Structural Graph Matching using the EM Algorithm and Singular Value Decomposition

Transcription

1 CCV2002: Th 5th sian Confrnc on Computr Vision, January 2002, lbourn, ustralia 1 Structural Graph atching using th E lgorithm and Singular Valu Dcomposition Bin Luo Univrsity of York,York, UK and nhui univrsity, PR China. luo@cs.york.ac.uk Edwin Hancock Univrsity of York,York, UK. rh@cs.york.ac.uk bstract This papr dscribs an fficint algorithm for inxact graph-matching. Th mthod is purly structural, that is to say it uss only th dg or connctivity structur of th graph and dos not draw on nod or dg attributs. W mak two contributions. Commncing from a probability distribution for matching rrors, w show how th problm of graph-matching can b posd as maximum liklihood stimation using th apparatus of th E algorithm. Our scond contribution is to cast th rcovry of corrspondnc matchs btwn th graph-nods in a matrix framwork. This allows us to fficintly rcovr corrspondnc matchs using singular valu dcomposition. W xprimnt with th mthod on both ral-world and synthtic data. Hr w dmonstrat that th mthod offrs comparabl prformanc to mor computationally dmanding mthods. 1 Introduction Graph-matching is a task of pivotal importanc in highlvl vision sinc it provids a mans by which abstract pictorial dscriptions can b matchd to on-anothr. Sinc th procss of liciting graph structurs from raw imag data is a task of som fragility du to nois and th limitd ffctivnss of th availabl sgmntation algorithms, graph-matching is invariably approachd by inxact mans [10] n importnnt ida hr has bn to us dit-distanc to compar graphs [10] and it has rcntly bn shown that th dit distanc is rlatd to th siz of th maximum common subgraph [2]. nothr powrful way to dal with inxactnss is to modl th structural rrors prsnt in th graphmatching problm in a probabilistic way. Wong and You [15] mad on of th first contributions hr by dfining an ntropy masur for structural graph-matching. Boyr and Kak [1] also adoptd an information thortic approach, but workd instad with attribut rlations. Using a probabilistic rlaxation framwork Christmas, Kittlr and Ptrou [3] hav dvlopd a statistical modl for pairwis attribut rlations. Working in th purly structural domain, Wilson and Hancock [14] hav drivd probability distributions for th rlational rrors that occur whn thr is significant graph corruption. Onc a masur of graph similarity is to hand thn th sarch for th st of corrspondnc matchs may b posd as an optmisation or nrgy minimisation problm. nothr important mthod which draws idas from th fild of mathmatics known as spctral graph thory [4], is to cast th graph-matching problm in a matrix stting and to us th ignvalus and ignvctors of th adjacncy matrix as a rprsntation of rlational structur. For instanc, Umyama has an igndcomposition mthod that matchs graphs of th sam siz [13]. Borrowing idas from structural chmistry, Scott and Longut- Higgins wr among th first to us spctral mthods for corrspondnc analysis [11]. Thy showd how to rcovr corrspondncs via singular valu dcomposition on th point association matrix btwn diffrnt imags. In kping mor closly with th spirit of spctral graph thory, yt smingly unawar of th rlatd litratur, Shapiro and Brady [12] dvlopd an xtnsion of th Scott and Longut-Higgins mthod which prforms multidimnsional scaling on th point-st proximity matrics to xtract a fatur-vctor for matching. Horaud and Sossa[8] hav adoptd a purly structural approach to th rcognition of lin-drawings. Thir rprsntation is basd on th immanntal polynomials for th