Matching patterns of line segments by eigenvector decomposition

Transcription

1 Title Mtching ptterns of line segments y eigenvector decomposition Author(s) Chn, BHB; Hung, YS Cittion The 5th IEEE Southwest Symposium on Imge Anlysis nd Interprettion Proceedings, Snte Fe, NM., 7-9 April 00, p Issued Dte 00 URL Rights 00 IEEE. Personl use of this mteril is permitted. However, permission to reprint/repulish this mteril for dvertising or promotionl purposes or for creting new collective works for resle or redistriution to servers or lists, or to reuse ny copyrighted component of this work in other works must e otined from the IEEE.; This work is licensed under Cretive Commons Attriution-NonCommercil-NoDerivtives 4.0 Interntionl License.

2 Mtching Ptterns of Line Segments y Eigenvector Decomposition H.B. Chn Deprtment of Electricl nd Electronic Engineering The University of Hong Kong Pokfulm Rod, Hong Kong hchn@eee.hku.hk Y.S. Hung Deprtment of Electricl nd Electronic Engineering The University of Hong Kong Pokfulm Rod, Hong Kong yshung@eee.hku.hk Astrct This pper presents n lgorithm for mtching line segments in two imges which re relted y n ffine trnsformtion. Imges re represented s ptterns of line segments. Ares defined etween line segments re used to descrie line pttern in the form of proimity mtri. Mtches re determined y compring the feture vectors otined from eigenvlue decomposition of the proimity mtrices. Relile mtches of line segments re otined in oth synthetic nd rel imges.. Introduction Feture mtching hs lwys een very difficult prolem in computer vision nd mny different solutions hve een proposed. Scott nd Longuet-Higgins [] hve proposed n lgorithm for mtching two D point ptterns. In their lgorithm, point ptterns re descried y their inter-imge distnces in the form of proimity mtri. However, it does not work well when lrge rottion is introduced etween imges. In order to overcome this prolem, Shpiro nd Brdy [] used intrimge distnce insted of inter-imge distnce. Ecellent results re otined for trnsltion nd rottion motions. In order to cope with ffine trnsformtions, Xue nd Teoh [3] used re rtios s entries of proimity mtrices. As re rtio is ffine invrint, the method works well for ffine trnsformtions. However, re rtios defined depend on the conve hull of the point set, therefore the result deteriortes if some outer most points re missing. All the ove methods re point-sed. Points re very sensitive to noise nd more difficult to etrct ccurtely thn lines. Prk, Lee nd Lee [4] proposed method for line mtching. However the method involves complicted preliminry tests, nd the proimity mtri is not ffine invrint. In this pper, n eigenvector pproch is proposed for line mtching etween two line ptterns. The method is new in tht res defined etween line segments re used to represent line ptterns. It works well for ptterns under ffine trnsformtion, even in cses with missing fetures.. The New Eigenvector Approch The rtio of two res is well-known ffine invrint. It cn e shown tht corresponding res under n ffine trnsformtion re relted y scle fctor. Consider three points (, y ), (, y ) nd ( 3, y 3 ), the re enclosed y them is 3 S y y y3 () Suppose (w, v ), (w, v ) nd (w 3, v 3 ) re the corresponding points of (, y ), (, y ) nd ( 3, y 3 ) fter pplying n ffine trnsformtion A: r r t A r r t () 0 0 Then, wi r r i t +, (i,,3), (3) vi r r yi t The re enclosed y (w, v ), (w, v ) nd (w 3, v 3 ) is S w v w v w3 v3 Fifth IEEE Southwest Symposium on Imge Anlysis nd Interprettion (SSIAI 0) /0 $ IEEE

