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Optical Flow

Optical Flow

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Optical Flow

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  1. Optical Flow Marc Pollefeys COMP 256 Some slides and illustrations from L. Van Gool, T. Darell, B. Horn, Y. Weiss, P. Anandan, M. Black, K. Toyama

  2. last week: polar rectification

  3. Last week: polar rectification

  4. Similarity measure (SSD or NCC) Optimal path (dynamic programming ) Last week: Stereo matching • Constraints • epipolar • ordering • uniqueness • disparity limit • disparity gradient limit • Trade-off • Matching cost (data) • Discontinuities (prior) (Cox et al. CVGIP’96; Koch’96; Falkenhagen´97; Van Meerbergen,Vergauwen,Pollefeys,VanGool IJCV‘02)

  5. Questions for assignment?

  6. Tentative class schedule

  7. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  8. Optical Flow:Where do pixels move to?

  9. Motion is a basic cue Motion can be the only cue for segmentation

  10. Motion is a basic cue Even impoverished motion data can elicit a strong percept

  11. Applications • tracking • structure from motion • motion segmentation • stabilization • compression • mosaicing • …

  12. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  13. Definition of optical flow OPTICAL FLOW = apparent motion of brightness patterns Ideally, the optical flow is the projection of the three-dimensional velocity vectors on the image 

  14. Caution required ! Two examples : 1. Uniform, rotating sphere  O.F. = 0 2. No motion, but changing lighting  O.F.  0 

  15. Caution required !

  16. Mathematical formulation I (x,y,t) = brightness at (x,y) at time t Brightness constancy assumption: Optical flow constraint equation : 

  17. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  18. The aperture problem 1 equation in 2 unknowns 

  19. The aperture problem 0

  20. The aperture problem

  21. Remarks

  22. Apparently an aperture problem

  23. What is Optic Flow, anyway? • Estimate of observed projected motion field • Not always well defined! • Compare: • Motion Field (or Scene Flow) projection of 3-D motion field • Normal Flow observed tangent motion • Optic Flow apparent motion of the brightness pattern (hopefully equal to motion field) • Consider Barber pole illusion

  24. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  25. Horn & Schunck algorithm Additional smoothness constraint : besides OF constraint equation term minimize es+ec 

  26. so the Euler-Lagrange equations are is the Laplacian operator Horn & Schunck The Euler-Lagrange equations : In our case , 

  27. Horn & Schunck Remarks : 1. Coupled PDEs solved using iterative methods and finite differences 2. More than two frames allow a better estimation of It 3. Information spreads from corner-type patterns 

  28. Horn & Schunck, remarks 1. Errors at boundaries 2. Example of regularisation (selection principle for the solution of illposed problems) 

  29. Results of an enhanced system

  30. Structure from motion with OF

  31. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  32. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow

  33. Optical Flow • Brightness Constancy • The Aperture problem • Regularization • Lucas-Kanade • Coarse-to-fine • Parametric motion models • Direct depth • SSD tracking • Robust flow • Bayesian flow