CIS 580 Spring Lecture 24

Transcription

1 CIS 58 Spring - Lctur 4 April 6, Rviw: pos rcovry from E: Thorm: A ral matrix is ssntial (i.. E = TR) iff it has two qual singular valus and on zro singular valu: E =U σ σ VT T = ± u (third column of U) ± R =U ± VT To summariz, whn trying to rcovr th motion paramtrs from E, thr is mor than on solution: if (T, R) is a solution, thn ( T, R T,π R) is a also a solution. This phnomnon is calld th twistd pair ambiguity: it can b provd that if p T (T Rp ) = thn p T (T R T,πRp ) = too. Nots and figurs by Matthiu Lcc. T anti-symmtric, R orthogonal R T,π is a rotation of π around T (R T = I + T ) Rodrigus formula for rotations: (axis n, n = ) R n (θ) = I + sin θû + ( cos θ)û This usful formula coms from th formulation of a rotation as th xponntial of an antisymmtric matrix: S drivation in Jan Gallir s book. Exampl: u = R n ( π6 ) = I + sin π 6 4 R n (θ) = θû = I +! θû +! θ û , θ = π 6 + ( cos π 6 ) R n (θ)p = p + sin θ(n p) + ( cos θ(n (n p))) W now apply this rsult to a rotation of π around T: R T (π)p =p + ( cos π)(t (T p)) =p + (T T pt T T T p) a (b c) = a T cb a T bc End of drivation in handout.

2 cis 58 spring - lctur 4 Obtaining a valid E-matrix Rcall that our stimation algorithm is th following: Estimat E by prforming SVD on th following systm, obtaind by stacking 8 quations coming from 8 pairs of matching points: } {{ } 8 9 with E = [ R Rcovr ( T, R) from E. 9 =, Our problm is that th output E of th SVD is not ncssarily an accptabl ssntial matrix: it dos not ncssarily vrify σ = σ and σ =. For an arbitrary E with σ σ and σ, how can w find th closst matrix E with σ = σ and σ =? W want to find argmin E E E F with E ssntial. W can also formulat th cost as follows: U σ σ VT U W us th following algbra rsults: σ σ σ VT. A B F = tr ((A B)T (A B)) = tr (A T A) + tr (B T B) tr (A T B) tr (B T A) F In othr words w want to projct E on th spac of ssntial matrics, for th Frobnius norm.. For Q orthogonal, QAQ T F = A F W hav th following: σ σ UT }{{} U P orthogonal σ =tr σ + tr σ σ σ σ σ σ () =σ(σ (p q + p q ) + σ (p q + p q )) V T V }{{} Q orthogonal F σ tr σ P σ σ σ Q } {{ } () bcaus of th minus sign, () has to b maximizd. P and Q ar orthogonal. p q + p q p q + p q σ, σ, σ >

3 cis 58 spring - lctur 4 Thrfor w can rwrit th cost to minimiz as follows: σ σ tr σ + tr σ σ (σσ + σσ ) =σ + σ + σ + σ σσ σσ =(σ σ ) + (σ σ ) + σ This xprssion is convx in σ and has to b minimizd, thrfor w obtain th solution by stting th drivativ to : σ = σ = σ + σ Th 8-point algorithm: summary W can now (at last!) writ th complt algorithm for camra motion stimation from ight pairs of matching points:. Build th homognous linar systm by stacking pipolar constraints p T i (T Rp i) =, i =,..., 8: Rcall th Hadamard product: p p = p x p p y p. R9 p z p (p i p i ) T.. Lt A (8 9) b th nullspac of A (if σ 8 giv up). [ = Udiag (σ, s, σ )VT. Thn us th following stimat of th ssntial matrix: ( σ E = Udiag + σ, σ + ) σ, V T 4. T = ±u R = UR Z,π/ V T or R = R T,π R 5. Try all pairs (T, R) to chck if rconstructd points ar in front of th camras λ p Rp + T (thr quations, two unknowns). W ar thn lft with a triangulation problm: λ p = λ Rp + T λ, λ =? Why ar thr thr quations but two unknowns? This is ovrconstraind bcaus in gnral two lins in spac do not intrsct. In our cas, w know that thy should intrsct if th pipolar constraint is satisfid (which mans p, T, p ar coplanar).

4 cis 58 spring - lctur 4 4 But th constraint p T (T Rp ) = is subjct to nois: th input points of th algorithm ar not prfct! Th problm of finding λ, λ can b formulatd as th following simpl linar systm: [ p Rp, rank bcaus p Rp W can solv th systm by hand: λ λ λ p Rp =T Rp = T λ (p Rp ) T (p Rp ) =(T Rp ) T (p Rp ) λ = (T Rp ) T (p Rp ) p Rp > Minimal problms in computr vision Using th 8-point is a bad ida for RANSAC, bcaus 8 points ar not th minimal numbr of corrspondncs ndd to solv for 5 unknowns. Tchnically th following systm obtaind by stacking only fiv pipolar constraints should b nough: p T (T RP ) = p T (T RP ) =. p T 5 (T RP 5) = This dfins fiv quations with fiv unknowns, but w hav no way to find fiv variabls that w could st as unknowns. Th trick is to dfin a polynomial systm. Lt s go back to th E-matrix systm: A solution = E = [ R. If w prform an SVD of A5 9 = UΣV T, w xpct by dfinition that σ 6 = σ 7 = σ 8 = σ 9 =. Thrfor, solutions ar in th null-spac, of which {v 6, v 7, v 8, v 9 } is an orthonormal basis. This mans that w can writ th following dcomposition of th vctor of ssntial matrix cofficints: = xv 6 + yv 7 + zv 8 + wv 9, x, y, z, w unknowns (w = ) In th 8-point algorithm, w dfind th cofficints of E as th unknowns and thrfor ndd mor quations, but thos 8 indpndnt cofficints of E containd rdundant information about th ral fiv unknowns that w wr intrstd in: th cofficints of R and T.

5 cis 58 spring - lctur 4 5 Our goal is now to find x, y, z such that [ is an E-matrix. Thorm: A ral matrix E is ssntial (i.. E = T R,antisym-orthogonal dcomposition) iff: dt(e) =, EE T E tr (EE T )E = Ths ar quations whr w can rplac = xv 6 + yv 7 + zv 8 + v 9, and w obtain quations with thr unknowns x, y, z. Th quations ar cubic in x, y, z. W us a linar algbra trick, th hiddn variabl limination trick : a (z)x + a (z)x y + a (z)xy + a (z)y +a 4 (z)x + a 5 (z)xy + a 6 (z)y +a 7 (z)x + a 8 (z)y + a 9 (z) = [ a (z) a 9 (z) x x y xy x xy y x Th tchniqu w just usd, which consists in prcomputing monomials of z and placing thm in linar forms, is calld lifting. It nabls thm to solv linar quations instad of polynomial ons. (to b continud) y