1 / 18

Organizing Optic Flow

Organizing Optic Flow. Cmput 610 Martin Jagersand. 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?.

sanura
Download Presentation

Organizing Optic Flow

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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!

More Related