Estimating the Fundamental Matrix by Transforming Image Points in Projective Space 1
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1 Estmatng the Fundamental Matrx by Transformng Image Ponts n Projectve Space 1 Zhengyou Zhang and Charles Loop Mcrosoft Research, One Mcrosoft Way, Redmond, WA 98052, USA E-mal: fzhang,cloopg@mcrosoft.com Ths paper proposes a novel technque for estmatng the fundamental matrx by transformng the mage ponts n projectve space. We therefore only need to perform nonlnear optmzaton wth one parameterzaton of the fundamental matrx, rather than consderng 36 dstnct parameterzatons as n prevous work. We also show how to preserve the characterstcs of the data nose model from the orgnal mage space. Key Words: fundamental matrx, eppolar geometry, projectve transformaton 1. INTRODUCTION Two perspectve mages of a sngle rgd object/scene are related by the so-called eppolar geometry, whch can be descrbed by a 3 3 sngular matrx known as fundamental matrx. Robust and accurate estmaton of the fundamental matrx s very mportant n many applcatons of 3D computer vson, and has been the focus of many researchers [4, 5, 6, 7]. Prevous work on nonlnear estmaton of the fundamental matrx requres the consderaton of 36 dstnct parameterzatons to account for the fact that an eppole may be at nfnty and an element of the eppolar transformaton may be equal to 0. Ths leads to a cumbersome mplementaton of the optmzaton procedure. Ths work proposes a novel technque for estmatng the fundamental matrx. Instead of usng 36 maps to parameterze the fundamental matrx, we transform the mage ponts n projectve space, and n turn we only need to perform nonlnear optmzaton wth one parameterzaton of the fundamental matrx. We also show how to preserve the characterstcs of the data nose model from the orgnal mage space. 2. PARAMETERIZATION OF THE FUNDAMENTAL MATRIX Ths secton revews the parameterzaton of the fundamental matrx. 1 Publshed n Computer Vson and Image Understandng, Vol.82, No.2, pages , May
2 2 ZHANG AND LOOP 2.1. Eppolar geometry The eppolar geometry exsts between any two camera systems. For a pont m n the frst mage, ts correspondence n the second mage, m 0, must le on the eppolar lne n the second mage. Ths s known as the eppolar constrant. Algebracally, n order for m and m 0 to be matched, the followng equaton must be satsfed: em 0T F em =0; (1) where F, known as the fundamental matrx, sa3 3 matrx of rank 2 (.e., det(f) =0), defned up to a scale factor; em and em 0 are homogeneous coordnates of ponts m and m 0,.e., em =[m T ; 1]T and em 0 =[m0t ; 1] T Parameterzaton based on the eppolar transformaton There are several possble parameterzatons for the fundamental matrx [4], e.g., we can express one row (or column) of the fundamental matrx as the lnear combnaton of the other two rows (or columns). One possblty s the followng: F = 2 4 a b ax by c d cx dy ax 0 cy 0 bx 0 dy 0 F (2) wth F 33 =(ax + by)x 0 +(cx + dy)y 0 : Here, x and y are the coordnates of the frst eppole, x 0 and y 0 are the coordnates of the second eppole, and a, b, c and d, defned up to a scale factor, parameterze the eppolar transformaton mappng an eppolar lne n the frst mage to ts correspondng eppolar lne n the second mage. However, ths parameterzaton does not work f an eppole s at nfnty. Ths s because n that case at least one of x; y; x 0 and y 0 has a value of nfnty. In order to overcome ths problem, INRIA group [1, 7] has proposed to use n total 36 maps to parameterze the fundamental matrx, as summarzed below maps to parameterze the fundamental matrx Let us denote the columns of F by the vectors c 1, c 2 and c 3. The rank-2 constrant on F s equvalent to the followng two condtons: 9 1 ; 2 such that c j0 + 1 c j1 + 2 c j2 =0 (3) 6 9 such that c j1 + c j2 =0 (4) for j 0 ;j 1 ;j 2 2f1; 2; 3g, where 1 ; 2 and are scalars. Condton (4), as a non-exstence condton, cannot be expressed by a parameterzaton: we shall only keep condton (3) and so extend the parameterzed set to all the 3 3-matrces of rank strctly less than 3. Indeed, the rank-2 matrces of, for example, the followng forms: [c 1 c 2 c 2 ] and [c c 3 ] and [c 1 c ] do not have any parameterzaton f we take j 0 =1. A parameterzaton of F s then gven by (c j1 ; c j2 ; 1 ; 2 ). Ths parameterzaton mples to dvde the parameterzed set among three maps, correspondng to j 0 =1, j 0 =2and j 0 =3.
