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Organizing Optic Flow

This lecture explores key questions in the study of optical flow, including comparisons of methodologies, density flow estimation, and temporal sampling requirements. It highlights notable works by Greg Hager on planar motion and Mike Black's research on low-dimensional subspaces for complex motion. The mathematical foundations are discussed, including warping functions, Jacobians, and error formulations. The lecture also addresses estimation techniques, local linearization, and the stabilization of different motion models, emphasizing practical applications such as image motion analysis.

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Organizing Optic Flow

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  1. Organizing Optic Flow Cmput 610 Martin Jagersand

  2. Last lecture:Questions to think about Compare the methods in the paper and lecture • Any major differences? • How dense flow can be estimated (how many flow vectore/area unit)? • How dense in time do we need to sample?

  3. Organizing different kinds of motion Two examples: • Greg Hager paper: Planar motion • Mike Black, et al: Attempt to find a low dimensional subspace for complex motion

  4. Remember last lecture: • Over determined equation system Im = Mu • Can be solved in e.g. least squares sense using matlab u = M\Im

  5. 3-6D Optic flow • Generalize to many freedooms (DOFs) Im = Mu

  6. Know what type of motion(Greg Hager, Peter Belhumeur) u’i = A ui + d Planar Object => Affine motion model: It = g(pt, I0)

  7. Mathematical Formulation • Define a “warped image” g • f(p,x) = x’ (warping function), p warp parameters • I(x,t) (image a location x at time t) • g(p,It) = (I(f(p,x1),t), I(f(p,x2),t), … I(f(p,xN),t))’ • Define the Jacobian of warping function • M(p,t) = • Consider “Incremental Least Squares” formulation • O(Dp, t+Dt) = || g(pt,It+Dt) – g(0,I0) ||2

  8. Estimating motion parameters • Model • I0 = g(pt, It ) (image I, variation model g, parameters p) • DI = M(pt, It) Dp (local linearization M) • Define an error • et+1 = g(pt, It+1) - I0 • Close the loop • pt+1 = pt - (MT M)-1MT et+1 where M = M(pt,It) M is N x m and is time varying!

  9. A Factoring Result Suppose I = g(It, p) at pixel location u is defined as I(u) = I(f(p,u),t) And= L(u)S(p) Then M(p,It) = M0 S(p) where M0 = M(0,I0)

  10. O(mN) operations G is m x N, e is N x 1 S is m x m Numerical Solution • In general, solve • [STG S] Dp = M0T et+1 where G = M0TM0 constant! • pt+1 = pt +Dp • If S is invertible, then • pt+1 = pt - S-T G et+1where G = (M0TM0)-1M0T

  11. G is constant! Local asymptotic stability! S is small and time varying Numerical Solution • In general, solve • [STG S] Dp = M0T et+1 where G = M0TM0 constant! • pt+1 = pt +Dp • If S is invertible, then • pt+1 = pt - S-T G et+1where G = (M0TM0)-1M0T

  12. G is constant! Local asymptotic stability! S is small and time varying • In general, solve • [STG S] Dp = M0T et+1 where G = M0TM0 constant! • pt+1 = pt +Dp • If S is invertible, then • pt+1 = pt - S-T G et+1 where G = (M0TM0)-1M0T Stabilization Revisited

  13. On The Structure of M u’i = A ui + d Planar Object -> Affine motion model: X Y Rotation Scale Aspect Shear

  14. Organizing flowfields • Express flow field f in subspace basis m • Different “mixing” coefficients a correspond to different motions

  15. Example:Image discontinuities

  16. Mathematical formulation Let: Mimimize objective function: = Where

  17. ExperimentMoving camera • 4x4 pixel patches • Tree in foreground separates well

  18. Experiment:Characterizing lip motion • Very non-rigid!

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