Laplacian matrix of th linconnctivity graph. By comparing th cofficints of th polynomials, thy ar abl to indx into a larg data-bas of lin-drawings. lthough formally lgant, th main limitation of ths matrix mthods is thir inability to cop with graphs of diffrnt sizs. This mans that thy can not b usd whn significant lvls of structural corruption ar prsnt. orovr, thr has bn littl attmpt to rndr thm robust using probabilistic or statistical mthods. Basd on ths obsrvations our aim in this papr is to cast th statistical matching of graphs into a matrix rprsntation and to xploit singular valu mthods to fficintly rcovr corrspondncs. W commnc by dvloping a liklihood function for th graph-matching problm. This trats th graph to b matchd (th data-graph) as obsrvd data and th st of corrspondncs with th availabl modl (th modl-graph) as hiddn variabls. ccordingly, w construct a mixtur modl ovr th st of corrspondncs btwn th nods of th data-graph and thos of th modlgraph. W adopt a Brnoulli modl for th probability distribution of th corrspondnc rrors ncountrd in matching th data-graph to th modl-graph. Th xistnc or othrwis of corrspondnc rrors is gaugd using th dg-consistncy of th pattrn of matchs. 2 Liklihood Function Our ovrall goal in this papr is to dvlop a maximum liklihood framwork for structural graph matching. In this sction w dvlop th liklihood function undrpinning our study. To commnc w must dfin som notation. W us th notation to dnot th graphs

2 > > 5 t m undr match, whr is th st of nods and is th st of dgs. Our aim in matching is to associat nods in a graph rprsnting " #$ #% data to # b & matchd ' against thos from th st in a graph () * rprsnting an availabl modl. Formally, th matching is rprsntd by a function from th nods in th data graph to thos in th modl graph. Suppos that th stat of match btwn th two graphs is rprsntd by th function +-, /.0 from th nods of th data-graph to thos of th modl-graph. W will us latin lttrs to dnot nods from th data-graph and grk lttrs to dnot nods from th modl-graph. Hnc, th statmnt & 98 8=:< 5;:< mans that th nod is assignd th labl or symbol. On of th goals in this papr is to show how th two graphs can b matchd using matrix factorisation mthods. W thrfor introduc som matrix notation to rprsnt th graphs. To this nd w dfin a > >?@> > matching matrix B1 34 whos lmnts ar assignmnt variabls which convy th following maning CEDGF IH&J if + 657K L8 (1) othrwis W rprsnt th structur of th two graphs using a >?N> >E?<> > adjacncy matrix O for th data graph and a > adjacncy matrix P for th modl graph. Th lmnts of th adjacncy matrix for th data graph ar dfind as follows O DGQ IH&J if 65SRGB: (2) othrwis whil thos for th modl graph ar dfind to b P FT IH&J 68UVW: if (3) othrwis Sinc w ar working with undirctd graphs, th two adjacncy matrics ar symmtric, i.. O OYX and P PZX. Having introducd th ncssary formalism, w now procd to dvlop our maximum liklihood framwork for graph-matching. W sk th matrix of assignmnt variabls that maximiss th conditional liklihood of th obsrvd data-graph givn th availabl modl graph. Hnc, w sk th matrix of assignmnt variabls which satisfis th condition L[\^]`_a[b c df > )g Nxt w construct a mixtur modl ovr th st of possibl corrspondncs. W follow th standard approach to constructing th liklihood function for a mixtur distribution. This involvs factorising th liklihood function ovr th obsrvd data (i.. th nods of th data-graph) and summing ovr th hiddn or unobsrvd variabls (i.. th corrsponding nods in th modl-graph). s a rsult w writ h > K ji D k lnm F7k po q whr D