Affine structure from motion

1. Affine structure from motion Marc Pollefeys COMP 256 Some slides and illustrations from J. Ponce, A. Zisserman, R. Hartley, Luc Van Gool, …

2. Last time: Optical Flow Ixu Ix u Ixu=- It It Aperture problem: • two solutions: • - regularize (smoothness prior) • constant over window • (i.e. Lucas-Kanade) Coarse-to-fine, parametric models, etc…

3. Tentative class schedule

4. AFFINE STRUCTURE FROM MOTION • The Affine Structure from Motion Problem • Elements of Affine Geometry • Affine Structure from Motion from two Views • A Geometric Approach • Affine Epipolar Geometry • An Algebraic Approach • Affine Structure from Motion from Multiple Views • From Affine to Euclidean Images • Structure from motion of multiple and deforming object Reading: Chapter 12.

5. Affine Structure from Motion Reprinted with permission from “Affine Structure from Motion,” by J.J. (Koenderink and A.J.Van Doorn, Journal of the Optical Society of America A, 8:377-385 (1990).  1990 Optical Society of America. • Given m pictures of n points, can we recover • the three-dimensional configuration of these points? • the camera configurations? (structure) (motion)

6. Orthographic Projection Parallel Projection

7. Weak-Perspective Projection Paraperspective Projection

8. Problem: estimate the m 2x4 matrices M and the n positions P from the mn correspondences p . i j ij The Affine Structure-from-Motion Problem Given m images of n fixed points P we can write j 2mn equations in 8m+3n unknowns Overconstrained problem, that can be solved using (non-linear) least squares!

9. If M and P are solutions, i j The Affine Ambiguity of Affine SFM When the intrinsic and extrinsic parameters are unknown So are M’ and P’ where i j and Q is anaffine transformation.

10. Affine Spaces: (Semi-Formal) Definition

11. 2 Example: R as an Affine Space

12. In General The notation is justified by the fact that choosing some origin O in X allows us to identify the point P with the vector OP. Warning:P+u and Q-P are defined independently of O!!

13. Barycentric Combinations • Can we add points? R=P+Q NO! • But, when we can define • Note:

14. Affine Subspaces

15. Affine Coordinates • Coordinate system for U: • Coordinate system for Y=O+U: • Affine coordinates: • Coordinate system for Y: • Barycentric • coordinates:

16. When do m+1 points define a p-dimensional subspace Y of an n-dimensional affine space X equipped with some coordinate frame basis? Rank ( D ) = p+1, where Writing that all minors of size (p+2)x(p+2) of D are equal tozero givestheequationsof Y.

17. Affine Transformations • Bijections from X to Y that: • map m-dimensional subspaces of X onto m-dimensional • subspaces of Y; • map parallel subspaces onto parallel subspaces; and • preserve affine (or barycentric) coordinates. • Bijections from X to Y that: • map lines of X onto lines of Y; and • preserve the ratios of signed lengths of • line segments. 3 In E they are combinations of rigid transformations, non-uniform scalings and shears.

18. Affine Transformations II • Given two affine spaces X and Y of dimension m, and two • coordinate frames (A) and (B) for these spaces, there exists • a unique affine transformation mapping (A) onto (B). • Given an affine transformation from X to Y, one can always write: • When coordinate frames have been chosen for X and Y, • this translates into:

19. Affine projections induce affine transformations from planes onto their images.

20. Affine Shape Two point sets S and S’ in some affine space X are affinely equivalentwhen there exists an affine transformation y: X X such that X’ = y ( X ). Affine structure from motion = affine shape recovery. = recovery of the corresponding motion equivalence classes.

21. Geometric affine scene reconstruction from two images (Koenderink and Van Doorn, 1991).

22. Affine Structure from Motion Reprinted with permission from “Affine Structure from Motion,” by J.J. (Koenderink and A.J.Van Doorn, Journal of the Optical Society of America A, 8:377-385 (1990).  1990 Optical Society of America. (Koenderink and Van Doorn, 1991)

23. The Affine Epipolar Constraint Note: the epipolar lines are parallel.

24. Affine Epipolar Geometry

25. The Affine Fundamental Matrix where

26. An Affine Trick.. Algebraic Scene Reconstruction

27. The Affine Structure of Affine Images Suppose we observe a scene with m fixed cameras.. The set of all images of a fixed scene is a 3D affine space!

28. has rank 4!

29. From Affine to Vectorial Structure Idea: pick one of the points (or their center of mass) as the origin.

30. What if we could factorize D? (Tomasi and Kanade, 1992) Affine SFM is solved! Singular Value Decomposition We can take

31. From uncalibrated to calibrated cameras Weak-perspective camera: Calibrated camera: Problem: what is Q ? Note:Absolute scale cannot be recovered. TheEuclideanshape (defined up to an arbitrary similitude) is recovered.

32. Reconstruction Results (Tomasi and Kanade, 1992) Reprinted from “Factoring Image Sequences into Shape and Motion,” by C. Tomasi and T. Kanade, Proc. IEEE Workshop on Visual Motion (1991).  1991 IEEE.

33. More examples Tomasi Kanade’92, Poelman & Kanade’94

34. More examples Tomasi Kanade’92, Poelman & Kanade’94

35. More examples Tomasi Kanade’92, Poelman & Kanade’94

36. Further Factorization work Factorization with uncertainty Factorization for indep. moving objects (now) Factorization for articulated objects (now) Factorization for dynamic objects (now) Perspective factorization (next week) Factorization with outliers and missing pts. (Irani & Anandan, IJCV’02) (Costeira and Kanade ‘94) (Yan and Pollefeys ‘05) (Bregler et al. 2000, Brand 2001) (Sturm & Triggs 1996, …) (Jacobs ‘97 (affine), Martinek & Pajdla‘01Aanaes’02 (perspective))

37. Structure from motion of multiple moving objects

38. Structure from motion of multiple moving objects

39. Shape interaction matrix Shape interaction matrix for articulated objects looses block diagonal structure Costeira and Kanade’s approach is not usable for articulated bodies (assumes independent motions)

40. Articulated motion subspaces Motion subspaces for articulated bodies intersect (Yan and Pollefeys, CVPR’05) (Tresadern and Reid, CVPR’05) Joint (1D intersection) (joint=origin) (rank=8-1) Hinge (2D intersection) (hinge=z-axis) (rank=8-2) Exploit rank constraint to obtain better estimate Also for non-rigid parts if (Yan & Pollefeys, 06?)

41. Student Segmentation Intersection Toy truck Segmentation Intersection Results

42. Articulated shape and motion factorization (Yan and Pollefeys, 2006?) Automated kinematic chain building for articulated & non-rigid obj. • Estimate principal angles between subspaces • Compute affinities based on principal angles • Compute minimum spanning tree

43. Structure from motion of deforming objects (Bregler et al ’00; Brand ‘01) Extend factorization approaches to deal with dynamic shapes

44. Representing dynamic shapes (fig. M.Brand) represent dynamic shape as varying linear combination of basis shapes

45. Projecting dynamic shapes (figs. M.Brand) Rewrite:

46. Dynamic image sequences One image: (figs. M.Brand) Multiple images

47. Dynamic SfM factorization? Problem: find J so that M has proper structure

48. Dynamic SfM factorization (Bregler et al ’00) Assumption: SVD preserves order and orientation of basis shape components

49. Results (Bregler et al ’00)

50. Dynamic SfM factorization (Brand ’01) constraints to be satisfied for M constraints to be satisfied for M, use to compute J hard! (different methods are possible, not so simple and also not optimal)