K-Mutual Nearest Neighbour Approach for Clustering Two-Dimensional Shapes Described by Fuzzy-Symbolic Features

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1 Engneerng Letters, 14:1, EL_14_1_24 (Advance onlne publcaton: 12 February 2007) K-Mutual Nearest Neghbour Approach for Clusterng Two-Dmensonal Shapes Descrbed by Fuzzy-Symbolc Features H.S.Nagendraswamy and D.S.Guru Abstract In ths paper, a new method of representng twodmensonal shapes usng fuzzy-symbolc features and a smlarty measure defned over fuzzy-symbolc features useful for clusterng shapes s proposed. A k-mutual nearest neghborhood approach for clusterng two-dmensonal shapes s presented. The proposed shape representaton scheme s nvarant to smlarty transformatons and the clusterng method explots the mutual closeness among shapes for clusterng. The feasblty of the proposed methodology s demonstrated by conductng experments on a consderably large database of shapes and also, ts valdty s tested by comparng wth the well known clusterng methodologes. Index Terms Fuzzy-Symbolc features, k-mutual nearest neghbour, Symbolc data analyss, Shape representaton, Shape clusterng. I. INTRODUCTION Clusterng plays a sgnfcant role n several exploratory pattern analyss, groupng, decson-makng, machne learnng stuatons, ncludng data mnng, document retreval, mage segmentaton and pattern classfcaton. In general, cluster analyss s of great mportance n classfyng a heap of nformaton, whch s n the form of data patterns. Clusterng of planar objects or mages of objects, accordng to the shapes of ther boundares along wth the nner shape detals can sgnfcantly mprove database searches n systems wth shape-based queres. For nstance, testng a query aganst prototypes of dfferent clusters to select a cluster and then testng aganst shapes only n that cluster s much more effcent than testng exhaustvely. In order to cluster shapes, we need to characterze shapes n terms of features, whch best captures the shape nformaton. The concept of symbolc data sets provdes a natural way of capturng shape nformaton and the concept of fuzzy sets effcently handles the uncertanty n the data. Many clusterng algorthms have been proposed for clusterng conventonal [12], [14], [15], Manuscrpt receved on Aprl 24, Ths work s a part of the UGC sponsored major research project No. F (SR). The UGC support s hghly apprecated. H.S.Nagendraswamy, Author, Research Scholar n the Department of Studes n Computer Scence, Unversty of Mysore, Mysore , INDIA (E-mal: swamy_hsn@yahoo.com) D.S.Guru, Author wth the Department of studes n Computer Scence, Unversty of Mysore, Mysore , INDIA (E-mal: guruds@lycos.com) [16] as well as symbolc data [5], [6], [8], [9], [10], [11], [14] and shown that the approaches based on fuzzy-symbolc data outperforms conventonal approaches. The reason s that the fuzzy-symbolc data provdes a natural and realstc analyss to problems as symbolc data are more unfed by means of relatonshps [2] and the fuzzy theory handles effcently the mprecseness. Even though the concept of fuzzy-symbolc approach s much explored n data analyss and clusterng, ts nherent capabltes n preservng shape propertes of an object have not yet been explored. To the best of our knowledge, there has been no attempt n proposng an approach based on fuzzy-symbolc data for shape analyss. Ths motvated us to thnk of clusterng two dmensonal shapes effectvely by descrbng shapes n terms of fuzzysymbolc features. However, there are several approaches for shape analyss based on conventonal data types. Moment based features [13], sgnature based centrodal profle [3], Fourer Descrptors [19], Curvature Scale Space (CSS) [18], Edtng shock graphs [21], polar representaton of the contour ponts about the geometrc centre of the shape [1] have been proposed. A detaled survey on shape analyss can be found n [4], [17]. It has been observed from the lterature survey that most of the conventonal agglomeratve clusterng technques use smlarty proxmty matrx for clusterng patterns. The agglomeratve clusterng merges two patterns, whch possess hgh smlarty value at each level and contnue the mergng process untl the desred number of clusters are obtaned or all the patterns are put nto one cluster. However, t has been shown that about 62% of the total numbers of ndvduals n random artfcal and natural populatons are n mutual pars. The concept of Mutual Nearest Neghbourhood