Optical flow and Tracking CISC 649/849 Spring 2009 University of Delaware

Outline Fusionflow Joint Lucas Kanade Tracking Some practical issues in tracking

What smoothing to choose?

Stereo Matching results…

Difficulties in optical flow Cannot directly apply belief propagation or graph cut – Number of labels too high Brightness variation higher than stereo matching

Can we combine different flows? ???

Formulation as a labeling problem Given flows x 0 and x 1, find a labeling y Combine the flows to get a new flow x f

Graph Cut formulation

Graph cut

Proposal Solutions Horn and Shunck with different smoothing Lucas Kanade with different window sizes Shifted versions of above

Discrete Optimization Choose one of the proposals randomly as initial flow field Visit other proposals in random order and update labeling Combine the proposals according to the labeling to give fused estimate

Continuous Optimization Some areas may have same solution in all proposals Use conjugate gradient method on the energy function to decrease the energy further Use bicubic interpolation to calculate gradient

Results

Recap… Lucas Kanade (sparse feature tracking) Horn Schunck (dense optic flow) assumes unknown displacement u of a pixel is constant within some neighborhood i.e., finds displacement of a small window centered around a pixel by minimizing: regularizes the unconstrained optic flow equation by imposing a global smoothness term computes global displacement functions u(x, y) v(x, y) by minimizing: λ: regularization parameter, Ω: image domain minimum of the functional is found by solving the corresponding Euler-Lagrange equations, leading to: denotes convolution with an integration window of size ρ differentiating with respect to u and v, setting the derivatives to zero leads to a linear system:

Limitations of Lucas-Kanade Tracking Tracks only those features whose minimum eigenvalue is greater than a fixed threshold Do edges satisfy this condition? Are edges bad for tracking? How can this be corrected?

Ambiguity on edges ?

Joint Lucas Kanade Tracking

Matrix Formulation

Iterative Solution

Joint Lucas Kanade Tracking For each feature i, 1. Initialize u i ← (0, 0) T 2. Initialize i For pyramid level n − 1 to 0 step −1, 1. For each feature i, compute Z i 2. Repeat until convergence: (a) For each feature i, i. Determine ii. Compute the difference I t between the first image and the shifted second image: I t (x, y) = I 1 (x, y) − I 2 (x + u i, y + v i ) iii. Compute e i iv. Solve Z i u′ i = e i for incremental motion u’ i v. Add incremental motion to overall estimate: u i ← u i + u′ i 3. Expand to the next level: u i ←  u i, where  is the pyramid scale factor

How to find mean flow? Average of neighboring features? – Too much variation in the flow vectors even if the motion is rigid Calculate an affine motion model with neighboring features weighted according to their distance from tracked feature

What features to track? Given the Eigen values of a window are e max and e min Standard Lucas Kanade chooses windows with e min > Threshold This restricts the features to corners Joint Lucas Kanade chooses windows with max(e min,K e max ) > Threshold where K<1.

Results LK JLK

Observations JLK performs better on edges and untextured regions Aperture problem is overcome on edges Future improvements – Does not handle occlusions – Does not account for motion discontinuities

Some issues in tracking Appearance change Sub pixel accuracy Lost Features/Occlusion

Further reading Joint Tracking of Features and Edges. Stanley T. Birchfield and Shrinivas J. Pundlik. CVPR 2008 FusionFlow: Discrete-Continuous Optimization for Optical Flow Estimation. V. Lempitsky, S. Roth, C. Rother. CVPR 2008 The template update problem, Matthews, L.; Ishikawa, T.; Baker, S. PAMI 2004