o > is th probability that data-graph 8 nod is in corrspondnc with th modl-graph nod undr th matrix of assignmnt variabls. In ordr to procd, q w rquir a modl for th obsr- D >. W commnc from th as- vation dnsity o sumption that th obsrvation dnsity is factorial ovr th paramtrs of th mixtur modl, i.. th st of assignmnt variabls. If this is th cas, thn w can writ (4) (5) K i Qrk l i Tsk * CEQqT Nxt w dvlop a modl for th probability distribution for th obsrvd st of corrspondncs btwn th nods of th data and th modl graphs givn th currnt st of as- q signmnt paramtrs, i. D > CEQqT. Our modl draws on th rcnt work of Wilson and Hancock [14] and assums that th obsrvd data-graph nods ar drivd from th modl-graph nods through a Brnoulli distribution. Th paramtr of this distribution is th probability of corrspondnc rror è t. Th ida bhind this modl is that th modal-graph nod D # can mit a symbol F drawn from th st of modl-graph nods. Th probability that this symbol is th corrct corrspondnc is J u è t whil th probability that it is in rror is è t. To gaug th corrctnss 5 of th mittd symbol, w chck whthr th nods 68UVv: R and of th data-graph ar matchd to a valid dg xw of th modl-graph. To tst for dg-consistncy, w mak us of th quantity O D y Q P FTCEQqT IH&J if 65SRGB: and 68UVW: othrwis (7) Using this switching proprty, th Brnoulli distribution bcoms CEQqT K z J{u ` }rx~g% r} 2t (6) S ƒ` }rx~g r} t (8) With th factorial assumption and th distribution rul to hand, th obsrvation dnsity bcoms K i Qrk l i Tsk J%u ` }rx~g r} 2t S ƒ` }rx~g r} t (9) This xprssion is xponntial in charactr. It can b rwrittn in as a natural xponntial function ˆ) L Œ S ƒ Ž whr Ž K N Z Gb7 ˆ m Qrk l and / Tsk O DGQ P FTCEQqT7Š (10) lk. Finally, th corrsponding log-liklihood function for th assignmnt matrix isk (11) D k m l %] F7k m Gb7 ˆ m Qrk l Tsk m O DGQ P FTCEQqT7Š2 Unfortunatly, bcaus of th mixtur structur th dirct stimation of th matrix of assignmnt variabls from th log-liklihood function is not tractabl in closd form. For this rason, in th nxt sction w xplain how th xpctation-maximisation algorithm may b usd instad. 3 Expctation-aximisation Having dvlopd our computational modl which poss th graph-matching problm in a maximum-liklihood framwork, in this sction w provid a concrt algorithm for rcovring th paramtrs of th undrlying mixturmodl. W choos to us th E algorithm originally introducd by Dmpstr, Laird and Rubin [5]. Th utility masur undrpinning th algorithm is th xpctd logliklihood function. Th basic ida undrlying th algorithm is to itrat btwn th intrlavd xpctation and maximisation stps until convrgnc is rachd. Expctation involvs updating th a postriori probabilitis of th missing data using th most rcntly availabl paramtr stimats. In th maximisation phas, th modl paramtrs ar rcomputd to maximis th xpctd valu of th incomplt data liklihood. 2

3 6 65 j Ž 5 œ 3.1 Expctd log-liklihood function For our graph-matching problm, maximisation of th xpctation of th conditional liklihood is quivalnt to maximising th wightd log-liklihood function " # $ %" (12) whr indicats th matrix of assignmnt variabls takn at itration & of th E algorithm. Hnc, th a postriori corrspondnc matching probabilitis computd at " ar usd to wight th itration &('*) contributions to th log-liklihood function. With th xpctd log-liklihood function to hand, th maximum-liklihood matrix of assignmnt variabls is th on which satisfis th condition itration &, i.. +*, -/.102, (13) On way to ralis th updat procss is by paralll itrativ local gradint ascnt. In th nxt sction w show how th xpctd log-liklihood function can b rcast in a matrix framwork. This allows us to ralis th updat procdur mor fficintly using singular valu dcomposition. 