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Graph Cut Algorithms for Binocular Stereo with Occlusions

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Graph Cut Algorithms for Binocular Stereo with Occlusions

Vladimir Kolmogorov, Ramin Zabih

- Traditional Stereo Methods
- Energy Minimization via Graph Cuts
- Stereo with Occlusions
- Voxel Labeling Algorithm
- Pixel Labeling Algorithm
- Results and Conclusions

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Traditional Stereo Problem

pixel correspondences labeling (disparity)

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

disparity

depth

disparity ~ depth

ground truth disparity

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- goal is to compute pixels correspondences
- traditional stereo problem pixel labeling problem
- advantage: can be solved by graph cuts
- problem is formulated as energy term
- new goal: find the minimizing labeling

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

we assign the label to pixel p when p of image I corresponds to p + in I‘

find labeling that minimizes

cost for assigning labels

smoothness term

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- data cost – gives penalty for different intensities

- smoothness term – gives penalty for discontinuities (Potts model)

other models:

absolute distance

quadratic

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Max-flow / Min-Cut

(Ford and Fulkerson Algorithm, Push-Relabel Method)

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

α-expansion

α-β-swap

Initial Labeling

- convex V vs. metric / semimetric
- α-β-swap move
- α-expansion move: assigning label α to an arbitrary set of pixels

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- treat input symmetrically
- scene elements only visible in single view
- physically correct scenes geometric constraints occlusions physically possible labelings
- introduce constraints in the problem formulation
- graph cuts perform unconstrained energy minimization

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- discrete scene of voxels
- voxel v is active when visible from both cameras
- uniqueness constraint – 1:1 correspondence of pixels

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

smoothness term

(Potts model)

matching penalty

(only active voxels)

occlusion penalty

set of occluded pixels

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

like traditional stereo but for both images e.g. Potts model

active ?

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- convert constrained into unconstrained minimization problem
- write as sum over pairs
- form of energy function = standard stereo problem

- minimization with α-expansion algorithm
- modified definition of α-expansion move for voxel labeling

(0=valid, else ∞)

uniqueness

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

traditional s.p.

voxel labeling

pixel labeling

ground truth

Tsukuba ref. image

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- efficient energy minimization polynominal time instead of exponential time
- traditional stereo algorithm is faster
- pixel labeling better than voxel labeling:
- prohibits ‚holes‘ in the scene
- allows to use other effective smoothness terms

- algorithms can be extended for multiple cameras

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Graph-cut-based stereo matching using image segmentation with symmetrical treatment of occlusions, 2006 TUW

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007

- M. Bleyer, M. Gelautz, „Graph-cut-based stereo matching using image segmentation with symmetrical treatment of occlusions“, 2007
- Y. Boykov, O. Veksler, R. Zabih, „Fast Approximate Energy Minimization via Graph Cuts“, 2001
- V. Kolmogorov, R. Zabih, „Graph Cut Algorithms for Binocular Stereo with Occlusions“,2005
- V. Kolmogorov, R. Zabih, „What energy functions can be minimized via graph cuts“, 2004
- V. Kolmogorov, R. Zabih, „Generalized multi-camera scene reconstruction using graph cuts“, July 2003
- V. Kolmogorov, R. Zabih, „Multi-camera Scene Reconstruction via Graph Cuts“, 2002
- S. Seits, C. Dyer, „Photorealistic Scene Reconstruction by Voxel Coloring“, 1997
- R.Szeliski, R. Zabih, „An Experimental Comparison of Stereo Algorithms“, 1999

Graph Cut Algorithms for Binocular Stereo with Occlusions

Math Basics for Vision and Graphis 2007