Affine Invariant Matching of Broken Boundaries Based on Differential Evolution

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1 Affne Invarant Matchn of Broken Boundares Based on Dfferental Evoluton Wuchao tu P.W.M. Tsan Department of Electronc Enneern Cty Unversty of Hon Kon Hon Kon Chna Abstract - Affne nvarant matchn of a par of contours can be posed as a search process to determne whether an affne transform ests between them. Past research has also demonstrated that the search process can be swftly accomplshed wth smple enetc alorthms. In addton the method can be etended to the matchn of framented obect contours wth the nteraton a contour reconstructon scheme. The latter enables a closed outermost boundary to be etracted from dscontnued ede ponts and sements. Despte the effectveness of ths approach the performance n terms of computaton tme and success rates s unfavorable for certan knds of obect contours. Ths paper proposes to overcome these problems by employn the dfferental evoluton alorthm n the search process. Epermental results reveal that the proposed method s faster than estn method based on enetc alorthms. Keywords: Affne Invarant Matchn Broken contours Dfferental evoluton Dstance transform Contour reconstructon e-sample M M d (y Genetc alorthm d (y Dstance transform F. 1. Essental buldn blocks of the parent method Decson 1 Introducton Applcaton of enetc alorthm (GA n shape matchn s an mportant research area n computer vson n the past two decades. Attempts that have been made nclude but are n lmted to the works reported n [1]-[5]. ecently t has been demonstrated n [6] (hereafter refer as the parent method that the matchn of framented contours can be accomplshed wth the nteraton of GA the mrant prncple and a contour nteraton scheme. Despte the effectveness of ths approach the processn tme and the success rates could devate rather snfcantly for dfferent contours. In ths paper we propose to overcome the shortcomns n the parent method by replacn the enetc alorthm wth dfferental evoluton (DE whle retann the remann parts of the process ntact. Epermental evaluaton reveals that our proposed method s faster and capable to match dfferent contours wth hher success rates. For clarty of eplanaton we would brefly descrbe the parent method wth the daram shown n Fure 1. To ben wth the matchn process can be dvded nto two staes and outlned as follows. 1.1 tae 1: preprocessn and scene contour reconstructon Let and represent the reference and scene contours respectvely. nce the reference contour s a closed boundary t s represented by an ordered sequence of M contour ponts ven by ( y ( y ( y 1 1 = o o o o o o (1 M M The scene contour s framented and s represented by a set of N ede ponts as {( y ( y ( y o o o o o N 1 o N 1 } where ( o o and ( o o = (2 are the rectanular coordnates. The reference contour s unformly re-sampled to a sequence of K ponts as ven by ( y ( y ( y 1 1 M = om om om om om K om K (3

2 y where ( om om = ( o M K o M K. A dstance map s constructed for each of the contours wth the dstance transform ven by ( 2 y 2 = ( o + ( y o d mn (4 < M ( 2 y 2 = ( o + ( y o d mn. (5 < N Net an so-contour on the dstance map d ( y (.e. pels wth dentcal values T s traced resultn n a sequence of Q ponts as ( y ( y ( y Q 1 Q 1 I = s s s s s s (6 such that d ( sk sk = T k< Q the dstance of I from and ( s y. T s a constant defnn s s an arbtrary start pont on the so-contour. Fnally a closed outermost boundary M s enerated from the framented scene contour by replacn each pont n I wth ts nearest pont n.e. ( y ( y ( y 1 1 M = om om om om omq om Q (7 om om and nearest to the pont where ( ( s s on the so-contour. 1.2 tae 2: contour matchn based on a smple enetc alorthm An affne transform A can be derved ven a par of correspondn non-collnear and non-overlappn pont trplets: P = [ s s1 s2 ] on M (the seed ponts and TP = [ t t1 t2 ] on M (the taret ponts. The affne transform A and ts nverse A ~ are appled to the reference and the scene contours to enerate a new par of contours as ( T = A (8 T ( ~ = A. (9 M = A A N 1 - ( ( d M 1 1 d M = N = = + y T where ( and ( (1 T. M s bounded between [1] representn the ftness of the ndvdual and reflectn the smlarty between the reference and scene contours. GA s employed to determne whether ven a fed set of seed ponts P there are a correspondn set of test ponts TP whch would lead to an affne transform relatn and. If affrmatve t can be concluded that the par of contours are matched to each her. As the contour M s normalzed to K=128 sample ponts n [6] the test pont can be represented by a 7-bt bnary number. 2 Proposed method based on Dfferental Evoluton As eplaned n secton 1 the performance of the parent method vares wth n the matchn of dfferent contour pars. To overcome ths problem we propose to adopt DE n place of GA n the parent method. To ben wth an ntal populaton of N ndvduals each comprsn a 21-bt bnary strn representn the three test ponts s created. Then the populaton underoes teratve evolutonary operatons: mutaton crossover and selecton. The teraton wll stop untl the mamum matchn score n the populaton eceeds a preset threshold or allowable number of eneratons has elapsed. The alorthm s epressed n the form of pseudocode and lsted n Table 1 and the natons for the alorthm are lsted n Table 2. To determne the smlarty of score s defned as and a matchn

$( ma M( M 1 N smatch = false = whle (<=MAX_GEN for = 1 to N do do {r1=randnt(1n} whle (r1== do {r2=randnt(1n} whle ((r2== (r2==r1 do {r3=randnt(1n} whle ((r3== (r3==r1 (r3==r2 MV = r1 + F ( r 2 r 3$ for =1 to 3 do rand = rand nt 13 ( rand 1 C = f ( rand tv = mv else tv = v end f end for f M ( TV M( + 1 = TV M TV M f ( ( = TV end f else + 1 = end f end for f M ( > Threshold smatch = true end = +1

for =1 to 3 do rand = rand nt 13 ( rand 1 C = f ( rand tv = mv else tv = v end f end for f M ( TV M( + 1 = TV M TV M f ( ( = TV end f else + 1 = end f end for f M ( > Threshold smatch = true end = +1

