# Examples. Epipoles. Epipolar geometry and the fundamental matrix

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1 Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon to) th mappd points x and x wi i in th sam pan π. This pan is cad th pipoar pan for C, C and. Givn a point x in imag 1, th pipoar pan π is dfind by th ray through x and C and th basin through C and C. A corrsponding point x thus has to i on th intrscting in btwn th pipoar pan π and imag pan 2. Th in is th projction of th ray through x and in imag 2 and is cad th pipoar in to x. pipoar pan π x x x C C pipoar in for x p. 1 Epipos Examps Th intrsction points btwn th bas in and th imag pans ar cad pipos. Th pipo in imag 2 is th mapping of th camra cntr C. Th pipo in imag 1 is th mapping of th camra cntr C. Sinc a pipoar pans intrsct both camra cntrs, a pipoar ins wi intrsct th pipos. π basin basin p. 3

2 Th fundamnta matrixf Th fundamnta matrixf Th fundamnta matrix F is th agbraic rprsntation of th pipoar gomtry. It dscribs th mapping x btwn a point x in on imag and its pipoar in in anothr imag. Lt P and P b th camra matrics for imag 1 and 2. Th ray in P 3 that is projctd onto th point x in imag 1 is (λ) = P + x + λc, whr P + is th psudo-invrs to P, i.. PP + = I, and PC = 0. Th in (λ) intrscts th points P + x and C. Ths points ar mappd into th othr camra P at P P + x and P C. Th pipoar in intrscts ths projctd points, i.. = (P C) (P P + x). Th point P C is th pipo, i.. th projction of th camra cntr in th othr camra. Th pipoar in can thus b writtn as = (P P + x) or whr = [ ] (P P + )x = Fx, F = [ ] (P P + ). p. 5 Examp Corrspondnc Assum th camra matrics corrspond to a caibratd stro rig with th word origin in camra cntr 1. P = K[I 0], P = K [R t]. Thn P + = " K 1 0, C = " 0 1 and F = [P C] P P + = [K t] K RK 1 = K [t] RK 1 = K R[R t] K 1 = K RK [KR t] Not that th pipos ar W can thus writ = P " R t 1 = KR t, = P " 0 1 = K t. Givn two camras with diffrnt camra cntrs, th fundamnta matrix F is a 3 3 homognous matrix with rank 2. For ach corrsponding point pair x x it satisfis x Fx = 0, sinc if x and x ar corrsponding points, thn x is on th pipoar in = Fx corrsponding to x, i.. x = 0 = x Fx. Simiary, = F x is th pipoar in in imag 1 corrsponding to th point x in imag 2. F = [ ] K RK 1 =... = K RK [], F = = K R K [ ]. x pipoar in for x p. 7

3 Th pipos Th numbr of dgrs of frdom Th pipoar in = Fx to ach point x (xcpt ) intrscts th pipo. Thus satisfis (Fx) = ( F)x = 0 for a x. This impis that F = 0 or F = 0. Th pipo is thus a nu vctor to F (in th ft nu-spac of F). Simiary, F = 0, i.. is a nu-vctor to F (in th right nu-spac of F). Th fundamnta matrix F has 7 dgrs of frdom: A 3 3 homognous matrix has 8 dgrs of frdom. Th constraint rank(f) = 2 or dt(f) = 0 rducs th numbr to 7. p. 9 Projktiv invarianc Th corrspondnc ration x Fx = 0 is invariant undr a homography in P 2. If ˆx = Hx and ˆx = H x thn x Fx = ˆx H FH 1ˆx = ˆx ˆFˆx, whr ˆF = H FH 1 is th fundamnta matrix corrsponding to ˆx ˆx. Th fundamnta matrix F is invariant undr a homography in P 3. Lt H b a 4 4 matrix corrsponding to a projctiv mapping of P 3. Thn th camra pairs (P,P ) and (PH,P H) hav th sam fundamnta matrix. Th points x = P = (PH)(H 1 ) and x = P = (P H)(H 1 ) ar corrsponding mappings of in th camras P and P and corrsponding mappings of H 1 in th camras PH and P H. Thus a homography H in P 3 dos affct th word points and camras P,P, but not F. This mans that th fundamnta matrix F dtrmins th camra matrics P,P up to a right mutipication by a 3D projctiv transformation. Canonica form Givn this ambiguity a canonica form for th camra pairs is dfind corrsponding to a fundamnta matrix whr th first camra P = [I 0] has cntr at th origin and word coordinat axs. If th scond camra is P = [M m] thn th fundamnta matrix F corrsponding to th canonica camras is F = [m] M. For finit camras P = K[I 0],P = K [R t] w hav F = [K t] K RK 1. p. 11