3.2 atrix Rprsntation To commnc, w not that whn th distribution function for th assignmnt variabls is substitutd from Equation (10) th xpctd log-liklihood function bcoms : ;=<>?@CB DFE <>?DHGJI KL ONPL Q%R KSPUTV >?D WXZY\[^] K/_(P4Q` >?@CB D Qba (14) c^dfgc^hi whr w hav introducd th matrix j k l" whos lmnts j ar st qual to th a postriori probability of corrspondnc match btwn th data-graph nod m and th modl-graph nod n at itration & of th E algorithm. Th critical quantity in dtrmining th updat dirction for maximum liklihood matchs is 7o kut kq/vg4s^w pqorps q s (15) In matrix form th xpctd log-liklihood is 7 k 1*x9yz t { j v {%} (16) 3.3 aximisation Th maximisation stp of th E algorithm can b statd as that of rcovring th st of corrspondnc indicators which +~, -/.+02,43 satisfis th condition x9yzt { j v {%} (17) In othr words, th utility masur gaugs th dgr of corrlation btwn th dg-sts of th two graphs undr th wightd prmutation structur inducd by th corrspondnc probabilitis. To locat th updatd st of corrspondnc indicators w us th xtrmum principal rportd by Scott and Longut-Higgins [11]. Thir rsult is as follows. Suppos that c^d 1f c^hi is a positiv dfinit c^df c^hƒ matrix. Thy hav shown how th x9yz orthogonal matrix that maximiss th quantity U { } may b found by prforming singular valu dcomposition. To do this thy prform c8d Jfˆc8d th matrix factorisation c 2 { c, whr is a orthogonal cchi fˆc^hƒ matrix, is a orthogonal matrix and Œ c^d fšc^hi is a matrix whos diagonal lmnts Œ p Ž if and whos diagonal lmnts p ar non-zro. Suppos that is th matrix obtaind from by making th diagonal lmnts p unity. Th matrix x9yz which maximiss U { } is c {. This xtrmum principl may b applid to our graph matching problm if w mak th substitution t { j v and prform th singular valu dcomposition t { j v c 2 { to obtain. This matrix satisfis th condition *, -/.102,43 x9yzt { j v 7 {%} (18) Providd that th matrix t { j v is positiv-dfinit, thn th lmnts of ar ral. lthough this xtrmum principl is usful, it is not ntirly suitd to our nds. Th rasons for this ar that th lmnts of can not b intrprtd as probabilitis sinc thy ar nithr guarantd to b positiv, nor ar thy normalisd. Furthrmor, thy can not b intrprtd as assignmnt indicators sinc thy ar not binary in natur. To ovrcom ths difficultis, w follow Scott and Longut- Higgins by tsting th lmnts of to obtain a matrix of If th lmnt binary corrspondnc indicators. p is th maximum valu for both th row and column that contains it, thn th assignmnt indicator w p is st to unity. Othrwis it is st to zro. s a rsult th updatd st of corrspondnc indicators is w k ) if k ~, -/.+02,43 q s q s othrwis (19) 3.4 Expctation In th xpctation stp of th E algorithm, th a postriori probabilitis of th hiddn data ar computd from th componnt dnsitis apparing in th mixtur-distribution. This is don by applying th Bays thorm. t itration & '*) w hav whr l" 1 š %" O š %" O c8d ) l" (20) (21) 4 Exprimnts In this sction of th papr, w provid som xprimntal valuation of th nw graph-matching tchniqu. Thr ar two aspcts to this study. W commnc with a snsitivity study using synthtic data. Th aim hr is to valuat how th nw mthod prforms undr controlld structural corruption and to compar it with som altrnativs rportd lswhr in th litratur. Th scond part of th study valuats th mthod on ral-world data. 4.1 Snsitivity Study Our snsitivity study is dividd into two parts. W compar our mthod with som altrnativ mthods for inxact graph-matching which rly on matrix factorisation tchniqus. Ths mthods do not work whn th graphs ar of diffrnt siz. Hr w kp th graphs of fixd qual siz and invstigat th ffct of corrupting th pattrn of dgs. 3

