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Fast and Accurate Optical Flow Estimation Primal-Dual Schemes and Second Order Priors Thomas Pock and Daniel Cremers CVPR Group, University of Bonn Collaborators:

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Presentation on theme: "Fast and Accurate Optical Flow Estimation Primal-Dual Schemes and Second Order Priors Thomas Pock and Daniel Cremers CVPR Group, University of Bonn Collaborators:"— Presentation transcript:

1 Fast and Accurate Optical Flow Estimation Primal-Dual Schemes and Second Order Priors Thomas Pock and Daniel Cremers CVPR Group, University of Bonn Collaborators: Christopher Zach, Markus Unger, Werner Trobin, and Horst Bischof

2 Variational Optical Flow – Short History Horn and Schunck 1981 Black and Anadan, Cohen 1993 Aubert Brox et al Bruhn et al.

3

4 Outline Model of Horn and Schunck TV-L 1 Model Fast Numerical Scheme Parallel Implementation 2 nd order Prior

5 The Model of Horn and Schunck [1] Regularization TermData Term (OFC) + Convex + Easy to solve - Does not allow for sharp edges in the solution - Sensitive to outliers violating the OFC [1] Horn and Schunck. Determinig Optical Flow. Artificial Intelligence, 1981

6 Can we do better? Replace quadratic functions by L 1 – norms Done by Cohen, Aubert, Brox, Bruhn,... +Allows for discontinuities in the flow field +Robust to some extent to outliers in the OFC +Still convex - Much harder to solve

7 How can we minimize this functional ? Compute Euler-Lagrange Equations Non-linear, non-smooth,...

8 Standard Approach Replace L 1 – norm by regularized variants (Charbonnier function) Example: Small epsilon: Nearly degenerated Large espilon: Smears edges

9 Our Approach(1) Introduce auxiliary variables and constraints Quadratic penalty

10 Our Approach(2) What do we gain? We solve a sequence of simpler problems Algorithm[3]: 1.For fixed (u´,v´), solve for(u,v) using Chambolles algorithm[4] 2.For fixed (u,v), solve for (u´,v´) using a 1D shrinkage formula 3.Goto 1. until convergence [2] Rudin, Osher and Fatemi. Nonlinear Total Variation Based Noise Removal Algorithms, 1992 [3] Zach, Pock and Bischof. A Duality Based Algorithm for Realtime TV-L 1 Optical Flow, DAGM 2007 [4] Chambolle. An Algorithm for Total Variation Minimization, D Problem ROF Model [2]

11 Implementation Numerical scheme can be easily parallelized We use state-of-the-art GPUs

12 Performance Evaluation Image SizeFrames per Second 128x x x51236 TV-L 1 Optical Flow Implemented in CUDA 2.0 Computed on Nvidia GeForce GTX Overall Iterations (5 Chambolle Iterations)

13 Results for TV-L 1 Ground Truth: Our Results: Input Image:

14 2 nd order Prior TV regularization favors piecewise constant flow fields (frontoparallel motion) Extension to piecewise affine flow fields? Approach of Cremers et al. [5] – Fixed number of regions Approach of Nir et al. [6] – Over-parametrized optical flow Our approach [7] – 2 nd order derivatives to regularize flow field [5] Cremers and Soatto, Motion Competition: A Variational Framework for Piecwise Parametric Motion Segmentation. [6] Nir, Bruckstein and Kimmel, Over-Parameterized Variational Optical Flow, IJCV 2007 [7] Trobin, Pock, Cremers and Bischof, An Unbiased Second-Order prPior for High-Accuracy Motion Estimation, DAGM 2008

15 2 nd -L 1 Optical Flow 2 nd order derivatives are not orthogonal We use a transformation due to Danielsson [8] Optimization – Similar strategy to TV-L 1 – 4 th order PDE [8] Danielsson and Lin, Efficient Detection of Second-Degree Variations in 2D and 3D Images, 2001.

16 Comparison Ground truthTV-L 1 2 nd -L 1

17 Results for 2 nd -L 1 Ground Truth: Our Results:

18 Conclusion TV-L 1 Optical Flow – Fast Numerical Scheme Parallel Implementation – Realtime Performance 2 nd order prior – Piecewise affine motion

19 Recent Application: Tracking

20 Why does it allow for discontinuities ? Total Variation has no bias against discontinuities

21 Evaluation of Optical Flow Methods [1] Baker, S., Scharstein, D., Lewis, J.P., Roth, S., Black, M., Szeliski, R.: A database and evaluation methodology for optical flow. ICCV 2007 Input Images Ground Truth


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