has been ntroduced [7] and successfully used for agglomeratve and dsaggregate clusterng. Thus, t s very approprate to cluster patterns by lookng at ther mutual nearness than just by lookng at ther proxmty matrx. In the proposed methodology, the concept of mutual nearest neghbourhood has been extended such that k patterns are clustered nto one class f they are k-mutually nearest neghbours. Wth ths backdrop, n ths paper, we propose a method of representng two-dmensonal shapes usng fuzzy-symbolc (mult nterval membershp values) features. Unlke other exstng shape representaton schemes, the proposed scheme s capable of utlzng both contour as well as regon nformaton. A smlarty measure defned over the fuzzy-

2 symbolc features for clusterng shapes s also descrbed. The concept of mutual nearest neghbourhood s extended for realstc and effectve clusterng of shapes. The proposed shape representaton scheme s shown to be nvarant to smlarty transformatons (rotaton, translaton, scale and reflecton) and the clusterng methodology s effectve. Several expermentatons have been conducted to demonstrate the success of the proposed methodology. Moreover, the proposed methodology s valdated by comparng wth the well known clusterng methodologes n lterature. The paper s organzed as follows: Secton 2 presents the proposed symbolc shape representaton scheme and the proposed smlarty measure. Secton 3 descrbes a novel shape clusterng procedure based on the concept of k-mutually nearest neghbours. Expermental results are presented n secton 4. A comparatve study of the proposed methodology wth the well known clusterng technques s gven n secton 5 followed by conclusons n secton 6. II. PROPOSED METHODOLOGY Ths secton explans n detal the proposed method of representng shapes and a smlarty measure for estmatng the degree of smlarty between two shapes useful for clusterng. A. Shape representaton The proposed shape representaton scheme extracts fuzzysymbolc features from a shape wth reference to the axs of least nerta, whch s unque to the shape. The axs of least nerta of a shape s defned as the lne for whch the ntegral of the square of the dstances to ponts on the shape boundary s a mnmum. It serves as a unque reference lne to preserve the orentaton of the shape. Preservng the orentaton of a shape s essental for extractng features, whch are nvarant to smlarty transformatons. However, for symmetrcal shapes (star lke shapes) more than one axs of least nerta can be expected. Because of shape symmetry, n spte of multple axs of least nerta, the extracted features from a shape wll not change drastcally. In addton, varaton n feature values due to rotaton and lmtatons of fnte precson arthmetc s recorded n our feature extracton methodology. But f the shape s major deformed or occluded then the axs of least nerta may change ts orentaton and hence the extracted feature vector may not agree wth the feature vector of the orgnal shape. Therefore, the proposed method of shape representaton s not ntended to handle major occlusons or deformatons. However, the method copes well wth mnor occlusons and deformatons. In order to extract fuzzy-symbolc features from a shape, frst, the slope angle θ of the axs of least nerta of a shape curve s estmated as descrbed n [22]. Once the slope angle θ s calculated, each pont on the shape curve s projected on to the axs of least nerta. S = s s, s,..., represent L ponts on the shape Let { 1, 2 3 s L } curve and ( y ) pont x, denote the Cartesan co-ordnates of the s for = 1, 2, 3,, L, Let C (, ) = C x C y denote the centrod of S. Gven a pont ( ) x, y on a shape curve, the coordnates of ts projected pont ( x, y ) on the axs of least nerta can be found as follows: Snce, the ponts ( C x, C y ) and ( x, y ) are lyng on the axs of least nerta, we have ( C y y ) = tanθ C y y = C ( C x x ) Smlarly, as the pont (, y ) pont ( x, y ), we have ( y y ) b ( x x ) b = tan x tanθ x tanθ. x s the projecton of the (1) ( 90 + θ ) y y = x tan( 90 + θ ) x tan( 90 + θ ). By combnng equatons (1) and (2), we get ( C tan( 90 θ ) tan θ ) y yb + xb + C x x =. ( tan( 90 + θ ) tan θ ) From eq. (1) we have y = C + ( x C ) tanθ y b Fg. 1. Shape of an object wth ts axs of least nerta and extreme ponts The two farthest projected ponts say E1 and E2 on the axs of least nerta are chosen as extreme ponts as shown n Fg. 1. The Eucldean dstance between these two extreme ponts