The multi-scale convexity concavity (MCC) representation pro-posed by Adamek and O’Connor (2004), where the curvature ismeasured taken into account the relative displacement of a con-tour point with respect to its position in the preceding scalelevel.

� The beam angle statistics (BAS) proposed by Arica and Vural(2003), which define the curvature at each contour point as arandom variable that draws its values from the angles betweeneach equally distant neighboring points at that point. Few ordermoments are computed for the random variable at each contourpoint.

� The curvature scale space (CSS) Mokhtarian and Mackworth,1986; which has been selected as the MPEG-7 standard for theboundary-based shape descriptor.

� The adaptive approach (AM) proposed by Bandera et al. (2000),which represents the curvature at each contour point using thek-slopes defined between this point and two neighboring points.The region-of-support of each point is estimated using an adap-tive method.

All these methods use dynamic programming to compare theshape descriptor with the ones stored in the databases. Table 1 pre-sents the results of these methods on the MPEG-7 CE-shape-1 partB test. Two versions of our approach have been tested. It can benoted that the normalization stage allows us to increase the perfor-mance of the recognition system. When the normalization stage isconducted, our results are better than the ones provided by the CSSand BAS approaches and similar to the ones provided by the DSWand MCC methods. However, the reported complexities of theselast two multi-scale approaches, when they are applied to a con-tour length of N points, are OðN2Þ for the DSW and OðN3Þ for theMCC (Alajlan et al., 2007). On the contrary, the complexity of ourshape descriptor is OðNÞ, as it is directly related to the contourlength.

In the retrieval experiment using the Kimia’s data set, eachshape in the data set is considered as a query and the first 10 re-trieved shapes, excluding the query, are determined. The correctnumber of retrieved shapes for each ranking, over all 99 shapes,are counted. As it is shown in Fig. 9, partial occlusion is the mainfactor of variation among shapes of the same category. Fig. 10shows the number of correct retrievals at different rankings. Itcan be mentioned that these results are quite similar to the onesreported by Alajlan et al. (2007) which use a multi-scale descriptordefined on the TAR representation.

5.2. Affine invariance test

Shapes included in the MPEG-7 CE-shape-1 dataset have beentaken from natural and man-made objects, cartoons and manuallydrawn contours. However, this dataset is not specially dedicated tothe problem of affine-invariant shape description. Therefore, toevaluate the proposed descriptor against affine transformations,

Table 1Comparison of the results of different curvature-based approaches on the MPEG-7 CE-sha

CSS (%) AM (%) BAS (%) Proposed (%)a

81.12 81.73 82.37 82.53

a Proposed approach without shape normalization stage.

we have repeated the experiment described by El Rube et al.(2005). In this test, the 70 original shapes of the MPEG-7 CE-shape-1 were transformed using

x0 ¼1:0 b

0:0 1:0

� �x ð26Þ

where the parameter b takes eight different values ([0, 0.4, 0.8, 1.5,2.0, 3.0, 4.0, 6.0])(El Rube et al., 2005). The performance of the pro-posed approach is evaluated using the precision-recall curves asillustrated in Fig. 11. Curves show the results provided by our pro-posal using and not using the normalization stage. Besides, curvesprovided by other two curvature-based approaches for shapedescription are included for comparison purposes: the combinedMTAR (El Rube et al., 2005) and the CSS (Mokhtarian and Mack-worth, 1986). Obtained results show that the normalization stageallows to achieve better performance under affine transformation.

5.3. Evaluation of the proposed incremental scheme

In this section, the efficiency and accuracy of our incrementalapproach is examined and compared to other related approaches.For these experiments, the MPEG-7 CE-shape-1 database is em-ployed. For every test, we firstly build an initial feature eigenspaceusing 10% of the total samples (140 samples), in which at least twoclasses data are ensured to be included (Raducanu and Vitrià,2007). These training samples are used for calculating eigenvectorsand eigenvalues through conventional NDA. The remaining train-ing data are randomly provided to the incremental classifier with-out any consideration about their class labels. Then, the eigenspaceis updated by the INDA computation introduced in Section 4. Allthe training samples are presented only once to learn. Since theperformance of incremental learning could depend on the se-quence of training samples, ten trials with different sequences oftraining samples are conducted to evaluate the average perfor-mance on every test. As it was aforementioned, we use a PentiumIV 2.6 GHz PC.

The proposed approach is compared with three related eigen-space models:

� Fixed eigenspace: An eigenspace is obtained by applying NDA toan initial training set, and it is fixed during the whole learningstages.

� Incremental NDA: An eigenspace model is incrementally updatedbased on the INDA algorithm proposed by Raducanu and Vitrià(2007).

pe-1 part B (Bull’s Eye test).

Proposed (%) MCC (%) DSW (%)

84.97 84.93 85.03

Fig. 11. Affine test precision-recall curves.