4 j m lk l m lk l n Inxact Graph atching W commnc by studying th ffct of controlld structural rror on th graphs bing matchd. Th graphs usd in our study ar th Dlaunay triangulations of randomly gnratd point-sts. Th ffcts of structural rror ar simulatd by dlting a prdfind fraction of randomly slctd nods and r-triangulating th rmaining points. W compar th prformanc of our nw matching mthod with thr altrnativs. Ths ar th dictionarybasd rlaxation schm of Wilson and Hancock [14], th quadratic assignmnt mthod of Gold and Rangarajan [7] and th non-quadratic graduatd assignmnt mthod of Finch, Wilson and Hancock [6]. Figur 1 compars th four algorithms. Hr w show th fraction of corrct corrspondncs as a function of th fraction of nods dltd from th graphs. Th main fatur to not is that th nw graph matching mthod dlivrs prformanc that is intrmdiat btwn th discrt rlaxation mthod and th non-linar graduatd assignmnt mthod. This is an intrsting obsrvation whn w compar th computational ovrhads associatd with th thr mthods. Positiv corrspondnc rat Non-quadratic assignmnt Discrt rlaxation Quadratic SVD Prcntag of cluttr Figur 1. Snsitivity study for graphs of diffrnt siz Factorisation thods In this subsction w provid comparison with two mthods for wightd graph-matching which shar with our own mthod th fatur of rlying on matrix factorisation. Th mthods slctd for this comparison ar Umyama s wightd graph-matching mthod which sks th prmutation matrix that minimiss quantity [13]. Th mthod prforms th singular valu dcompositions and, whr th s ar orthogonal matrics and th s ar diagonal matrics. Onc ths factorisations hav " bn prformd, th rquird prmutation matrix is. Shapiro and Brady s [12] wightd graph-matching mthod which uss th modal structur of th two wightd adjacncy matrics and. Th modal structur of th two adjacncy graphs is found by solv- $# ing th ignvalu quation % '& % # % &. whr % is # th (*),+ ignvalu of th adjacncy matrix and % is th corrsponding ignvctor. Th ignvctors ar ordrd according to th siz of th associatd ignvalus and ar usd as th columns of th modal matrix -. /0# 1 2 # 342 # This procdur is rpatd to construct a scond modal matrix - for th modl-graph adjacncy matrix. Th column indx of ths two modal matrics rfrs to th ordr of th ignvalus whil th row-indx is th indx of th nods in th graphs. Shapiro and Brady find corrspondncs by locating pairs of rows which hav minimum distanc, i.. 9;:=< >@?BC if D?FEHGJILK4 N >PORQTS U VXWXY Y Z[]\_^a`Jb_cXdZfg\ DXh `Jb_c;Y Y i othrwis (22) Ths two mthods rly on wightd adjacncy matrics rathr than th binary ons dfind arlir. To conduct our xprimnts, w hav gnratd random 2D point-sts. W us th positions of ths points to gnrat th wights of th adjacncy matrix. Suppos that k and k rprsnt th co-ordinat vctors associatd with th nods indxd o and p. Th wight associatd with th dg3zy conncting th nods is n rqhsat4uv wx m n (23) Ths two mthods ar not ffctiv whn th graphs undr study contain diffrnt numbrs of nods. To compar with our mthod w hav thrfor kpt th numbr of points fixd and hav addd Gaussian rrors to th point positions. Th paramtr of th nois procss is th standard dviation of th positional jittr. In our xprimnts, w xprss this paramtr as a fraction of th avrag minimum distanc btwn points (th rlativ standard dviation). It is important