Shape Representation and Indexing Based on Region Connection Calculus and Oriented Matroid Theory

Transcription

1 Shpe Representtion nd Indexing Bsed on Region Connection Clculus nd Oriented Mtroid Theory Ernesto Stffetti 1, Antoni Gru 2, Frncesc Serrtos 3, nd Alerto Snfeliu 1 1 Institute of Industril Rootics (CSIC-UPC) Llorens i Artigs 4-6, Brcelon Spin {estffetti,snfeliu}@iri.upc.es 2 Deprtment of Automtic Control, Technicl University of Ctloni Pu Grgllo 5, Brcelon Spin ntoni.gru@upc.es 3 Deprtment of Computer Engineering nd Mthemtics Rovir i Virgili University, Av. Pisos Ctlnes 26, Trrgon Spin frncesc.serrtos@etse.urv.es Astrct. In this pper novel method for indexing views of 3D ojects is presented. The topologicl properties of the regions of the views of set of ojects re used to define n index sed on the region connection clculus nd oriented mtroid theory. Both re formlisms for qulittive sptil representtion nd resoning nd re complementry in the sense tht wheres the region connection clculus encodes informtion out connectivity of pirs of connected regions of the view, oriented mtroids encode reltive position of the disjoint regions of the view nd give locl nd glol topologicl informtion out their sptil distriution. This indexing technique is pplied to 3D oject hypothesis genertion from single views to reduce cndidtes in oject recognition processes. 1 Introduction In this pper we present new method for indexing views of 3D ojects which is pplied to 3D oject hypothesis genertion from single views to reduce cndidtes in 3D oject recognition processes. Given set of views of different 3D ojects, the prolem of oject recognition using single view ecomes the prolem of finding suset of the set of regions in the imge with reltionl structure identicl to tht of memer of the set of views. The stndrd wy to reduce the complexity of shpe mtching is sudividing the prolem into hypothesis genertion followed y verifiction. To e of interest for oject recognition, hypothesis genertion should e reltively fst lthough imprecise procedure in which severl possile cndidtes for mtching re generted. In this wy the verifiction cn e crried out using more complex, nd therefore, slower procedure [1] over reduced numer of I. Nyström et l. (Eds.): DGCI 2003, LNCS 2886, pp , c Springer-Verlg Berlin Heidelerg 2003

2 268 Ernesto Stffetti et l. DC(, ) EC(, ) PO(, ) TPP(, ) NTPP(, ) EQ(, ) () () (c) (d) (e) (f) Fig. 1. Some of the 8 possile reltive positions of two regions nd the corresponding descriptions using the formlism of the region connection clculus. The other two cn e otined from (d) nd (e) interchnging with. In sitution () is disconnected from, in() is externlly connected to, in sitution (c) is prtilly overlpped to, in(d) is tngentil proper prt of, in(e) is non-tngentil proper prt of nd, finlly, in sitution (f) nd coincide. cndidtes. The hypothesis genertion cn e crried out very efficiently if it is formulted s n indexing prolem where the set of views of the set of 3D ojects re stored into tle tht is indexed y some function of the views themselves. In this pper n indexing technique tht comines the region connection clculus nd oriented mtroid theory is presented. More precisely, the type of connectivity etween connected regions of the views is descried y mens of the formlism of the region connection clculus [2], wheres the topologicl properties of the disconnected regions of the views re encoded into dt structure clled set of cocircuits [3]. The set of cocircuits, tht re one of the severl comintoril dt structure referred to s oriented mtroids, encode incidence reltions nd reltive position of the elements of the imge nd give locl nd glol topologicl informtion out their sptil distriution. Resoning with the region connection clculus is sed on composition tles, while oriented mtroids permit lgeric techniques to e used. These two descriptions merged re used s n index of the dtse. This indexing method is employed to the hypothesis genertion for 3D oject recognition from single views tht cn e regrded s qulittive counterprt of the geometric hshing technique [4]. For nother pproch to shpe representtion nd indexing sed on comintoril geometry see [5]. The region connection clculus nd oriented mtroids re introduced in Section 2 wheres Section 3 descries the proposed indexing method. In Section 4 some experimentl results re reported nd Section 5 contins the conclusions. 2 Qulittive Sptil Representtion Qulittive resoning is sed on comprtive knowledge rther thn on metric informtion. Mny methods for shpe representtion nd nlysis re sed on extrcting points nd edges which re used to define projectively invrint descriptors. In this pper, insted of points, regions of the imges re tken into ccount. The motivtion ehind this choice is tht the regions of n imge cn e more relily extrcted thn vertices nd edges. In the following sections two formlisms for qulittive representtion nd resoning re descried: the first

