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Motion from normal flow

Motion from normal flow. The aperture problem. Depth discontinuities. Optical flow difficulties. Translational Normal Flow. In the case of translation each normal flow vector constrains the location of the FOE to a half-plane. Intersection of half-planes provides FOE.

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Motion from normal flow

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  1. Motion from normal flow

  2. The aperture problem Depth discontinuities Optical flow difficulties

  3. Translational Normal Flow • In the case of translation each normal flow vector constrains the location of the FOE to ahalf-plane. • Intersection of half-planes provides FOE.

  4. Egoestimation from normal flow • Idea: choose particular directions: patterns defined on the sign of normal flow along particular orientation fields • positive depth constraint • 2 classes of orientation fields: copoint vectors and coaxis vectors

  5. Optical flow and normal flow

  6. Optical flow and normal flow

  7. Coaxis vectors with respect to axis (A,B,C)

  8. Coaxis vectors

  9. Translational coaxis vectors

  10. Translational coaxis vectors h passes through FOE and (Af/C, Bf/C), defined by 2 parameters

  11. Rotational coaxis vectors

  12. Rotational coaxis vectors

  13. Rotational coaxis vectors g passes through AOR and (Af/C, Bf/C), defined by 1 parameter

  14. Combine translation and rotation Positive + positive  positive Negative + negative negative Positive + negative don’t know (depends on structure)

  15. Coaxis pattern translational rotational combined

  16. g-vectors: Translation

  17. g-vectors: Rotation

  18. g-vectors: Translation and Rotation

  19. Three coaxis vector fields alpha beta gamma

  20. Copoint vectors

  21. copoint vectors

  22. Copoint vectors defined by point (r,s)

  23. Translational copoint vectors

  24. Translational copoint vectors k passes through FOE and (r,s) defined by 1 parameter

  25. Rotational copoint vectors

  26. Rotational copoint vectors l passes through AOR and (r,s), is defined by 2 parameters

  27. translational component rotational component

  28. Three coaxis vector fields (a) (b) (c)

  29. a,b,c : positive and negative a,b,g vectors c,d,e: Fitting of a,b,g patterns g: Separation of (a,b,g) coaxis pattern h: Separation of (x0, y0) copoint pattern

  30. Optical illusion

  31. What is the Problem? • Flow can be accurately estimated in an image patch corresponding to a smooth scene patch, • But erroneous flow estimates are obtained for image patches corresponding to scene patches containing discontinuities Image Flow Scene structure Discontinuities 3D Motion

  32. Depth variability constraint • Errors in motion estimates lead to distortion of the scene estimates. • The distortion is such that the correct motion gives the “smoothest” (least varying) scene structure.

  33. Depth estimation • Scene depth can be estimated from normal flow measurements:

  34. Visual Space Distortion • Wrong 3D motion gives rise to a rugged (unsmooth) depth function (surface). • The correct 3D motion leads to the “smoothest” estimated depth.

  35. correct motion incorrect motion Inverse depth estimates

  36. The error function • A normal flow measurement: For an estimate • The error function to be minimized:

  37. The error function • Estimated normal flow • The error function to be minimized: • Global parameters: • Local parameter: locally planar patches:

  38. Error function evaluation • Given a translation candidate , each local depth can be computed as a linear function of the rotation . • We obtain a second order function of the rotation; its minimization provides both the rotation and the value of the error function.

  39. Is derived from image gradients only Brightness consistency: Flow: Planar patch:

  40. Handling depth discontinuities • Given a candidate motion, the scene depth can be estimated and further processed to find depth discontinuities. • Split a region if it corresponds to two depth values separated in space.

  41. The algorithm • Compute spatio-temporal image derivatives and normal flow. • Find the direction of translation that minimizes the depth-variability criterion. • Hierarchical search of the 2D space. • Iterative minimization. • Utilize continuity of the solution in time; usually the motion changes slowly over time.

  42. Depth variation small? Divide image into small patches YES Search in the 2D space of translations Use the error NO Distinguish between two cases For each candidate 3D motion, using normal flow measurements in each patch, compute depth of the scene. existence of a discontinuity at the patch wrong 3D motion For each image patch Use the error Split the patch and repeat the process The Algorithm

  43. 3D reconstruction • Comparison of the original sequence and the re-projection of the 3D reconstruction.

  44. 3D model construction

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