to strss that th mthods compard hr us diffrnt rprsntations of th arrangmnt of th points. Th Shapiro and Brady, and Umyama mthods us th wightd adjacncy matrix. Our mthod, on th othr hand, uss a binary adjacncy matrix to rprsnt th Dlaunay triangulation of th points. Corrct corrspondnc rat(%) LUO(Wightd) LUO(Binary) UEY SHPIRO Rlativ position dviation Figur 2. Comparison of th four igndcomposition mthods for graphs with th sam numbr of nods. In Figur 2 w show th fraction of corrct corrspondncs as a function of th rlativ standard dviation for our nw mthod (bold curv), Umyama s [13] mthod (solid curv) and th mthod of Shapiro and Brady [12] (dottd curv). Th main fatur to not is that our mthod outprforms th two altrnativs. Thr is littl to distinguish th prformanc of th Shapiro and Brady [12], and Umyama [13] mthods. Both fail abruptly onc th rlativ standard dviation xcds 0.2, i.. th nois standard dviation is gratr than 20% of th avrag closst point distanc. Our mthod, on th othr hand, dgrads almost linarly with th nois standard dviation. Howvr, it must b strssd that th rsults ar not compltly comparabl. In th cas of Shapiro and Brady, and Umyama [13], w 4

5 ar masuring th snsitivity of th mthod to nois on th ntris of th wightd adjacncy matrics. In th cas of our mthod, w ar masuring th snsitivity of th mthod to rrors in th dg-sts of th graphs usd for matching. Finally, w illustrat th rsults obtaind whn w apply our mthod to th wightd adjacncy matrix rathr than th binary adjacncy matrix. Th dot-dashd curv in Figur 1 shows th fraction of corrct corrspondncs as a function of th rlativ standard dviation of th point-position jittr. Th mthod prforms considrably bttr than th Shapiro and Brady, and Umyama mthods. Howvr, thr is littl to distinguish its prformanc from that obtaind with th binary adjacncy matrix. 4.2 Ral-world data W commnc our ral-world valuation of th graphmatching mthod on imags of indoor scns. Hr w ar concrnd with matching th Dlaunay triangulations of cornr-faturs. W us th cornr dtctor rcntly rportd by [9] to xtract point faturs. Figur 3 shows two xampls of th indoor imags usd in our study. Suprimposd on th imags ar th dtctd cornrs and thir associatd Dlaunay triangulations. Th two imags ar takn from diffrnt viwpoints. Thr is rotation, scaling and prspctiv distortion prsnt. orovr, svral of th objcts in th scn ar at diffrnt dpths and mov rlativ to on-anothr. s a rsult thr ar significant structural diffrncs in th two Dlaunay graphs. Figur 4 shows th corrspondncs btwn th cornrs as lins btwn th two imags. ftr chcking by hand, th fraction of corrct corrspondncs is 77%. Figur 5. Dlaunay graphs ovrlayd on th toy hous imags. Figur 3. Tst imags ovrlayd with Dlaunay graphs. Figur 6. Corrspondncs for pairs of imags with incrasing diffrnc in viwing angl. th imags. Thr ar clarly significant structural diffrncs in th graphs. Figur 6 shows th rsults obtaind whn pairs of imags in th squnc ar matchd. Th rsults ar summarisd in Tabl 1. Hr w list th numbr of dtctd cornrs in th imags bing matchd, th numbr of cornrs that ar in corrct corrspondnc, th numbr of cornrs that ar in rror, and th numbr of cornrs for which thr ar no corrspondncs (i.. thr is no row and column maximum). Th mthod braks down aftr th 4th imag in th squnc. To provid som comparison, w hav slctd a pair of imags which contain th sam numbr of cornr points (imag 2 and imag 4). lthough th numbr of cornrs is th sam, thr ar diffrncs in th both idntitis of th dtctd points and thir structural arrangmnt. For ths imags w compar th matchs rturnd by th unwightd and wightd vrsions of our algorithm (rfrrd to as Luo), th mthod of Umyama and th mthod of Shapiro and Brady. Th rsults ar shown in Figurs 7 and 8 and th numbrs of corrct matchs ar summarisd in Figur 4. Corrspondncs. W hav prformd our xprimnts using imags takn from th CU/VSC modl-hous squnc. Th imags usd in our study ar shown in Figur 5 and corrspond to diffrnt camra viwing dirctions. Th dtctd cornr faturs and thir Dlaunay triangulations ar ovrlayd on 5