3 Shpe Representtion nd Indexing 269 OUT(, ) P-INS(, ) INS(, ) () () (c) Fig. 2. Some of the possile positions of convex region with respect to the convex hull of non-convex one. one is sed on the region connection clculus nd the second one is derived from oriented mtroid theory. 2.1 Region Connection Clculus For sptilly extended ojects we cn qulittively distinguish the interior, the oundry, nd the exterior of the oject, without tking into ccount the concrete shpe or size of the oject. A set theoreticl nlysis of the possile reltions etween ojects sed on the ove prtition is provided y [6]. The reltion etween ojects tht they exmine is the intersection etween their oundries nd interiors. This setting is sed on the distinction of the vlues empty nd non-empty for the intersection. Some vrints of this theory were developed y Cohn nd his coworkers in series of ppers (see for exmple [2]). In this work the distinction etween interior nd the oundry of n oject is ndoned, nd eight topologicl reltions derived from the single inry reltion connected to re tken into ccount. Some of them re represented in Fig. 1. Some of these reltions, nmely those of Fig. 1.d nd Fig. 1.e, re not symmetricl nd, following the nottion of [2], their inverses re denoted TPPi(, ) nd NTTPi(, ), respectively. Furthermore in [2] the theory is extended to hndle concve ojects y distinguishing the regions inside nd outside of the convex hull of the ojects. A convex oject cn e inside, prtilly inside or outside the convex hull of non-convex one (Fig. 2). If oth regions re non-convex 23 reltions etween them cn e defined. These reltions permit qulittive description of rther complex reltions, such s tht represented in Fig. 3. Moreover, y mens of this formlism clled region connection clculus it is possile, for instnce, to infer the reltive position of two regions knowing their position with respect to third one. Resoning with the region connection clculus is essentilly sed on composition tles. 2.2 Oriented Mtroids Oriented mtroid theory [3], [7], [8] is rod setting in which the comintoril properties of geometricl configurtions cn e descried nd nlyzed. It

4 270 Ernesto Stffetti et l. Fig. 3. With the formlism of the region connection clculus the reltion etween these two disconnected non-convex regions, where is prtilly inside the convex hull of nd vice vers, is denoted y P-INS P-INSi DC(, ). provides common generliztion of lrge numer of different mthemticl ojects usully treted t the level of usul coordintes. In this section oriented mtroids will e introduced over rrngements of points using two comintoril dt structures clled chirotope nd set of cocircuits, which represent the min tools to trnslte geometric prolems into this formlism. In the strction process from the concrete configurtion of points to the oriented mtroid, metric informtion is lost ut the structurl properties of the configurtion of points re represented t purely comintoril level. Oriented Mtroids of Arrngements of Points. Given point configurtion in R d 1 whose elements re the columns of the mtrix P =(p 1,p 2,...,p n ), the ssocited vector configurtion is finite spnning sequence of vectors {x 1, x 2,..., x n } in R d represented s columns of the mtrix X =(x 1,x 2,...,x n ) where ech point p i is represented in homogeneous coordintes s x i = ( p i ) 1. To encode the comintoril properties of the point configurtion we cn use dt structure clled chirotope [8], which cn e computed y mens of the ssocited vector configurtion X. The chirotope of X is the mp χ X : {1, 2,..., n} d {+, 0, } (λ 1,λ 2,...,λ d ) sign ([x λ1,x λ2,...,x λd ]) tht ssigns to ech d-tuple of vectors of the finite configurtion X sign + or depending on whether it forms sis of R d hving positive or negtive orienttion, respectively. This function ssigns the vlue 0 to those d-tuples tht do not constitute sis of R d. The chirotope descries the incidence structure etween the points of X nd the hyperplnes spnned y the sme points nd, t the sme time, encodes the reltive position of the points of the configurtion with respect to the hyperplnes tht they spn. Consider the point configurtion P represented in Fig. 4 whose ssocited vector configurtion X is given in Tle 1. Tle 1. Vector configurtion tht corresponds to the plnr point configurtion represented in Fig. 4. x 1 =(0, 3, 1) T x 2 =( 3, 1, 1) T x 3 =( 2, 2, 1) T x 4 =(2, 2, 1) T x 5 =(3, 1, 1) T x 6 =(0, 0, 1) T