defnes the length of the axs of least nerta. Once the axs of least nerta of a shape s obtaned, keepng t as a unque reference lne, fuzzy-symbolc features are extracted from the shape. A fxed number say n, of equdstant ponts called feature ponts on the axs of least nerta are found (Fg. 1). The number of feature ponts defnes the dmenson of the feature vector assocated wth a shape. At every feature pont chosen, an magnary lne perpendcular to the axs of least nerta s drawn. It s nterestng to note that ths perpendcular lne ntersects the shape curve at several ponts. Generally, the number of such ntersectng ponts s even when the drawn perpendcular lne s not tangental to a shape curve at any pont. The number of such ntersectng ponts s exactly two f the shape s fully convex wthout havng holes nsde. In the case of shapes wth holes or wth concave parts, the perpendcular lne ntersects the shape curve at more than two ponts and the number of such ntersectng ponts s always even. x b (2)

3 Once the ntersectng ponts on the shape curve are dentfed, we extract fuzzy-symbolc features by usng the concept of fuzzy equlateral trangle membershp functon. In ths work, we have consdered equlateral trangle membershp functon formed by any three non-collnear ponts. Let θ1,θ 2 and θ 3 be the nner angles of a trangle, n the orderθ1 θ 2 θ 3, and let U be the unverse of trangles;.e., o {, θ, θ ) θ θ θ 0; θ + θ + } U = θ θ 180 (3) ( = We can defne an approxmate equlateral trangle for any trplet of angles fulfllng the constrants gven n equaton (3) usng the followng membershp functon [20] μ ( θ 3 θ, θ, θ ) = 1 1 ( θ ) (4) Thus the membershp functon μ θ, θ, ) assgns a value ( 1 2 θ 3 between 0 and 1 to a fuzzy set of equlateral trangles. For nstance, the lne drawn perpendcular to the axs of least nerta at the feature pont f p ntersects the shape curve at four ponts P 1, P 2, P 3 and P 4 as shown n Fg. 1. A trangle s formed by consderng the frst two farthest ntersectng ponts P 1, P 4 and the feature pont (f p +1), then t s approxmated to an equlateral trangle and ts membershp value ( μ 1) s calculated usng Eq. (4). Smlarly, we form a trangle by consderng the next two farthest ntersectng ponts P 2, P 3 and the feature pont (f p +1). The equlateral trangle membershp value ( μ 2 ) assocated wth ths trangle s also calculated. Thus { μ 1, μ 2 } s consdered to be the feature value at the feature pont f p. The reason behnd consderng the farthest ntersectng ponts whle formng a trangle s that t measures the span length of the ntersectng ponts and also the problem of sequencng feature values n case of multple values at any partcular feature pont can be avoded. Notce that the feature value s not necessarly a sngleton (crsp) but multvalued The set of all extracted multvalued type features (because of n feature ponts chosen on the axs of least nerta) forms the feature vector of dmenson n to represent the shape. In our work the value for n s set to 15 and s emprcally chosen. Unlke conventonal shape representaton schemes, where a shape feature vector s a collecton of sngle valued data, the proposed shape feature vector s a collecton of multvalued type data. The proposed representaton scheme s capable of capturng the nternal detals of shapes n addton to capturng the external boundary detals. Let S be the shape to be represented and n s the number of feature ponts chosen on ts axs of least nerta. Then the feature vector F representng the shape S, s n general of the F = f f, f,...,, where f = { μ, μ, μ,..., μ } for form [ ] 1, 2 3 f n l l1 l2 l3 lk some l k 1. It shall be observed that as the feature extracton process s based on the axs of least nerta and fuzzy equlateral trangle membershp value, the feature vector F representng the shape S s nvarant to shape smlarty transformatons. The angles of the trangle formed by the ntersectng ponts at a feature pont f p wth respect to feature pont (f p +1) on the axs of least nerta are scalng nvarant. Thus the membershp value does not change even f the shape s scaled up or scaled down unformly. The rotatonal nvarance s automatcally acheved, as the axs of least nerta s nvarant to rotaton. However, due to rotaton the sequence n whch the features are extracted may be dfferent. A way of allevatng ths problem s to sequence the features such that the frst feature corresponds to the extreme pont, whch s nearer to the centrod of the shape. An