3 ESTIMATING THE FUNDAMENTAL MATRIX IN PROJECTIVE SPACE 3 If we construct a 3-vector such that 1 and 2 are the j th 1 and j th 2 coordnates and 1 s the j th 0 coordnate, then t s obvous that ths vector s the egenvector of F, and s thus the eppole n the case of the fundamental matrx. Usng such a parameterzaton mples to compute drectly the eppole whch s often a useful quantty, nstead of the matrx tself. To make the problem symmetrcal and snce the eppole n the other mage s also worth beng computed, the same decomposton as for the columns s used for the rows, whch now dvdes the parameterzed set nto 9 maps, correspondng to the choce of a column and a row as lnear combnatons of the two columns and two rows left. A parameterzaton of the matrx s then formed by the two coordnates x and y of the frst eppole, the two coordnates x 0 and y 0 of the second eppole and the four elements a, b, c and d left by c 1, c 2, l j1 and l j2, whch n turn parameterze the eppolar transformaton mappng an eppolar lne of the frst mage to ts correspondng eppolar lne n the second mage. The parameterzaton shown n (2) corresponds to the case 0 =3and j 0 =3. At last, to take nto account the fact that the fundamental matrx s defned only up to a scale factor, the matrx s normalzed by dvdng the four elements (a; b; c; d) by the largest n absolute value. We have thus n total 36 maps to parameterze the fundamental matrx. 3. A NOVEL TECHNIQUE FOR ESTIMATING THE FUNDAMENTAL MATRIX As sad n the last secton, n order to deal wth all possbltes, we have to consder 36 dstnct parameterzatons, and ths s a burden n mplementng the optmzaton procedure. In ths secton, we propose to transform the mage ponts n a projectve space and work only wth one parameterzaton such as (2) whle ensurng the value of a s largest so we can set a =1. The dea s to fnd a projectve transformaton n each mage, denoted by P and P 0, such that n the transformed mage space the frst element of the fundamental matrx has the largest value and the eppoles are not at nfnty. Let the mage ponts n the transformed space be ebm = P ~m and ebm 0 = P0 ~m 0 ; (5) then the fundamental matrx n the transformed space s gven by bf = P 0 T FP 1 : (6) Gven an ntal estmate of matrx F 0 obtaned for example wth Hartley s normalzed 8-pont algorthm [3], we compute the eppoles e 0 and e 0 0. The matrces P and P 0 are 3 3 permutaton matrces determned as follows: 1. Intalze P and P 0 to be dentty matrces. 2. Fnd the poston of the largest element of F 0, denoted by ( 0 ;j 0 ). (Index of a vector or a matrx starts wth 0 as n C++) 3. If j 0 6= 0, permute rows 0 and j 0 of matrx P and permute elements 0 and j 0 of eppole e If 0 6= 0, permute rows 0 and 0 of matrx P 0 and permute elements 0 and 0 of eppole e If je 0 [1]j > je 0 [2]j, permute elements 1 and 2 of eppole e 0 and permute rows 1 and 2 of matrx P.
4 4 ZHANG AND LOOP 6. If je 0 0 [1]j > je0 0 [2]j, permute elements 1 and 2 of eppole e0 0 and permute rows 1 and 2 of matrx P 0. Steps 3 and 4 ensure that the frst element of the fundamental matrx n the transformed space has the largest value, whle steps 5 and 6 ensure that the eppoles are not at nfnty. Note that P 1 = P T and P 0 1 = P 0T because they are permutaton matrces. The reader s referred to [2, page 109] for effcently representng a permutaton matrx wth an nteger vector. We can now use the parameterzaton (2) to estmate b F from the transformed mage ponts ebm and ebm 0. The fundamental matrx n the orgnal mage space s gven by F = P 0T b FP. 4. PRESERVING INFORMATION OF THE ORIGINAL NOISE DISTRIBUTION Ponts are detected n the orgnal mage space. All nonlnear optmzaton crtera (see [7]) are derved from the nose dstrbuton of these ponts. One reasonable assumpton s that the mage ponts are corrupted by ndependent and dentcally dstrbuted Gaussan nose wth mean zero and covarance matrces gven by Λm = Λm 0 = ff2 dag (1; 1) ; (7) where ff s the nose level, whch s usually unknown. Ths assumpton s clearly no longer reasonable n the transformed mage space. In the followng, we show how to estmate the transformed fundamental matrx whle preservng nformaton of the orgnal nose dstrbuton, and we wll llustrate ths by usng the gradent-weghted crteron. From (1), (5) and (6), we see that the transformed fundamental matrx and the orgnal mage ponts are related by f em 0T P 0T b FP em =0: (8) The least-squares technque produces an optmal soluton f each term has the same varance. Therefore, we can estmate the transformed fundamental matrx by mnmzng the followng weghted sum of squares (the ch-square χ 2 ): mn b F X where ff f s the varance of f, and ts computaton s gven below. Let us ntroduce the notaton Z = dag (1; 1; 0) ; f 2 =ff2 f ; (9) then from (7), the covarances of the homogeneous coordnates of the mage ponts are gven by Λ em = Λ em 0 = ff2 Z : (10) Under frst order approxmaton, the varance of f s then gven by ff 2 em T Λ em 0 = ff 2 (^l T PZP T^l + ^l 0T P 0 ZP 0T^l 0 ) Λ em 0
