1. Parallel and Distributed Graph Cuts by Dual Decomposition
Petter Strandmark Fredrik Kahl
Parallel and Distributed Graph Cuts by Dual Decomposition
Lund University

2. Applications of Graph Cuts
Image denoising
Stereo estimation
Shape fitting from point clouds
Segmentation

3. Graph Cuts 3 3 1 1 1 2 1 2 1 1 1 2 2 2 1 1 S T 5 1 3 1 1 1 2 1 1 2 1 2 1 4 3
Minimum cut: 4

4. Previous work Delong and Boykov, CVPR 2008
Implementation of push-relabel
Excellent speed-up for 2-8 processors
Method of choice for dense 3D graphs
CUDA-cuts: Vineet and Narayanan, CVGPU CVPR 2008
Push-relabel on GPU
Not clear what range of regularization can be used
L1-norm: Bhusnurmath and Taylor, PAMI 2008
Solves continuous problem on GPU
Not faster than augmenting paths on single processor

5. Previous work Liu and Sun, CVPR 2010 Our approach
” Parallel Graph-cuts by Adaptive Bottom-up Merging”
Splits large graph into several pieces
Augmenting paths found separately
Pieces merged together and search trees reused
Our approach
Graph split into several pieces
Solutions constrained to be equal with dual variables
Shared memory not required
See Komodakis et al. in ICCV 2007 for dual decomposition

6. Dual decomposition
Dualize the constraint!
Two separate problems!

7. Decomposition of graphs= 1 2 3 3 1 4 2 = = ≠ S T

8. Decomposed Linear Program
Global solution Û ?
Decomposed
Min-cut Problem
Original
Min-cut Problem Û Û Û Û
Linear Program
Decomposed Linear Program
Dual Linear Program
Zero duality gap
Dual function has a maximum such that the constraints are met
Global solution guaranteed!

9. Integer graphs
Theorem: If the graph weights are even integers, there exists an integer vector maximizing the dual function. This means that the dual problem can be solved without floating point arithmetic.

10. Solution procedure = Begin with a graph Split into two parts- 1 2 3
Begin with a graph
Split into two parts
Constrained to be equal on the overlap =
Independent problems!

11. Multiple splits

12. Multiple splits (3D)

13. Results Berkeley segmentation database 301 images 2 processors

14. Convergence 1152 × 1536 Iteration 1 2 3 4 5 ... 10 11 Differences 108
105 30 33 16 9
Time (ms)
245
1.5
1.2
0.1
0.08
0.07
0.47
1152 × 1536

15. Regularization
Easy problem: 230 ms
Hard problem: 4 s

16. ”Worst case” scenario S T
This choice of split severes all possible s/t paths
Parallel approach still 30% faster

17. Multiple computers

18. Multiple computers LUNARC cluster
401 × 396 × seconds 4 computers
95 × 98 × 30 × connectivity 12.3 GB 4 computers
512 × 512 × connectivity 131 GB 36 computers
Not much data need to be exchanged, 54kB in the first example
4D MRI data
3D CT
data

19. Conclusions Dual decomposition allows: Open source Faster processing
Solving larger graphs
Open source
C++/Matlab
Python