added advantage of the proposed scheme s that unlke other contour traversal technque based approaches, the proposed method s also nvarant to flppng transformaton. Generally, flppng nvarance s dffcult to acheve n shape analyss f a method s senstve to the drecton of contour traversal because the drecton of contour traversal changes when an objectshape s flpped over. Snce the axs of least nerta of a shape flps along wth the shape about the same lne by preservng the startng pont, the proposed representaton method s nvarant to flppng. In practce, however, feature values defned at a feature pont on the axs of least nerta are not exactly same but they le wthn certan range due to fnte precson arthmetc. Thus, each component of the feature value defned at a feature pont s approxmated by nterval data type as fl l1 + l1 l2 + l2 + l k l k = {[ μ... μ ],[ μ... μ ],...,[ μ... μ ]}, where μ and + μ represent, respectvely, the mnmum and the maxmum varatons of the respectve component. These mnmum and maxmum values shall be computed by examnng a shape n all possble orentatons and wth dfferent scalng factors. Thus, n a more general format, the symbolc feature vector F assocated wth a shape S s a collecton of n mult-ntervalvalued data type aganst the crsp data type n case of conventonal shape analyss technques. In practce, for the purpose of creatng a shape database, we generated a large number of samples for each shape wth dfferent orentatons and scalng factors. However, only one representatve vector, whch s an aggregaton of all, s stored n the database. B. A Smlarty measure Smlarty s the fundamental noton n clusterng shapes, whch defnes the closeness or farness of two shapes. A measure of smlarty between two shapes drawn from the same feature space s essental to most of the clusterng technques. In ths subsecton, we present a smlarty measure for estmatng the degree of smlarty between two shapes descrbed by mult nterval valued features, whch s used for clusterng of shapes. Let f Ak = μ, μ +, μ, μ +,..., + μ, μ Ak Ak Ak Ak Ak p Ak p and f Bk = μ, μ, μ, μ,..., μ, μ be the Bk Bk Bk Bk Bk q Bk q k th feature component of two shapes A and B descrbed by n mult nterval valued feature vectors F A = [ f A1, f A2, f A3,..., f An ] F B = [ f B1, f B2, f B3,..., f Bn ] + respectvely. Here μ s the lower lmt and μ s the upper lmt of the nterval. Snce we use the concept of mutual smlarty between shapes for clusterng, we frst estmate the degree of smlarty S from the shape A to the shape and B the degree of ( ) A B

4 S B A from the shape B to the shape A. Then the mutual smlarty value (MSV) between the shapes A and B s defned to be the average of S A B and S B A. Snce the k th feature component n a feature vector s of type mult nterval valued, the degree of smlarty from A to B wth respect to ther l th nterval s estmated based on degree of overlappng between ther lower and upper lmts of ther ntervals. If the nterval μ, μ + descrbng the l Ak l Ak l th nterval of the k th feature of A s contaned n the nterval + μ, μ descrbng the l Bk Bk l l th nterval of the k th feature of B then the degree of smlarty from A to B s taken as 1, otherwse t s gven by the average degree of smlarty between ther respectve lower and upper lmts. Thus, the degree of smlarty from A to B wth respect to the l th nterval of ther k th feature s gven by f ( μ ) ( ) Ak μ and l Bk μ l Ak μ l Bkl l SAk Bk = 1 + [ ] (5) η + η otherwse 2 smlarty ( ) Smlarly, degree of smlarty from B to A wth respect to the l th nterval of ther k th feature s gven by f ( μ ) ( ) Bk μ and l Ak μ l Bk μ l Akl l SBk Ak = 1 + [ ] (6) η + η otherwse 2 where η = 1+ μ 1 Ak l μ Bk l, and * β β s the normalzng factor, whch s set to 10. The mutual smlarty value (MSV) between A and B wth respect to l th nterval s gven by It shall be notced that f the ntervals are one and the same then the MSV between them s 1 (maxmum); otherwse the smlarty value depends on the extent to whch the ntervals are separated. More the extent to whch they are separated less shall be the degree of smlarty. It may so happen that n some cases, the number of ntervals (l) n the k th feature of A and B are not the same. Let the number of ntervals present n the k th feature of A be p and the number of feature values present n the k th feature of B be q and p<q. In such stuatons we choose the frst p components out of q from B for the computaton of smlarty; for