Structure from Motion, Feature Tracking, and Optical Flow

1. 04/15/10 Structure from Motion, Feature Tracking, and Optical Flow Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Many slides adapted from Lana Lazebnik, Silvio Saverse, who in turn adapted slides from Steve Seitz, Rick Szeliski, Martial Hebert, Mark Pollefeys, and others

2. Last class • Estimating 3D points and depth • Triangulation from corresponding points • Dense stereo • Projective structure from motion

3. This class • Factorization method for structure from motion • Feature tracking • Optical flow (dense tracking)

4. Structure from motion under orthographic projection 3D Reconstruction of a Rotating Ping-Pong Ball • Reasonable choice when • Change in depth of points in scene is much smaller than distance to camera • Cameras do not move towards or away from the scene C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method.IJCV, 9(2):137-154, November 1992.

5. Start with an affine camera model x a2 X a1 • Affine projection is a linear mapping + translation in inhomogeneous coordinates Projection ofworld origin Does this ever happen?

6. Get rid of b by shifting to centroid 3d normalized point 2d normalized point (observed) Linear (affine) mapping

7. Suppose we know 3D points and affine camera parameters … then, we can compute the observed 2d positions of each point 3D Points (3xn) Camera Parameters (2mx3) 2D Image Points (2mxn) What rank is the matrix of 2D points?

8. What if we instead observe corresponding 2d image points? Can we recover the camera parameters and 3d points? cameras (2m) points (n)

9. Factorizing the measurement matrix AX Source: M. Hebert

10. Factorizing the measurement matrix • Singular value decomposition of D: Source: M. Hebert

11. Factorizing the measurement matrix • Singular value decomposition of D: Source: M. Hebert

12. Factorizing the measurement matrix • Obtaining a factorization from SVD: Source: M. Hebert

13. Factorizing the measurement matrix • Obtaining a factorization from SVD: Source: M. Hebert

14. Affine ambiguity • The decomposition is not unique. We get the same D by using any 3×3 matrix C and applying the transformations A → AC, X →C-1X • That is because we have only an affine transformation and we have not enforced any Euclidean constraints (like forcing the image axes to be perpendicular, for example) Source: M. Hebert

15. Eliminating the affine ambiguity • Orthographic: image axes are perpendicular and of unit length a1 · a2 = 0 x |a1|2 = |a2|2= 1 a2 X a1 Source: M. Hebert

16. Solve for orthographic constraints • Solve for L = CCT • Recover C from L by Cholesky decomposition: L = CCT • Update A and X: A = AC, X = C-1X Three equations for each image i where ~ ~

17. Algorithm summary • Given: m images and n tracked features xij • For each image i, center the feature coordinates • Construct a 2m ×n measurement matrix D: • Column j contains the projection of point j in all views • Row icontains one coordinate of the projections of all the n points in image i • Factorize D: • Compute SVD: D = U W VT • Create U3 by taking the first 3 columns of U • Create V3 by taking the first 3 columns of V • Create W3 by taking the upper left 3 × 3 block ofW • Create the motion and shape matrices: • M = U3W3½ and S = W3½V3T(or M = U3 and S = W3V3T) • Eliminate affine ambiguity Source: M. Hebert

18. Dealing with missing data • So far, we have assumed that all points are visible in all views • In reality, the measurement matrix typically looks something like this: One solution: • solve using a dense submatrix of visible points (as in last lecture) • Iteratively add new cameras cameras points

19. Reconstruction results C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method.IJCV, 9(2):137-154, November 1992.

20. Recovering motion • Feature-tracking • Extract visual features (corners, textured areas) and “track” them over multiple frames • Optical flow • Recover image motion at each pixel from spatio-temporal image brightness variations (optical flow) Two problems, one registration method B. Lucas and T. Kanade. An iterative image registration technique with an application to stereo vision. In Proceedings of the International Joint Conference on Artificial Intelligence, pp. 674–679, 1981.

21. Feature tracking • Last problem required corresponding points in images • If motion is small, tracking is an easy way to get them

22. Feature tracking • Challenges • Need good features to track • Points may appear or disappear: need to be able to add/delete tracked points • Some points may change appearance over time (e.g., due to rotation, moving into shadows, etc.) • Drift: small errors can accumulate if appearance model is updated

23. Feature tracking • Given two subsequent frames, estimate the point translation I(x,y,t–1) I(x,y,t) • Key assumptions of Lucas-Kanade Tracker • Brightness constancy: projection of the same point looks the same in every frame • Small motion: points do not move very far • Spatial coherence: points move like their neighbors

24. The brightness constancy constraint Hence, • Brightness Constancy Equation: I(x,y,t–1) I(x,y,t) Linearizing right hand side using Taylor expansion: Image derivative along x