5 p2 p1 p6 Shpe Representtion nd Indexing 271 p5 p3 p4 Fig. 4. A plnr point configurtion. Tle 2. Chirotope of the plnr point configurtion represented in Fig. 4. χ(1, 2, 3)=+χ(1, 2, 4)=+χ(1, 2, 5)=+χ(1, 2, 6)=+χ(1, 3, 4)=+ χ(1, 3, 5)=+χ(1, 3, 6)=+χ(1, 4, 5)=+χ(1, 4, 6) = χ(1, 5, 6) = χ(2, 3, 4)=+χ(2, 3, 5)=+χ(2, 3, 6)=+χ(2, 4, 5)=+χ(2, 4, 6)=+ χ(2, 5, 6) = χ(3, 4, 5)=+χ(3, 4, 6)=+χ(3, 5, 6)=+χ(4, 5, 6)=+ Tle 3. Set of cocircuits of the plnr point configurtion represented in Fig. 4. (0, 0, +, +, +, +) (0,, 0, +, +, +) (0,,, 0, +, ) (0,,,, 0, ) (0,,, +, +, 0) (+, 0, 0, +, +, +) (+, 0,, 0, +, +) (+, 0,,, 0, ) (+, 0,,, +, 0) (+, +, 0, 0, +, +) (+, +, 0,, 0, +) (+, +, 0,,, 0) (+, +, +, 0, 0, +) (, +, +, 0,, 0) (,, +, +, 0, 0) The chirotope χ X of this vector configurtion is given y the orienttions listed in Tle 2. The element χ(1, 2, 3) = + indictes tht in the tringle formed y p 1, p 2, nd p 3 these points re counterclockwise ordered. These orienttions cn e rerrnged in n equivlent dt structure clled set of cocircuits of X shown in Tle 3. In this plnr cse, the set of cocircuits of X is the set of ll prtitions generted y the lines pssing through two points of the configurtion. For exmple, (0, 0, +, +, +, +) mens tht the points p 3, p 4, p 5, nd p 6 lie on the hlf plne determined y the line through the points p 1 nd p 2. Reversing ll the signs of the set of cocircuits we otin n equivlent description of the plnr rrngement of points. Besides chirotopes nd cocircuits there re severl dt structures cple of encoding the topologicl properties of point configurtion. In [8] their definitions cn e found nd it is shown tht ll of them re equivlent nd re referred to s oriented mtroids. Oriented Mtroid of Arrngements of Regions. Consider segmented view of 3D oject. Extrcting the oriented mtroid of view is not strightforwrd since the regions tht form the imge cnnot e reduced to points, tking for instnce their centroids, without losing essentil topologicl informtion for