the remanng (p-q) components of B, (for whch there are no correspondng components n A for the computaton of smlarty), we assume the degree of smlarty to be zero. Thus the overall degree of smlarty between the shape A and the shape B wth respect to ther k th feature component s gven by (7) (8) Thus the total degree of smlarty between the shape A and B wth respect to all the n features s gven by where n s the number of feature ponts chosen on the axs of least nerta. In our approach, feature values descrbng the shape are n the range 0 and 1. Snce, the k th feature component smlarty between shape A and B s n the range 0 and 1, the total degree of smlarty between the shape A and the shape B s also n the range 0 and 1. III. CLUSTERING OF SHAPES In order to cluster shapes descrbed by mult nterval-valued features, we frst compute the degree of mutual smlarty among all the shapes usng the proposed smlarty measure as explaned n secton IIB and a smlarty matrx s obtaned. A smlarty poston matrx of sze (r x r) for all r shapes s created based on the descendng order of ther smlartes. The poston matrx gves the nearness poston of shapes n terms of ther smlarty rank. The concept of k-mutual nearest neghbor, based on ther poston n the poston matrx, s employed to cluster shapes. In a poston matrx, f the shape S s the k th nearest neghbor of the shape S j, and the shape S j s also the k th nearest neghbor of the shape S, then S and S j are sad to be k-mutually nearest neghbors. Ths dea of k- mutually nearest neghbors s successfully appled n our clusterng procedure. Accordng to ths dea, f m shapes are put nto one cluster then all those m shapes must be k-mutually nearest neghbors. The value of k s set to 2 ntally and ncremented by 1 each tme untl we get the desred number of clusters or all shapes are put nto a sngle cluster. Thus the proposed method of clusterng shapes can be algorthmcally expressed as follows. Algorthm: Clusterng-of-Shapes Input : Shapes descrbed by fuzzy-symbolc features (S 1,S 2, S 3,, S r ). Output : Clusters of shapes (C 1, C 2,,C m ). Method: 1. Obtan smlarty matrx of sze r x r usng the proposed smlarty measure. 2. Create a smlarty poston matrx of sze (r x r) for all the shapes based on the descendng order of ther smlartes. 3. Let C={C 1, C 2,,C r }, ntally contan r number of clusters each wth an ndvdual shape. 4. Set the number of clusters (noc) to r. 5. Set k to Merge two clusters C u and C v, f all the shapes n C u and C v are k-mutual nearest neghbors. 7. Decrement the number of clusters noc by Increment k by Repeat steps 6 to 8 untl the desred number of clusters s obtaned or all the shapes are put nto sngle a cluster. Algorthm Ends (9)

5 IV. EXPERIMENTAL RESULTS We have conducted several experments on varety of data sets to valdate the feasblty of the proposed methodology. In ths secton we present the clusterng results on a consderably large database of shapes. Experment 1: In order to llustrate the proposed method of shape clusterng, we frst consder a sample database of shapes consstng 9 shapes of 3 classes. The shapes to be clustered are frst subjected to feature extracton process as descrbed n secton IIA and then the smlarty measure descrbed n secton IIB s appled on the extracted features to obtan a smlarty matrx. Based on the smlarty matrx, a smlarty poston matrx s obtaned. The smlarty poston matrx gves the closeness of the shapes n terms of ther poston. The proposed k-mutual nearest neghbor approach s then appled on the smlarty poston matrx to obtan clusters as explaned n the algorthm Clusterng-of-Shapes. Fg. 2 shows the shapes and ther smlarty values, Table I shows the smlarty poston matrx and Fg. 3 shows varous stages of clusterng for dfferent values of k. shapes belong to the same category are very near and are mutual nearest neghbors. Fg. 2 Smlarty matrx TABLE I SIMILARITY POSITION MATRIX In the example, we have consdered 3 categores of shapes vz., Dog, Hand and Man. One can observe a sgnfcant varaton among shapes belongng to the same category. The smlarty matrx (Fg. 2) shows that the shapes belong to the same category possess more smlarty value than the shapes belong to the outsde category. Smlar observaton can be made from the smlarty poston matrx (Table 1) that the Fg. 3. Varous stages of clusterng From Fg. 3, we can observe that the shapes Dog-1 and Dog- 2, the shapes Hand-1 and Hand-3 and the