4 symmtry and th fundamnta matrix Canonica camra pairs givnf A non-zro matrix F is th fundamnta matrix corrsponding to th camra pair P,P iff P FP is skw symmtric. Th condition that P FP is skw symmtrica is quivant to that P FP = 0 for a. With x = P and x = P this bcoms x Fx = 0, which is th dfining quation for th fundamnta matrix. Lt F b a fundamnta matrix and S an arbitrary skw symmtric matrix. Dfin th camra matrics P = [I 0] and P = [SF ], whr is th ft pipo of F, F = 0 and assum that P is a vaid camra matrix (has rank 3). Thn F is th fundamnta matrix corrsponding to (P,P ). Chck by vrifying that P FP = [SF ] F[I 0] = [ F S F 0 F 0 ] = [ F S F ] is skw symmtric. p. 13 Canonica camra pairs givnf In ordr for th matrix P to hav rank 3 s has to b non-zro, whr s is a nu-vctor of S = [s]. A working choic is s = ading to th camra pairs P = [I 0] and P = [[ ] F ]. Th most gnra formuation for a canonica camra pair is P = [I 0], P = [[ ] F + v λ ], whr v is an arbitrary 3-vctor and λ is a scaar. Normaizd coordinats Study a camra matrix P = K[R t] and t x = P b an arbitrary point in th imag. If th camra caibration matrix K is known w may appy its invrs on th point x and gt ˆx = K 1 x. Thn ˆx = [R t] is th projction of xprssd in normaizd coordinats. Th camra matrix K 1 P = I[R t] is cad a normaizd camra matrix and has camra caibration matrix K = I. p. 15

5 Th ssntia matrixe Th numbr of dgrs of frdom for Study a normaizd camra pair P = [I 0], P = [R t]. Th fundamnta matrix corrsponding to normaizd camra pairs is cad th ssntia matrix and is on th form E = [t] R = R[R t]. Th dfining quation for th ssntia matrix is Th ssntia matrix E = [t] R has 5 dgrs of frdon; 3 rotation angs in R, 3 mnts in t, but arbitrary sca. Th fwr dgrs of frdom corrspond to on additiona constraint; a 3 3 matrix is an ssntia matrix if two of its singuar vaus ar qua and th ast is zro. ˆx Eˆx = 0, xprssd in normaizd imag coordinats for th corrsponding points x x. Substitution with ˆx and ˆx givs x K EK 1 x = 0 ading to F = K EK 1 or E = K FK. p. 17 h numbr of dgrs of frdom fore Study th factorization E = [t] R = SR, whr S is skw symmtric. W wi us th matrics W = and Z = that ar orthogona and skw symmtric, rspctivy. Not that Z = diag(1,1, 0)W Cacuation of th camra matrics from Lt th first camra matrix b P = [I 0]. In ordr to cacuat th othr camra matrix P it is ncssary to factoriz E into a product SR by a skw symmtric and a rotation matrix. Givn S = [t] and R, P is givn by P = [R t]. Lt E has th singuar vau dcomposition E = U diag(1, 1, 0)V. Ignoring sign, thr ar two possib factorizations E = SR: S = UZU, R = UWV or R = UW V A skw symmtric matrix S can b writtn as S = kuzu, whr U is orthogona. Thus up to sca and S = U diag(1,1, 0)WU E = SR = U diag(1,1,0)(wu R), which is a singuar vau dcomposition of E with two singuar vaus qua and th third qua to zro. p. 19 Th factorization givs th t part of th camra matrix P up to sca from S = [t]. If w choos t = 1 w gt a unit basin. Furthrmor St = 0 St = UZU t = U ( UZ ) t = U [ u 2 u 1 0 ] t = 0 givs that t = u 3, whr u i is th i:th coumn of U. Th sign of E and hnc t can howvr not b dtrmind which ads to 4 diffrnt possibiitis for th scond camra P.

6 Th 4 camra factorizations ofe Givn a singuar vau dcomposition of E = U diag(1, 1, 0)V and a canonica camra 1 P = [I 0], thr ar 4 atrnativ camra pairs: P = [UWV + u 3 ], P = [UWV u 3 ], P = [UW V + u 3 ], P = [UW V u 3 ]. Ths 4 options hav gomtric intrprtations; basin rvrsa and rotation by th scond camra 180 around th basin. A (a) B B (b) A A B B A (c) (d) p. 21

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