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1.Feature Tracking and Optical Flow Computer Vision Jia-Bin Huang, Virginia Tech Many slides from D . Hoiem

2.Administrative Stuffs HW 1 due 11:59 PM Sept 19 Submission through Canvas Regular office hour by Jia-Bin 3:00- 4:00 PM, Friday Sept 16 Bonus office hour by Akrit 10:30 AM – 11: 30 AM, Monday Sept 19 HW 1 Competition: Edge Detection Submission link Leaderboard

3.Administrative Stuffs HW 1 due 11:59 PM Sept 19 Submission through Canvas Regular office hour by Jia-Bin 3:00- 4:00 PM, Friday Sept 16 Bonus office hour by Akrit 10:30 AM – 11: 30 AM, Monday Sept 19 HW 1 Competition: Edge Detection Submission link Leaderboard

4.This class: 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 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 , 1981. Two problems, one registration method

5.Feature tracking Many problems, such as structure from motion require matching points If motion is small, tracking is an easy way to get them

6.Feature tracking - Challenges Figure out which features can be tracked Efficiently track across frames Some points may change appearance over time (e.g., due to rotation, moving into shadows, etc.) Drift: small errors can accumulate as appearance model is updated Points may appear or disappear: need to be able to add/delete tracked points

7.Feature tracking Given two subsequent frames, estimate the point translation 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 I ( x , y , t ) I ( x , y , t+1 )

8.Brightness Constancy Equation: Take Taylor expansion of I(x+u, y+v, t+1) at (x,y,t) to linearize the right side: The brightness constancy constraint I ( x , y , t ) I ( x , y , t+1 ) So: Image derivative along x Difference over frames

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

10.The aperture problem Actual motion

11.The aperture problem Perceived motion

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

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

14.Solving the ambiguity… 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 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.

15.Least squares problem: Solving the ambiguity…

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

17.Conditions for solvability Optimal (u, v) satisfies Lucas-Kanade equation Does this remind you of anything? When is this solvable? I.e., what are good points to track? A T A should be invertible A T A should not be too small due to noise eigenvalues 1 and 2 of A T A should not be too small A T A should be well-conditioned 1 / 2 should not be too large ( 1 = larger eigenvalue) Criteria for Harris corner detector

18.Eigenvectors and eigenvalues of A T A relate to edge direction and magnitude The eigenvector associated with the larger eigenvalue points in the direction of fastest intensity change The other eigenvector is orthogonal to it M = A T A is the second moment matrix ! (Harris corner detector…)

19.Low-texture region gradients have small magnitude small l 1 , small l 2

20.Edge gradients very large or very small large l 1 , small l 2

21.High-texture region gradients are different, large magnitudes large l 1 , large l 2

22.The aperture problem resolved Actual motion

23.The aperture problem resolved Perceived motion

24.Dealing with larger movements: Iterative refinement Initialize ( x’,y ’) = ( x,y ) Compute ( u,v ) by Shift window by (u, v): x’ = x’+u ; y’= y’+v ; Recalculate I t Repeat steps 2-4 until small change Use interpolation for subpixel values 2 nd moment matrix for feature patch in first image displacement I t = I(x’, y’, t+1) - I(x, y, t) Original (x,y) position

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

26.Shi- Tomasi feature tracker Find good features using eigenvalues of second-moment matrix (e.g., Harris detector or threshold on the smallest eigenvalue ) Key idea: “good” features to track are the ones whose motion can be estimated reliably Track from frame to frame 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.

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

28.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

29.Implementation issues Window size Small window more sensitive to noise and may miss larger motions (without pyramid) Large window more likely to cross an occlusion boundary (and it’s slower) 15x15 to 31x31 seems typical Weighting the window Common to apply weights so that center matters more (e.g., with Gaussian)