25. The brightness constancy constraint Can we use this equation to recover image motion (u,v) at each pixel? • How many equations and unknowns per pixel? • One equation (this is a scalar equation!), two unknowns (u,v) The component of the motion perpendicular to the gradient (i.e., parallel to the edge) cannot be measured • If (u, v ) satisfies the equation, so does (u+u’, v+v’ ) if gradient (u,v) (u+u’,v+v’) (u’,v’) edge

26. The aperture problem Actual motion

27. The aperture problem Perceived motion

28. The barber pole illusion http://en.wikipedia.org/wiki/Barberpole_illusion

29. The barber pole illusion http://en.wikipedia.org/wiki/Barberpole_illusion

30. Solving the ambiguity… B. Lucas and T. Kanade. An iterative image registration technique with an application to stereo vision. In Proceedings of the International Joint Conference on Artificial Intelligence, pp. 674–679, 1981. • How to get more equations for a pixel? • Spatial coherence constraint • Assume the pixel’s neighbors have the same (u,v) • If we use a 5x5 window, that gives us 25 equations per pixel

31. Solving the ambiguity… • Least squares problem:

32. Matching patches across images • Overconstrained linear system Least squares solution for d given by • The summations are over all pixels in the K x K window

33. Conditions for solvability • Optimal (u, v) satisfies Lucas-Kanade equation • When is This Solvable? • ATA should be invertible • ATA should not be too small due to noise • eigenvalues 1 and 2 of ATA should not be too small • ATA should be well-conditioned • 1/ 2 should not be too large (1 = larger eigenvalue) Does this remind you of anything? Criteria for Harris corner detector

34. Edge • gradients very large or very small • large l1, small l2

35. Low-texture region • gradients have small magnitude • small l1, small l2

36. High-texture region • gradients are different, large magnitudes • large l1, large l2

37. The aperture problem resolved Actual motion

38. The aperture problem resolved Perceived motion

39. Dealing with larger movements: Iterative refinement • Initialize (u,v) = (0,0) • Compute (u,v) by • Shift window by (u, v) • Repeat steps 2-3 until small change • Only It changes per iteration It = I(x, y, t-1) - I(x+u, y+v, t) 2nd moment matrix for feature patch in first image displacement

40. Dealing with larger movements: coarse-to-fine registration upsample run iterative L-K . . . image J image I image 1 image 2 Gaussian pyramid of image 1 (t) Gaussian pyramid of image 2 (t+1) run iterative L-K

41. Shi-Tomasi feature tracker • Find good features using eigenvalues of second-moment matrix • Key idea: “good” features to track are the ones whose motion can be estimated reliably • From frame to frame, track with Lucas-Kanade • This amounts to assuming a translation model for frame-to-frame feature movement • Check consistency of tracks by affine registration to the first observed instance of the feature • Affine model is more accurate for larger displacements • Comparing to the first frame helps to minimize drift J. Shi and C. Tomasi. Good Features to Track. CVPR 1994.

42. Tracking example J. Shi and C. Tomasi. Good Features to Track. CVPR 1994.

43. Summary of KLT tracking • Find a good point to track (harris corner) • Use intensity second moment matrix and difference across frames to find displacement • Iterate and use coarse-to-fine search to deal with larger movements • When creating long tracks, check appearance of registered patch against appearance of initial patch to find points that have drifted

44. Optical flow Vector field function of the spatio-temporal image brightness variations Picture courtesy of Selim Temizer - Learning and Intelligent Systems (LIS) Group, MIT

45. Motion and perceptual organization • Sometimes, motion is the only cue

46. Motion and perceptual organization • Even “impoverished” motion data can evoke a strong percept G. Johansson, “Visual Perception of Biological Motion and a Model For Its Analysis", Perception and Psychophysics 14, 201-211, 1973.

47. Motion and perceptual organization • Even “impoverished” motion data can evoke a strong percept G. Johansson, “Visual Perception of Biological Motion and a Model For Its Analysis", Perception and Psychophysics 14, 201-211, 1973.

48. Uses of motion • Estimating 3D structure • Segmenting objects based on motion cues • Learning and tracking dynamical models • Recognizing events and activities • Improving video quality (motion stabilization)

49. Motion field • The motion field is the projection of the 3D scene motion into the image What would the motion field of a non-rotating ball moving towards the camera look like?

50. Optical flow • Definition: optical flow is the apparent motion of brightness patterns in the image • Ideally, optical flow would be the same as the motion field • Have to be careful: apparent motion can be caused by lighting changes without any actual motion • Think of a uniform rotating sphere under fixed lighting vs. a stationary sphere under moving illumination