Minimum cut/maximum flow algorithms on graphs have emerged as an increasingly useful tool for exactor approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-flow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of computer vision. The goal of this paper is to provide an experimental comparison of the efficiency of min-cut/max flow algorithms for applications in vision. We compare the running times of several standard algorithms, as well as a new algorithm that we have recently developed. The algorithms we study include both Goldberg-Tarjan style "push -relabel" methods and algorithms based on Ford-Fulkerson style "augmenting paths." We benchmark these algorithms on a number of typical graphs in the contexts of image restoration, stereo, and segmentation. In many cases, our new algorithm works several times faster than any of the other methods, making near real-time performance possible. An implementation of our max-flow/min-cut algorithm is available upon request for research purposes.

/pdf/an-experimental-comparison-of-min-cut-max-flow-algorithms-2823y6q313.pdf

An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision

After [10, 15, 12, 2, 4] minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-flow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of computer vision. The goal of this paper is to provide an experimental comparison of the efficiency of min-cut/max flow algorithms for energy minimization in vision. We compare the running times of several standard algorithms, as well as a new algorithm that we have recently developed. The algorithms we study include both Goldberg-style "push-relabel" methods and algorithms based on Ford-Fulkerson style augmenting paths. We benchmark these algorithms on a number of typical graphs in the contexts of image restoration, stereo, and interactive segmentation. In many cases our new algorithm works several times faster than any of the other methods making near real-time performance possible.

/pdf/an-experimental-comparison-of-min-cut-max-flow-algorithms-22jytmna6l.pdf

An Experimental Comparison of Min-cut/Max-flow Algorithms for Energy Minimization in Vision

In this paper we describe a new technique for general purpose interactive segmentation of N-dimensional images. The user marks certain pixels as "object" or "background" to provide hard constraints for segmentation. Additional soft constraints incorporate both boundary and region information. Graph cuts are used to find the globally optimal segmentation of the N-dimensional image. The obtained solution gives the best balance of boundary and region properties among all segmentations satisfying the constraints. The topology of our segmentation is unrestricted and both "object" and "background" segments may consist of several isolated parts. Some experimental results are presented in the context of photo/video editing and medical image segmentation. We also demonstrate an interesting Gestalt example. A fast implementation of our segmentation method is possible via a new max-flow algorithm.

/pdf/interactive-graph-cuts-for-optimal-boundary-region-247fxh4mh8.pdf

Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images

In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.

/pdf/what-energy-functions-can-be-minimized-via-graph-cuts-inkqpz6gaf.pdf

What energy functions can be minimized via graph cuts

In this paper we describe a new technique for general purpose interactive segmentation of N-dimensional images. The user marks certain pixels as “object” or “background” to provide hard constraints for segmentation. Additional soji constraints incorporate both boundary and region information. Graph cuts are used to find the globally optimal segmentation of the N-dimensional image. The obtained solution gives the best balance of boundary and region properties among all segmentations satishing the constraints. The topology o$our segmentation is unrestricted and both “object” and “background” segments may consist of several isolatedparts. Some experimental results are presented in the context ofphotohideo editing and medical image segmentation. We also demonstrate an interesting Gestalt example. A fast implementation of our segmentation method is possible via a new mar-$ow algorithm in [2].

/pdf/for-optimal-boundary-region-segmentation-of-objects-in-n-d-18y28ma3d3.pdf

for Optimal Boundary & Region Segmentation of Objects in N-D Images

Voxel occupancy is one approach for reconstructing the 3-dimensional shape of an object from multiple views. In voxel occupancy, the task is to produce a binary labeling of a set of voxels, that determines which voxels are filled and which are empty. In this paper, we give an energy minimization formulation of the voxel occupancy problem. The global minimum of this energy can be rapidly computed with a single graph cut, using a result due to D. Greig et al. (1989). The energy function we minimize contains a data term and a smoothness term. The data term is a sum over the individual voxels, where the penalty for a voxel is based on the observed intensities of the pixels that intersect it. The smoothness term is the number of empty voxels adjacent to filled ones. Our formulation can be viewed as a generalization of silhouette intersection, with two advantages: we do not compute silhouettes, which are a major source of errors; and we can naturally incorporate spatial smoothness. We give experimental results showing reconstructions from both real and synthetic imagery. Reconstruction using this smoothed energy function is not much more time consuming than simple silhouette intersection; it takes about 10 seconds to reconstruct a one million voxel volume.

/pdf/exact-voxel-occupancy-with-graph-cuts-2su0bkxalr.pdf

Exact voxel occupancy with graph cuts

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Exact voxel occupancy with graph cuts