Differential Techniques Second-order differential In (x, t)I(x, On./Iu(x, l(x,)n(x,) xt 0 Stronger restriction than first-order derivatives on permissible motion field VI(x.,1)v+l1(x,)=0 Can be combined with 1st order in isolation or together(over determined system) Velocity estimation from 2nd-order methods are often assumed be to sparser and less accurate than estimation from 1st-order methods
Differential Techniques • Second-order differential • Stronger restriction than first-order derivatives on permissible motion field • Can be combined with 1st order in isolation or together (overdetermined system) • Velocity estimation from 2nd -order methods are often assumed be to sparser and less accurate than estimation from 1st -order methods 1 2 ( , ) ( , ) ( , ) 0 ( , ) ( , ) ( , ) 0 xx yx tx xy yy tx I t I t v I t I t I t v I t + = x x x x x x ( , ) ( , ) 0 t + = I t I t x v x
Differential Techniques Additional constraints Fits the measurements in each neighborhood to a local model for 2d velocity Using least squares minimization or Hough transform Global smoothness
Differential Techniques • Additional constraints – Fits the measurements in each neighborhood to a local model for 2d velocity • Using least squares minimization or Hough transform – Global smoothness
Differential Techniques I(x, t)must be differentiable Temporal smoothing at the sensors is needed to avoid aliasing Numerical differentiation must be done carefully If aliasing can not be avoided in image acquisition apply differential techniques in a coarse-to-fine manner
Differential Techniques • must be differentiable – Temporal smoothing at the sensors is needed to avoid aliasing – Numerical differentiation must be done carefully • If aliasing can not be avoided in image acquisition – Apply differential techniques in a coarse-to-fine manner I t ( , ) x
Horn and schunck Combine gradient constraint with a global smoothness term, minimizing 「(Vv+4)2+(+)k n=0.5 instead of 2=100 Ⅰ(Ⅰv+Ⅰ uk+=ukx v=10=0 a2+2+ k+1 k(1n+1v+1) a2+2+Ⅰ
Horn and Schunck • Combine gradient constraint with a global smoothness term, minimizing ( ) ( ) 2 2 2 2 D t 2 2 + + + I I u v d v x 0 0 v u = = 0 1 222 1 222 ( ) ( ) k k k k x x y t x y k k k k y x y t x y I I u I v I u u I I I I u I v I v v I I + + + + = − + + + + = − + + = = 0.5 instead of 100
Horn and schunck Relatively crude form of numerical differentiation can be source of error Spatiotemporal smoothing Gaussian prefilter with o= 1.5 pixels in space and 1. 5 frames in time 4-point central differences for differentiation mask;(-1,8.0,-81)
Horn and Schunck • Relatively crude form of numerical differentiation can be source of error • Spatiotemporal smoothing – Gaussian prefilter with 1.5 pixels in space and 1.5 frames in time • 4-point central differences for differentiation – mask 1 ( 1,8,0, 8,1) 12 − − =