6 Tabl 2. From ths rsults it is clar that th nw mthod rturns considrably bttr matchs. Imags Cornrs Corrct Fals Unmatchd hous 1 30 hous hous hous hous hous Tabl 1. Summary of xprimntal rsults for th hous squnc imags. thods Corrct Fals Unmatchd Luo(Wightd) Luo(Unwightd) Umyama Shapiro Tabl 2. Summary of th comparison of th thr matching algorithms. Figur 7. Corrspondncs from th Umyama (lft) and Shapiro (right) algorithms. Figur 8. Corrspondncs from th unwightd (lft) and wightd (right) variants of our algorithm. 5 Conclusions Our main contributions in this papr ar twofold. First, w hav cast th problm of graph-matching into a maximum liklihood framwork by constructing a mixtur modl ovr th st of hiddn corrspondncs and adopting a Brnoulli modl for th distribution of dg-matching rrors. Scond, w hav usd th apparatus of th E algorithm to show how th problm of stimating th corrspondnc indicators may b cast into a compact matrix stting. This allows us to us singular valu dcomposition to stimat th corrspondnc indicators in th -stp. Th rsult is an fficint algorithm that can b usd to accuratly match inxact graphs undr considrabl lvls of structural corruption. Whn viwd from th prspctiv of rcnt work on matrix-basd graph-matching, th important contribution of this papr is to show how point-sts of diffrnt sizs can b matchd using singular valu dcomposition. Rfrncs [1] K. Boyr and. Kak. Structural Stropsis for 3D Vision. IEEE PI, 10: , [2] H. Bunk. Error corrcting graph matching: On th influnc of th undrlying cost function. IEEE PI, 21: , [3] W.J. Christmas, J. Kittlr, and. Ptrou. Structural matching in computr vision using probabilistic rlaxation. IEEE PI, 17(8): , [4] F.R.K. Chung. Spctral Graph Thory. mrican athmatical Socity Ed., CBS sris 92, [5].P. Dmpstr, N.. Laird, and D.B. Rubin. aximum-liklihood from incomplt data via th E algorithm. J. Royal Statistical Soc. Sr. B (mthodological), 39:1 38, [6].. Finch, R.C. Wilson, and E.R. Hancock. n nrgy function and continuous dit procss for graph matching. Nural Computation, 10(7): , [7] S. Gold and. Rangarajan. graduatd assignmnt algorithm for graph matching. IEEE PI, 18(4): , [8] R. Horaud and H. Sossa. Polyhdral objct rcognition by indxing. Pattrn Rcognition, 28(12): , [9] B. Luo,.D.J. Cross, and E.R. Hancock. Cornr dtction via topographic analysis of vctor potntial. Pattrn Rcognition Lttrs, 20: , [10]. Sanfliu and K.S. Fu. distanc masur btwn attributd rlational graphs for pattrn rcognition. IEEE Trans. Systms, an and Cybrntics, 13(3): , ay [11] G.L. Scott and H.C. Longut-Higgins. n lgorithm for ssociating th Faturs of 2 Imags. Procdings of th Royal Socity of London Sris B-Biological, 244(1309):21 26, [12] L.S. Shapiro and J.. Brady. Fatur-basd Corrspondnc - n Eignvctor pproach. Imag and Vision Computing, 10: , [13] S. Umyama. n ign dcomposition approach to wightd graph matching problms. IEEE PI, 10: , [14] R.C. Wilson and E.R. Hancock. Structural matching by discrt rlaxation. IEEE T-PI, 19(6): , Jun [15].K.C Wong and You. Entropy and distanc of random graphs with application to structural pattrn rcognition. IEEE PI, 7: ,