6 e1 e2 IS,T l2 IS,T RS,T 272 Ernesto Stffetti et l. oject recognition. Therefore, the convex hull [9] of ech region is employed to represent the region itself. Then, pirs of the resulting convex polygons re considered nd the oriented mtroid is computed sed on the sptil loction of the other convex regions of the imge with respect to the two lines rising in merging the convex hulls of pirs disconnected regions. Consider, for instnce, the ordered pir of convex regions (S, T ) of Fig. 5.. It is esy to see tht the convex hull of these two plnr convex disconnected polygonl regions is polygon whose set of vertices is included in the union of the set of vertices of S nd T. On the contrry, the set of edges of the convex hull of S nd T is not included in the union of their set of edges. Indeed, two new ridging edges, e 1 nd e 2, pper s illustrted in Fig. 5.. Actully, efficient lgorithms for merging convex hulls re sed on finding these two edges [10]. LS,T T U l1 Z S V () () (c) Fig. 5. Steps of encoding of the comintoril properties of view of n oject into chirotope. Consider the two lines l 1 nd l 2 tht support e 1 nd e 2. These two lines divide the imge into three or four zones depending on the loction of their intersection point with respect to the imge. Let R S,T, L S,T (Fig. 5.) e, respectively, the rightmost nd leftmost zones with respect to l 1 nd l 2 nd I S,T the zone of the imge comprised etween them. Since, R S,T, L S,T nd I S,T cn e univoclly determined from the ordered couple of region (S, T ), the loction of region U with respect to the regions (S, T ) of the imge is encoded into chirotope using the following rule + if U L S,T, χ(s, T, U) = 0 if U I S,T, if U R S,T. It hs een implicitly ssumed tht U is completely contined into either R S,T L S,T or I S,T ut, in generl, it elongs to more tht one of them. In this cse, since the rtio of res is n ffine invrint, introducing n pproximtion, we cn choose the sign sed on which region contins the lrgest portion of the re of U. For instnce, if regions U, V nd Z re locted s in Fig. 5.c we hve tht χ(s, T, U) = +,χ(s, T, V )= 0 nd χ(s, T, Z) =. 2.3 Invrince of the Representtion Consider 3D point configurtion nd one of its views. The comintoril structure of the 3D point configurtion nd tht of its 2D perspective projection re

7 Shpe Representtion nd Indexing 273 relted in the following wy: if x 0 represents in homogeneous coordintes the center of the cmer, p 0, we hve tht sign[ x i, x j, x k ] = sign[x i,x j,x k,x 0 ] (1) where x i, x j nd x k re the homogeneous coordintes of the 3D points p i, p j nd p k, nd x i, x j nd x k re those of the corresponding points in the view, p i, p j nd p k. Eqution (1) cn e regrded s projection eqution for chirotopes. It is esy to see tht, wheres the mtrix tht represents in homogeneous coordintes the vertices of projected set of points is coordinte-dependent, n oriented mtroid is coordinte-free representtion. Moreover, the representtion of oject views sed on oriented mtroid is topologicl invrint, tht is, n invrint under homeomorphisms. Roughly speking, this mens tht the oriented mtroid tht represents the rrngement of points of view of n oject does not chnge when the points undergo continuous trnsformtion tht does not chnge ny orienttion of the chirotope. Doe to this property this representtion is roust to discretiztion errors of the imge s well s to smll chnges of the point of view tht does not chnge ny orienttion of the chirotope. Since projective trnsformtions cn e regrded s specil homeomorphisms, we cn ssert tht the representtion of the projected set of points sed on oriented mtroids is projective invrint. However, since ffine nd Eucliden trnsformtions re specil projective trnsformtions, the oriented mtroid of the projected set of points of view of n oject does not chnge under rottions, trnsltions, nd ffine trnsformtions of the plnr rrngement of points themselves. These considertions cn e extended to the cse in which oriented mtroids represent rrngements of plnr regions. Since the rtio of res is not invrint under projective trnsformtions this representtion will e invrint only under ffine nd Eucliden trnsformtions of the views. 3 Indexing Views of 3D Ojects The process of indexing dtse of views of set of ojects strts with some preliminry choices, nmely the fetures used to chrcterize the regions of the segmented views of the set of 3D ojects. Suppose tht hue nd re re used to chrcterize ech region. Another prmeter to choose is the numer of levels in which the hue is quntized nd the numer of regions hving the sme hue tht will e tken into ccount. These choices, of course, depend on the properties of the views of the dtse. Then, the views re segmented ccording to these choices nd the convex hull of ech region is computed. As consequence, the resulting imges re compositions of convex polygonl regions tht cn e disconnected or prtilly or completely overlpped. In Fig. 6 re represented two views of two ojects in which hue quntiztion with 6 levels W, R, Y, G, B nd N hs een pplied nd only the two iggest regions with the sme hue vlue re tken into ccount.

