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

21. The aperture problem

22. Remarks

23. Apparently an aperture problem 

24. 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

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

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

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

29. 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 

30. 

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

32. Results of an enhanced system

33. Structure from motion with OF

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

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

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