shapes Man-1 and Man-3 are 2-mutually nearest neghbors (k = 2), hence, they are clustered n the frst level tself. We can also observe that they are vsually more smlar than the other shapes. In the second level, the shapes Dog-1, Dog-2 and Dog-3 are put nto cluster 1, the shapes Hand-1, Hand-2 and Hand-3 are put nto cluster 2 and the shapes Man-1, Man-2 and Man-3 are put nto cluster 3 as they are 3-mutually nearest neghbors (k = 3). No clusters are formed for k = 4, 5, 6 and 7, ths shows the clear margn between shapes belongng to same category and shapes belongng to outsde the category. Shapes Dog-1, Dog- 2, Dog-3, Man-1, Man-2 and Man-3 are clustered nto one group when k = 8 and fnally all the shapes are put nto one cluster when k = 9. Experment 2: In order to demonstrate the success of the proposed shape clusterng methodology, we have conducted an experment on a consderably large database of shapes. Fg. 4 shows the 350 shapes categorzed nto 35 classes, each class wth 10 example shapes. So t s expected to obtan 35 clusters from the proposed clusterng methodology. Fgure 5 (a-d) shows the clusters obtaned for the shapes shown n Fg. 4. We have shown only the last 4 levels of the clusters obtaned by the proposed methodology. An deal system s the one, whch produces all the 35 true clusters for k=10 as there are 10 shapes n each category. But because of large ntra class varaton among shapes, we cannot expect all the true clusters for k=10. We can observe from Fg. 5(a) that 26 true clusters are formed for k=73. APPLE and HEAD, FISH and LEAF are merged together at ths level to produce 2

6 clusters nstead of 4 clusters. Ths s because the shapes n these categores are vsually very smlar to each other. That s APPLE shapes are smlar to HEAD shapes and FISH category shapes are smlar to LEAF category shapes. Ths can be observed n Fg. 4. Shapes n the categores PLANE, HAND, HUMAN, MOMENTO and DOG are not completely formed ther clusters at ths level because shapes n these categores possess large ntra class varatons as observed n Fg. 4. Totally, we obtaned 38 clusters at ths level. In the next level, we have obtaned 37 clusters for k=82. Shape categores BELL and BUTTERFLY are merged at ths level and formed sngle group as shown n Fg. 5(b). No changes were observed n the clusters belongs to PLANE, HAND, HUMAN, MOMENTO and DOG categores. In the next level, for k=84, 36 clusters are realzed. At ths level CHOPPER and RABBIT are merged together to form a sngle cluster as they appear to be smlar and they are all mutually nearest neghbors for k=84 as shown n Fg. 5(c). Fnally, we obtaned 34 clusters for k=88 as shown n Fg. 5(d). Shape categores HAND and DOG, respectvely, formed ther true clear clusters. However, shape categores PLANE, HUMAN and MOMENTO stll could not form ther complete clusters. PLANE category results wth 2 clusters wth shapes {101, 102, 103, 108, 109} and {104, 105, 106, 107, 110}. HUMAN category results wth 2 clusters wth shapes {141, 142, 143, 144, 145, 146, 147, 148} and {149, 150}. Shapes {149, 150} are qute more deformed wth wdespread legs. Snce our method of feature extracton measures such span nformaton, these shapes are not merged wth all the other shapes n ts category. However, these shapes are not merged wth any other shape categores. Ths shows that the proposed method of clusterng s effectve and realstc n nature. Smlarly, shapes n the category MOMENTO formed 2 clusters wth shapes {281, 282, 283, 284, 289, 290} and {285, 286, 287, 288}. Smlar argument s true for ths category also. We have not encountered any mss groupng at any level. Ths s because all the shapes to be clustered nto one group must be mutually nearest neghbors. So only those shapes, whch belong to the same category, agree n ther structure and they only can be mutually nearest neghbors. Thus the proposed mutually nearest neghbor clusterng technque avods possble mss groupng, whch mght arse n case of smple agglomeratve clusterng technque. In smple agglomeratve clusterng technque, we could not get clear clusters for APPLE, HEAD, GREEBLE, BELT1, BELT2, FISH, LEAF and MOMENTO categores. Snce the proposed method of shape descrpton s based on the axs of least nerta, whch depends on the global structure of the shape, more shape deformaton or occluson mght change the orentaton of the axs of least nerta yeldng feature vectors, whch are entrely dfferent. Therefore the proposed method of shape clusterng s not ntended to consder major shape deformatons or occlusons. But one of the strength of the proposed shape descrpton s that t takes care of nternal detals of shapes such as holes, whch plays a sgnfcant role n most of the shape analyss applcatons. Fg. 4. Set of 35 categores of shapes each wth 10 examples