8 G2 B2 B2 274 Ernesto Stffetti et l. Let (W, R, Y, G, B, N) e the ordered tuple of hue levels considered. For exmple, lels G 1 nd G 2 in Fig. 6 denote, respectively, the first nd the second regions of the views with the iggest re hving the sme hue vlue G. The type of connection etween the existing regions is descried using the formlism of the region connection clculus. For ech pir of disconnected regions the set of cocircuits is computed. This is done for ech view of the dtse nd this informtion is comined into unique index tle whose entries re sptil comintions of fetures nd whose records contin list of the views in which ech comintion is present. W Y N G1 R N B1 W B1 G1 Oject 1 Oject 2 Fig. 6. Two views of two ojects whose topologicl properties re indexed in Tle 4. In Tle 4 the index of the topologicl properties of the two views v 1,1 nd v 1,2 of the ojects represented in Fig. 6 is reported. In the first column the reltion etween ordered couples of regions is descried in terms of the region connection clculus. The symol for certin couple (S, T ) indictes tht no view contins two regions hving fetures S nd T. This is the cse of the regions R nd Y. When S nd T re disconnected, the corresponding cocircuit is present in the index. The symol in correspondence with certin feture indictes tht no region with tht feture is present in the views listed in the record. For exmple, the cocircuit WR contins in the column Y ecuse no region with the Y feture is present in v 1,1. If (S, T ) is couple of connected regions, the corresponding row of the index is empty ecuse the cocircuit cnnot e computed. 3.1 Hypothesis Genertion for Oject Recognition Given dtse of views of set of 3D ojects nd view v i of one of them, not necessrily contined in the dtse, its set of cocircuits is computed. Ech cocircuit is used to ccess the tle tht constitutes the index of the dtse. Then the views tht est mtch v i re selected sed on the numer of correspondences they hve with v i in terms of cocircuits. It is esy to see tht this method for hypothesis genertion, tht cn e regrded s qulittive version of the geometric hshing technique [4], is lso roust to prtil occlusions of the ojects. Indeed, if region of n imge is

9 Shpe Representtion nd Indexing 275 Tle 4. Index of the topologicl properties of the two views v 1,1 nd v 1,2 of the two ojects represented in Fig. 6. Connection W R Y G 1 G 2 B 1 B 2 N Ojects WR DC v 1,1 WY DC v 1,2 WG 1 NTPP v 1,1 WG 1 DC v 1,2 WG 2 DC v 1,1 WB 1 DC v 1,1 WB 1 NTPP v 1,2 WB 2 DC v 1,1 WB 2 NTPPi v 1,2 WN DC v 1,1 WN DC v 1,2 RY RG 1 NTPP v 1,1 B 2N DC v 1,1 B 2N DC v 1,2 occluded, the set of cocircuits cn still e computed nd therefore, the numer of correspondences with the views of the dtse cn still e clculted. In this cse, oviously, its selectivity decreses. 4 Experimentl Results The method hs een fully implemented nd experiments with different sets of 3D ojects hve een crried out to vlidte it. Sixteen views of ech oject with ngulr seprtion of 22.5 degrees hve een used for the experiments. These imges hve een segmented using the segmenttion method descried in [11]. Then, the index of the lerning set of eight views per oject tken t the ngles 0, 45, 90, 135, 180, 225, 270 nd 315 hs een creted. In the recognition process the set of cocircuits of ech imge of the test set composed y the eight views not used in the lerning process tht is, the views tken t ngles: 22.5, 67.5, 115.5, 157.5, 202.5, 247.5, nd degrees, hs een clculted. The experimentl results re encourging nd currently we re refining the method introducing distnce mesure etween set of cocircuits. 5 Conclusions In this pper new method for indexing dtse of views of 3D oject hs een presented. It is sed on the comintion of two qulittive representtions derived from the region connection clculus nd oriented mtroid theory. This comintion of qulittive representtions chrcterizes the locl nd glol topology of the regions of n imge, is invrint under ffine nd Eucliden trnsformtion of the views, intrinsiclly roust to discretiztion errors of the imge nd insensitive to smll displcements of the point of view.

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