3 r r 0 r r 0 t t y y ( r r r r ) S αs 3 y3 (4) From eqution (4), S S is constnt α determined y A. In our lgorithm, imges re represented s line segments. First, edges re etrcted from imges. The thinned edges re represented s chins nd stright lines segments re then fitted to the chins under prescried error tolerence. Figure. Segment i nd segment j Assuming nd re two line ptterns to e mtched. Suppose there re M nd N segments in nd respectively. Consider segment i nd segment j, oth eisting in the sme line pttern, s shown in Figure, where the two segments re represented y solid lines. The centroid of the two line segments is denoted C(i,j). Let Q e the re enclosed y segment i nd the centroid, nd Q e the re enclosed y segment j nd the centroid. The entry G (, ) of n re mtri G is defined to e the verge re ( Q + Q ). Hence G is squre nd symmetric. As mentioned ove, corresponding res under ffine trnsformtion re relted y fctor α. Elements in G nd G re ctully corresponding res rrnged in different orders. Scling G y α will turn the corresponding entries in the two re mtrices into ffine invrints. Since the elements in the two re mtrices differ y scling fctor α, their eigenvlues lso differ y α fter eing rrnged in decresing order. To clculte α, only the first k (k<min(m,n)) most significnt eigenvlues re used k i k α λ λ (5) i m Proimity mtrices H re formed y pplying Gussin weightings to the elements of the scled re mtrices: H ep( α mn Gmn / σ ), H ep( G / σ ), for m, n M, i, j N (6) The purpose of Gussin weightings is to loclize the interctions etween segments. Since lrger G mens greter distnce etween segments i nd j, the Gussin weightings mke entries in G with greter vlues less dominting. σ controls the size of neighourhood of interctions. Since G is squre, positive definite nd symmetric, it cn e decomposed y n eigenvlue decomposition into T G VDV with V [ E E E3... E m ] where the columns V re the eigenvectors of G nd D is digonl mtri contining the eigenvlues in decresing order. The rows of V re etrcted s feture vectors F s follow: T T T V [ F, F,... F ] T m After trunction nd sign correction, these feture vectors re used to clculte the distnce Z [3], defined s Z Fi F j for i M, j N (7) If distnce Z is zero, it indictes perfect mtch of segment i in pttern with segment j in pttern. In prctice ny Z elow prescried threshold nd is the minimum for oth row i nd column j of Z is ccepted s mtch. 3. Eperimentl Results Here we demonstrte our lgorithm y using two sets of imges: the first set is cropped from n imge trees ville in Mtl imge processing toolo, the other set contins o. The edges of the imge trees (shown in Figure ) is etrcted, cropped nd divided into 33 line segments (shown in Figure 3). Dots represent the end points of line segments; circles represent the midpoints of line segments. A second pttern is formed y pplying n ffine trnsformtion to the pttern of line segments in Figure 3 (shown in Figure 4). The two ptterns of line segments re therefore relted y n ffine m Fifth IEEE Southwest Symposium on Imge Anlysis nd Interprettion (SSIAI 0) /0 $ IEEE

4 trnsformtion. All 33 segments re mtched correctly. The result is shown in Figure 5, where o nd + re used to mrk mtched segment midpoints of the two ptterns nd solid lines represent correspondences of segments. Line segments re not shown for etter visuliztion. Figure 5. Mtched line segments of trees Figure. Trees In the second emple, imge of o with tringles on its fces re tken y digitl cmer. These imges re tken t different ngles nd re not ffinely relted. We will refer to the first imge s o nd the second imge s o, which re shown in Figures 6 nd 7 respectively. Ptterns of line segments re etrcted from these imges (shown in Figures 8 nd 9), where dots represent the end points of line segments, nd circles represent the midpoints of line segments. As seen in the figures, there re 8 segments etrcted form oth o nd o. The mtched result is shown in Figure. 0. Symols re used with the sme menings s in Figure. 5 ecept tht the two ptterns of segments re drwn in dotted nd dshed lines respectively. All 8 segments re mtched correctly. Therefore it cn e seen tht the method works well even in cses with rel imges which re not ectly relted y ffine trnsformtions. 4. Conclusion Figure 3. Pttern of line segments etrcted from trees We hve proposed new line mtching lgorithm mking use of res defined etween line segments. These res re then scled to e ffine invrint. Proimity mtrices re clculted from these res. Mtches re otined y compring feture vectors otined from eigenvlue decomposition of the proimity mtrices. Relile mtches of line segments re otined in oth synthetic nd rel imges. Figure 4. Pttern of line segments otined y pplying n ffine trnsformtion to Figure 3 Fifth IEEE Southwest Symposium on Imge Anlysis nd Interprettion (SSIAI 0) /0 $ IEEE

5 Figure 6. Bo Figure 9. Pttern of line segments etrcted from Bo Figure 7. Bo Figure 0. Mtched line segments of o 5. References [] G.L. Scott nd H.C. Longuet-Higgins, An lgorithm for ssociting the fetures of two ptterns, Proceeding of Royl Society London, B, vol. 44, pp. -6, 99. [] L.S. Shpiro nd J.M. Brdy, Feture-sed correspondence: n eigenvector pproch, Imge nd Vision Computing, vol. 0, no. 5, pp , 99. [3] Z. Xue nd E.K. Teoh, A novel eigenvector pproch to pose nd correspondence estimtion, Systems, Mn, nd Cyernetics, 000 IEEE Interntionl Conference on, vol., pp , 000. Figure 8. Pttern of line segments etrcted from Bo [4] S.H. Prk, K.M. Lee nd S.U. Lee, A line feture mtching technique sed on n eigenvector pproch, Computer Vision nd Imge Understnding, vol. 77, pp , 000. Fifth IEEE Southwest Symposium on Imge Anlysis nd Interprettion (SSIAI 0) /0 $ IEEE