Petter Strandmark Fredrik Kahl Lund University

Image denoising Stereo estimation Segmentation Shape fitting from point clouds

S T Minimum cut: 4 1 1

 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

 Liu and Sun, CVPR 2010  ” 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

Two separate problems! Dualize the constraint!

= = = ≠ S T ½ ½

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

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.

 Begin with a graph  Split into two parts  Constrained to be equal on the overlap = Independent problems!

Berkeley segmentation database 301 images 2 processors 4 processors

Iteration Differences Time (ms) × 1536

Easy problem: 230 ms Hard problem: 4 s

ST This choice of split severes all possible s/t paths Parallel approach still 30% faster

 LUNARC cluster  401 × 396 × seconds 4 computers  95 × 98 × 30 × 1980-connectivity 12.3 GB4 computers  512 × 512 × connectivity 131 GB36 computers Not much data need to be exchanged, 54kB in the first example 4D MRI data 3D CT data

 Dual decomposition allows:  Faster processing  Solving larger graphs  Open source  C++/Matlab  Python