Fig. 12. Corners detected over three distorted shapes belonging to the same object.

A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320 1319

� Batch NDA: The eigen-feature space is updated every time aquery input is misclassified by applying NDA to all the trainingsamples given so far.

In every test, the sequence of training samples provided to thefour approaches is the same. Besides, in order to make the assess-ment under the same conditions, at each learning stage, the dimen-sion of the eigenspace is set to the same value as the one providedby the proposed approach. Therefore, the classification accuracy ofthe Batch NDA gives a target value for the proposed approach(Kuncheva, 2004). In all tests, the number of eigenvectors is chosenat each learning stage to ensure that the projection of the data ontothis reduced set covers at least 90% of the data’s spread,PK

i¼1ki=P

iki > 0:9. The k-nearest neighbor is always conductedwith a k value equal to 3. The threshold values Ut and Ui have beenfixed to 0.4 and 0.02, respectively (see Section 5.4).

Table 2 shows the experimental results of the above threeeigenspace models and the proposed approach when the k-nearestneighbor method is adopted as classifier. As seen from this Table,the fixed eigenspace model provides the worst results. This resultshows that the features chosen from the initial training sampleswere inappropriate to classify the remaining training samples cor-rectly; therefore, it is effective to update the feature space to adaptto the change of data distributions. Besides, there is no significantdifference in classification accuracy between the Raducanu and Vi-trià (2007)’s proposal and the proposed approach. Finally, compar-ing with Batch NDA, the proposed approach presents a little lowerclassification accuracy. However, this difference is not significant.That is, the proposed method ensures the quasi-optimality of theobtained eigenspace and it does not keep past training samples. Fi-nally, the computation costs of the approaches are estimated bymeasuring the time and dividing it by the number of input sam-ples. Table 2 shows that the CPU time of Fixed eigenspace is verylow, as it mainly corresponds to the time required to build the ini-tial eigenspace. The computation costs of the proposed approach is

Table 2Performance comparison with other eigenspace models.

Fixed eigenspace INDA (Raducanu and Vitrià (2007))

Classification accuracy (%)67.3 87.3

Time (s)0.11 0.62

significantly reduced against both the Raducanu and Vitrià’s (2007)proposal and Batch NDA.

5.4. Estimation of parameters

The proposed approach presents two main parameters to ad-just. These parameters are:

� The threshold value Ut which determines the noise level toler-ated by the adaptive curvature detector.

� The threshold value Ui employed by the incremental modeanalysis.

The threshold value Ut was obtained by testing the behaviour ofthe adaptive curvature function in a corner detection task (Banderaet al., 2000; Trazegnies et al., 2002). These tests have proven thatUt equal to 0.4 works correctly in most cases. Fig. 12 shows the cor-ners detected over three shapes which belong to the same object.These shapes have been affected by different distortions (scalingand rotation). It can be noted that corners are correctly detectedin spite of distortions.

The modified mode analysis strongly depends on the accep-tance threshold Ui. However, in our hybrid approach, this is not acritical issue and the only condition imposed to this threshold isthat it must provide an over-classification of the sample space. Ob-

Batch NDA Proposed approach

89.6 87.2

0.79 0.43

1320 A. Bandera et al. / Pattern Recognition Letters 30 (2009) 1310–1320

tained results show that the data set is processed correctly for Ui

ranging from 0.01 to 0.03. If Ui is less than 0.01, then the incremen-tal mode analysis will generate an excessively high number of clus-ters and it is not interesting to employ this grouping algorithm. IfUi is greater than 0.03, then the number of prototypes is reducedbut the classification accuracy also decreases.

6. Conclusions and future work

This paper has described a planar shape recognition system. Atthe representation stage, an affine-invariant closed-contour shaperepresentation that employs the triangle-area representation tomeasure the convexity/concavity of each contour point is de-scribed. The proposed method uses an adaptive algorithm to deter-mine the region-of-support associated to every contour point. Thisalgorithm allows to define a region-of-support which changescovariantly with the affine transform. The proposed shape descrip-tor is invariant to affine transformation and it is also robust againstmoderate amounts of noise. The proposed global incrementalscheme combines the Incremental Non-parametric DiscriminantAnalysis (INDA) and incremental mode analysis to simultaneouslyconduct feature selection and classifier learning. This scheme en-ables a classifier to learn incrementally, on-line, and in one-pass.Experimental results show that the system works correctly withno a priori information about the nature of the input patterns.

The nature of this algorithm makes it specially suitable to dealwith applications where a huge number of samples are on-line pro-vided to the classifier. Thus, we are actually testing its applicabilityto a shape-based robot mapping framework with promising results.

Acknowledgements

We would like to thank Dr. Adamek and Dr. O’Connor for pro-viding MCC results shown in Section 5. The authors would also liketo thank the anonymous referees for their suggestions and insight-ful comments.

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