5 ESTIMATING THE FUNDAMENTAL MATRIX IN PROJECTIVE SPACE 5 where ^l b 0 = F ebm and ^l = F b T ebm 0 are eppolar lnes n the transformed mage space. Snce multplyng each term by a constant does not affect the mnmzaton, the problem (9) becomes mn b F X ^l T PZPT^l + ( ebm 0T b F ebm ) 2 ^l 0T P 0 ZP 0T^l 0 : (11) The denomnator s smply the gradent of f. The mnmzaton can be conducted by means of, for example, the Levenberg-Marquardt algorthm. 5. EXPERIMENTAL RESULT In ths secton, we show an example wth real data as dsplayed n Fg. 1. The automatc robust mage-matchng algorthm descrbed n [8] was used to fnd pont matches across the two vews. 217 matches have been found, whch are shown n Fg. 1. The matchng algorthm also provdes an estmate of the fundamental matrx based on the least-medansquares (LMedS) technque. Ths estmate s used as the ntal guess for the experment descrbed n ths secton. FIG. 1. Automatcally matched ponts and four pars of eppolar lnes. A pont match s ndcated by a green lne segment, wth one end, ndcated by a red dot, beng the pont n the current mage, and the other end beng the matched pont n the other mage. TABLE 1 Comparson of dfferent methods, n pxels eppolar dstances eppole 1 eppole 2 Method mage 1 mage 2 (u, v) (u, v) Intal Method Method Method We have compared four methods, all based on the gradent-weghted crteron. ffl Method 1: only one parameterzaton of the fundamental matrx, as shown n (2), s used. ffl Method 2: the novel technque descrbed n Sect. 3 s used; however, the orgnal data nose nformaton s not preserved.
6 6 ZHANG AND LOOP ffl Method 3: the novel technque together wth the preservaton of the orgnal nose nformaton as descrbed n Sect. 4 s used. ffl Method 4: 36 dstnct parameterzatons, as descrbed n [7], are used. Not surprsngly, Method 4 produced exactly the same result as Method 3, and the results are therefore not shown. The comparson of the other three methods s shown n Table 1, where we show the average dstances from a pont to ts correspondng eppolar lne n each mage, and the poston of both eppoles. The frst row dsplays the results provded by the LMedS technque durng mage matchng. We see that Method 1 gves qute a good result, but t s also clear that the parameterzaton used has dffculty to represent eppoles located far away from the mage. Method 2 does not produce very good result because t does not take nto account approprately the orgnal nose nformaton. Method 3 gves the best result. 6. CONCLUSION In ths paper, we have proposed a novel technque for estmatng the fundamental matrx. We transform the mage ponts n the projectve space, and therefore only need to perform nonlnear optmzaton wth one parameterzaton of the fundamental matrx, rather than consderng 36 maps of parameterzaton as n prevous work. In addton, we have presented a method for preservng the characterstcs of the data nose model from the orgnal mage space. The mplementaton of the proposed technque s much easer than wth the prevous approach, and the same qualty has been obtaned. REFERENCES 1. G. Csurka, C. Zeller, Z. Zhang, and O. Faugeras. Characterzng the uncertanty of the fundamental matrx. Computer Vson and Image Understandng, 68(1):18 36, October G.H. Golub and C.F. van Loan. Matrx Computatons. The John Hopkns Unversty Press, Baltmore, Maryland, 3 edton, R.I. Hartley. In defence of the 8-pont algorthm. In Proceedngs of the 5th Internatonal Conference on Computer Vson, pages , Boston, MA, June IEEE Computer Socety Press. 4. Quang-Tuan Luong. Matrce Fondamentale et Calbraton Vsuelle sur l Envronnement-Vers une plus grande autonome des systèmes robotques. PhD thess, Unversté de Pars-Sud, Centre d Orsay, December Quang-Tuan Luong and Olver Faugeras. The fundamental matrx: theory, algorthms, and stablty analyss. The Internatonal Journal of Computer Vson, 17(1):43 76, January Phl Torr, Andrew Zsserman, and Steven Maybank. Robust detecton of degenerate confguratons for the fundamental matrx. In Proceedngs of the 5th Internatonal Conference on Computer Vson, pages , Boston, MA, June IEEE Computer Socety Press. 7. Z. Zhang. Determnng the eppolar geometry and ts uncertanty: A revew. The Internatonal Journal of Computer Vson, 27(2): , Z. Zhang, R. Derche, O. Faugeras, and Q.-T. Luong. A robust technque for matchng two uncalbrated mages through the recovery of the unknown eppolar geometry. Artfcal Intellgence Journal, 78:87 119, October 1995.
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