Lucas and Kanade Weighted least squares Fixed velocity in a small neighborhood Minimizing 2W(x)/VI(x, t)V+I, (x,t)] →AW2Ay=Aw2b 4=VI(x)…,VI(xn)2 w=diag[W(x1),.w(xr)
Lucas and Kanade • Weighted least squares • Fixed velocity in a small neighborhood 2 2 ( ) ( , ) ( , ) W I t I t t + x Minimizing x x v x
Lucas and Kanade When A'W2A is nonsingular, 1W24]-1AW2b 4W24 ∑W(x)l2(x)∑W2(x)2(x)l2(x) WP(x1(x(x)∑W2x1(x)」 2W(x)u(x)[vn(x)-s(x) Weighted least squares estimates of v from estimates of normal velocities Confidence measure ll(x)=‖VI(x.,)‖
Lucas and Kanade • When is nonsingular, Weighted least squares estimates of v from estimates of normal velocities Confidence measure
Lucas and Kanade Spatiotemporal smoothing Gaussian prefilter with 1.5 pixels-frames 4-point central differences for differentiation mask I (-1,8,0,-8,1) Spatial neighborhood 5x5 pixels Window function W(x) -(0.0625,025,0.375,0.25,00625)
Lucas and Kanade • Spatiotemporal smoothing – Gaussian prefilter with 1.5 pixels-frames • 4-point central differences for differentiation – mask • Spatial neighborhood 5x5 pixels • Window function W(x) – (0.0625, 0.25, 0.375, 0.25, 0.0625) 1 ( 1,8,0, 8,1) 12 − −
Lucas and Kanade Identify the unreliable estimates by eigenvalues of a'W2A If n>t,n, =[AW241-1AW2b If >t,n<t, compute normal velocity T=1.0 VESh L(x, t) X (I)= n(xt)= VI(x, tl V/(x. From ls minimization Otherwise, do not compute velocity
Lucas and Kanade • Identify the unreliable estimates by eigenvalues of – If – If , compute normal velocity • v=sn • From LS minimization – Otherwise, do not compute velocity 1 2 , 1 2 , ( , ) ( , ) ( , ) t I t s t I t − = x x x ( , ) ( , ) ( , ) t t I t = x n x x
N age First to use second-order derivatives to measure optical flow Basic measurements and global smoothness Oriented smoothness constraint VIv+l'+ 1+26 (a-)2+(c212-1L)2+6(2+2+e2+2)dd 6=1.0a=0.5 Attenuates the variation of the flow vv in the direction perpendicular to the gradient
Nagel • First to use second-order derivatives to measure optical flow – Basic measurements and global smoothness – Oriented smoothness constraint – Attenuates the variation of the flow in the direction perpendicular to the gradient