7 (a) (b) (c) (d) Fg. 5 Last four levels of clusters obtaned by the proposed methodology for shapes shown n Fg. 4

8 V. DISCUSSIONS AND COMPARATIVE STUDY In order to show the feasblty of the proposed smlarty measure for symbolc patterns and the proposed mutually nearest neghbourhood clusterng technque, we have conducted several experments on varety of data sets. In ths secton we present the clusterng results on a few well-known data sets. A. Experment 1 The frst experment s conducted on Ichno s fat ol data, whch has been used by several researchers as a typcal example of a data set nvolvng nterval-valued features. It s composed of eght patterns descrbed by four nterval-valued features [14]. The proposed smlarty measure s employed on the data set to estmate the degree of smlarty and the smlarty matrx s obtaned (see Table II). Based on the smlarty matrx, a poston matrx, whch gves the relatve nearness of patterns n the descendng order of smlarty, s created (see Table III). The concept of k-mutual nearest neghborhood s employed on the poston matrx and the dendrogram representaton of the cluster formed s shown n the Fg. 6. A dendrogram s a specal type of tree structure that provdes a convenent pcture of a herarchcal clusterng [15]. It conssts of layers of nodes, each representng a cluster and lnes connectng nodes to represent clusters whch are nested nto one another. Cuttng dendrogram horzontally creates a clusterng. From Fg. 6, we can observe that the patterns {1, 2}, {3, 4}, {5, 6} and {7, 8} are 2-mutually nearest neghbors (k=2). That means, the pattern 2 s n the second poston accordng to smlarty rank for the pattern 1. Smlarly, the pattern 1 s n the second poston accordng to smlarty rank for the pattern 2. One can notce ths fact from the smlarty matrx shown n Table II and the poston matrx shown n Table III. Thus the patterns {1, 2} are clustered n the frst level tself (dendrogram). The above argument s true for the patterns {3, 4}, {5, 6} and {7, 8}. The Patterns {3, 4} and {5, 6} are 5- mutually nearest neghbors (See Table III) and hence they are clustered n the second level (for k=5) as shown n the dendrogram (Fg. 6). As there are no patterns, whch are mutually nearest neghbors for k=3 and k=4, no patterns are merged for k=3 and k=4. The patterns {3, 4, 5, 6} and {1, 2} are 6-mutually nearest neghbors and are clustered at the level 3. No clusters are formed for (k=7) and fnally we get a sngle cluster at the level 4 for k=8. We can cut the dendrogram at any level and realze the clusters dependng on the desred number of clusters. TABLE II SIMILARITY MATRIX FOR FAT OIL TABLE III POSITION MATRIX OBTAINED BASED ON THE SIMILARITY RANK Pattern Smlarty Postons Number P 1 P 1 P 2 P 5 P 4 P 6 P 3 P 7 P 8 P 2 P 2 P 1 P 4 P 3 P 6 P 5 P 7 P 8 P 3 P 3 P 4 P 6 P 5 P 2 P 1 P 7 P 8 P 4 P 4 P 3 P 6 P 2 P 5 P 1 P 8 P 7 P 5 P 5 P 6 P 3 P 1 P 4 P 2 P 8 P 7 P 6 P 6 P 5 P 3 P 4 P 2 P 1 P 8 P 7 P 7 P 7 P 8 P 3 P 6 P 5 P 2 P 4 P 1 P 8 P 8 P 7 P 6 P 5 P 3 P 2 P 4 P 1 Fg. 6.Dendrogram representaton of the clusters formaton at varous levels for Fat-Ol data by the proposed methodology

9 B Experment 2 We have also conducted an experment on the data set gven n [14] on mcrocomputers. The data set descrbes a group of mcrocomputers consstng of 12 patterns. Each pattern has fve features. Two of the features are qualtatve (Dsplay and Mcroprocessor) and the rest are quanttatve (RAM, ROM and Keys). The smlarty proxmty matrx for ths data set s gven n the Table IV. The dendrogram representaton of the clusters formed at varous levels s shown n Fg. 7. The proposed algorthm resulted n two clusters {1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12} and {7}. In order to valdate the correctness, we have compared the results of the proposed method wth that of other avalable methodologes. For ths purpose we have consdered sx methodologes whch are lsted n Table V. Table V summarzes the results obtaned through the applcatons of all the seven methodologes ncludng the proposed methodology on two dfferent data sets vz., Fats