3 Table 1 Pseudo-code for the DE alorthm n shape matchn et the eneraton number = Establsh an ntal populaton PP. ( ma M( M 1 N smatch = false = whle (<=MAX_GEN for = 1 to N do do {r1=randnt(1n} whle (r1== do {r2=randnt(1n} whle ((r2== (r2==r1 do {r3=randnt(1n} whle ((r3== (r3==r1 (r3==r2 MV = r1 + F ( r 2 r 3 for =1 to 3 do rand = rand nt 13 ( rand 1 C = f ( rand tv = mv else tv = v end f end for f M ( TV M( + 1 = TV M TV M f ( ( = TV end f else + 1 = end f end for f M ( > Threshold smatch = true end = +1 end whle Table 2 Natons of Table 1 Varables Descrptons + The eneraton of the DE Z alorthm + N Z The populaton sze = The -th ndvdual vector of the v v populaton at eneraton { v } PP 1 = 2 3 { } N The ndvdual havn the hhest matchn score found so far The populaton of eneraton F The scale factor wth F [1] C M ( MAX_GEN ( L U rand nt 3 Epermental results The crossover rate wth C [1] The matchn score of an ndvdual based on Eq. (1. mamum of allowable eneratons a functon producn a random nteer wthn [L U] The model contours of two obects: (A hammer and (B scssors as shown n the frst column of Table 3 are employed to evaluate the performance of the proposed method. Each model contour s matched aanst two of ts varants (columns 2 and 3 of Table 3 captured n real world envronment. Ne that the obects n columns 2 and 3 are broken n many places ehbtn ll-formed boundares. Table 3 eference contours and scene contours eference cene 1 cene 2 A A1 A2

B B1 B2 Table 4 Iso-contours and outermost boundares for the unknown scene obects n Table 3 1 98 96 94 92 9 88 86 84 Parent method DE method A1 A2 B1 B2 cene 1 cene 2

A1 A2 B1 B2 F. 3. Comparson of mean eneratons requred n matchn the scene obects wth the parent scheme and proposed scheme The matchn process s outlned as follows.

Net determnaton of the matchn score between each scene contour and ts correspondn reference contour s conducted wth DE (crossover rate C=.8 scale factor F=.

The success rates attaned n matchn each par of scene and reference contours wth the proposed scheme and the parent method n [6] are lsted n F. 2.

The converence speed n terms of mean eneraton requred to attan successful match s depcted n F 3 where a faster converence speed n the DE method can be ned for all the

4 Conclusons Althouh affne nvarant matchn of broken contours can be effectvely realzed wth smple enetc alorthms and contour reconstructon the performance can devate

4 B B1 B2 Table 4 Iso-contours and outermost boundares for the unknown scene obects n Table Parent method DE method A1 A2 B1 B2 cene 1 cene 2 F. 2. Comparson of mean success rates (% n matchn the scene obects wth the parent scheme and proposed scheme A1 A2 B1 B Parent method DE method A1 A2 B1 B2 F. 3. Comparson of mean eneratons requred n matchn the scene obects wth the parent scheme and proposed scheme The matchn process s outlned as follows. To ben wth the method descrbed n secton 1.1 s employed to reconstruct the outermost boundares of the framented scene contours and the results are shown n Table 4. Net determnaton of the matchn score between each scene contour and ts correspondn reference contour s conducted wth DE (crossover rate C=.8 scale factor F=.11 and populaton sze N=1. The par of contours s consdered as matched once the matchn score eceeds a threshold (a value of.625 s adopted n the eperment. The success rates attaned n matchn each par of scene and reference contours wth the proposed scheme and the parent method n [6] are lsted n F. 2. It can be seen that the DE method acheved enerally hher success rates than the parent method. The converence speed n terms of mean eneraton requred to attan successful match s depcted n F 3 where a faster converence speed n the DE method can be ned for all the test obects compared wth the parent scheme. 4 Conclusons Althouh affne nvarant matchn of broken contours can be effectvely realzed wth smple enetc alorthms and contour reconstructon the performance can devate rather snfcantly for dfferent knds of obect contours. Ths paper proposes to allevate ths problem by employn dfferental evoluton alorthm to conduct the search process. Epermental results show the proposed method s superor to the parent method n terms of success rate and converence speed. 5 eferences [1] H.. Lm.H. Cherah. An optmzaton approach to shape matchn and reconton. Computers & Electrcal Enneern Vol. 24 ssue

5 [2] L. Zhan W. Xu C. Chan. Genetc alorthm for affne pont pattern matchn Pattern econton Letters Vol. 24 ssue [3] X.Q. Yan W.F. L D.F. Chen. Imae reconton based on evolutonary alorthm Proceedns of Internatonal Conference on Machne Learnn and Cybernetcs IEEE vol pp Nov. 22. [4] P.W.M. Tsan. A enetc alorthm for nvarant reconton of obect shapes from broken boundares Pattern econton Letters Vol. 18 ssue [5] P.W.M. Tsan T.Y.F. Yuen. Affne nvarant matchn of broken boundares based on an enhanced enetc alorthm and dstance transform IET Computer Vson Vol. 2 ssue [6] P.W.M. Tsan W.C. tu. Affne nvarant matchn of broken boundares based on smple enetc alorthm and contour reconstructon Pattern econton Letters Vol. 31 ssue

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