and Ols data and Mcrocomputer data, as the results on these two data sets are studed by all the sx methodologes. It can be notced n Table V that the fats and ols patterns are ether grouped nto 2 clusters or nto 3 clusters. The methods [6], [9], [11], [14] have grouped the patterns nto 2 clusters ({1,2,3,4,5,6},{7,8}) and the methods [5], [8] have grouped the patterns nto 3 clusters ({1,2}, {3,4,5,6}, {7,8}) based on ther own cluster ndcator functon whch acts as a stoppng crteron. The entres not avalable n the Table V denote that the correspondng result has not been shown n the respectve research work. We have not computed the same durng expermentaton as those methodologes requre a pror knowledge of the number of samples n each pattern, whch s ndeed a real drawback of those approaches. Authors [14] TABLE IV SIMILARITY MATRIX FOR MICROCOMPUTER DATA Fg. 7. Dendrogram representaton of the clusters formed at varous levels for Mcrocomputer data by the proposed methodology

10 have gven the clusterng of samples grouped nto 2 clusters and as well as the samples grouped nto 3 clusters. When our method s employed on the fats and ols data and the dendrogram (Fg. 6) s cut at the level of 3 clusters, the results are same as that of all the methods whch yeld 3 clusters [5], [8], [11], [14] and when agglomeraton s allowed to contnue up to 2 clusters then the result obtaned s exactly same as that of the methods whch yeld 2 clusters [6], [9], [11], [14]. Ths shows consstency n the results of all the consdered and our method on the fats and ols data. It can also be notced from Table V that the results obtaned on Mcrocomputer data through all the 6 approaches are entrely dfferent except the results of the methods [9], [11] and [14]. In the work [6], t s stated that no consstency can be expected on Mcrocomputer data. However, our method has resulted wth 2 clusters, whch are same as that of the methods [9], [11] and [14] encouragng ther results. VI CONCLUSIONS In ths paper, a new smlarty measure useful for clusterng shapes descrbed by fuzzy-symbolc features s proposed. The concept of k-mutually nearest neghbours used for clusterng shapes provdes a realstc nsght nto the closeness among the shapes n a cluster. One can accept the dea that two set of shapes are grouped together to form a sngle cluster only when all the shapes n both the clusters are k-mutually nearest neghbours and ths fact can be revealed by the results of the proposed methodology on the shape database and also on the standard data sets. The effcacy of the proposed methodology s expermentally establshed and ts valdty s tested by comparng wth the well-known methodologes. Methodology TABLE V RESULTS BASED COMPARISION Fats and ols Mcrocomputer Descrpton at 2 Clusters level Descrpton at 3 Clusters level Descrpton at 2 cluster level Descrpton at level more than or equal to 3 clusters Ichno and Yaguch (1994) {1,2,3,4,5,6} {7,8} {1,2} {3,4,5,6} {7,8} {1,2,3,4,5,6,8,910,11,12} {7} {1,2,3,9,10} {4,5,6,8,11,12} {7} Gowda and Rav (1995(a)) Gowda and Dday (1991) Not avalable {1,2} {3,4,5,6} {7,8} Not avalable {1,2} { 3,4,5,6} {7,8} Not avalable {1,2,4,6,8,9,10,11,12} {3} {7} {5} Not avalable { 1,2,4,10,11} {7} {3,9} {5,6,12} {8} Gowda and Dday (1992) {1,2,3,4,5,6} {7,8} Not avalable Not avalable {1,2,10,11} {7} {3,9} {4,5,6,8,12} Gowda and Rav (1995(b)) {1,2,3,4,5,6} {7,8} Not avalable {1,2,3,4,5,6,8,9,10,11,12} {7} Not avalable Guru et al. (2004) {1,2,3,4,5,6} {7,8} {1,2} {3,4,5,6} {7,8} {1,2,3,4,5,6,8,9,10,11,12} {7} {1,2,3,4,9,10,11} {4,5,6,12} {7} {8} Proposed method {1,2,3,4,5,6} {7,8} {1,2} {3,4,5,6} {7,8} {1,2,3,4,5,6,8,910,11,12} {7} {1,2,3,4,8,9,10,11} {7} {5,6,12} REFERENCES [1] Berner T and J. A. Landry, "A new method for representng and matchng shapes of natural objects". Pattern Recognton, Vol. 36, pp [2] Bock H.H and E. Dday (Eds.), Analyss of symbolc data. Sprnger Verlag. [3] Daves E.R., Machne Vson: Theory, Algorthms and Practcaltes. Academc press, New York [4] Dengsheng Zhang and Guojun Lu, "Revew of shape representaton and descrpton technques". Pattern Recognton, Vol. 37, pp [5] Gowda K.C and E. Dday, Symbolc Clusterng usng a new dssmlarty measure. Pattern Recognton, Vol 24, No 6, pp [6] Gowda K.C and Dday E., Symbolc Clusterng usng a new smlarty measure. IEEE Transactons on System, Man and Cybernetcs. Vol. 22, No 2, pp

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