Abstract. 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.

Abstract. Many problems in computer vision can be naturally phrased in terms of energy minimization. In the last few years researchers have developed a powerful class of energy minimization methods based on graph cuts. These techniques construct a specialized graph, such that the minimum cut on the graph also minimizes the energy. The minimum cut in turn is efficiently computed by max flow algorithms. Such methods have been successfully applied to a number of important vision problems, including image restoration, motion, stereo, voxel occupancy and medical imaging. However, each graph construction to date has been highly specific for a particular energy function. In this paper we address a much broader problem, by characterizing the class of energy functions that can be minimized by graph cuts, and by giving a general-purpose construction that minimizes any energy function in this class. Our results generalize several previous vision algorithms based on graph cuts, and also show how to minimize an interesting new class of energy functions.

Algorithms for discrete energy minimization are of fundamental importance in computer vision. In this paper, we focus on the recent technique proposed by Wainwright et al. [33]--tree-reweighted max-product message passing (TRW). It was inspired by the problem of maximizing a lower bound on the energy. However, the algorithm is not guaranteed to increase this bound--it may actually go down. In addition, TRW does not always converge. We develop a modification of this algorithm which we call sequential tree-reweighted message passing. Its main property is that the bound is guaranteed not to decrease. We also give a weak tree agreement condition which characterizes local maxima of the bound with respect to TRW algorithms. We prove that our algorithm has a limit point that achieves weak tree agreement. Finally, we show that, our algorithm requires half as much memory as traditional message passing approaches. Experimental results demonstrate that on certain synthetic and real problems, our algorithm outperforms both the ordinary belief propagation and tree-reweighted algorithm in [33]. In addition, on stereo problems with Potts interactions, we obtain a lower energy than graph cuts.

Several new algorithms for visual correspondence based on graph cuts [7,14,17]

Abstract-Among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel-labeling tasks such as depth or texture computation. It has been known for decades that such problems can be elegantly expressed as Markov random fields, yet the resulting energy minimization problems have been widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: For example, such methods form the basis for almost all the top-performing stereo methods. However, the trade-offs among different energy minimization algorithms are still not well understood. In this paper, we describe a set of energy minimization benchmarks and use them to compare the solution quality and runtime of several common energy minimization algorithms. We investigate three promising recent methods-graph cuts, LBP, and tree-reweighted message passing-in addition to the well-known older iterated conditional mode (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching, interactive segmentation, and denoising. We also provide a general-purpose software interface that allows vision researchers to easily switch between optimization methods. The benchmarks, code, images, and results are available at http://vision.middlebury.edu/MRF/.

After [15], [31], [19], [8], [25], [5], 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 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.

Abstract. Many problems in computer vision can be naturally phrased in terms of energy minimization. In the last few years researchers have developed a powerful class of energy minimization methods based on graph cuts. These techniques construct a specialized graph, such that the minimum cut on the graph also minimizes the energy. The minimum cut in turn is efficiently computed by max flow algorithms. Such methods have been successfully applied to a number of important vision problems, including image restoration, motion, stereo, voxel occupancy and medical imaging. However, each graph construction to date has been highly specific for a particular energy function. In this paper we address a much broader problem, by characterizing the class of energy functions that can be minimized by graph cuts, and by giving a general-purpose construction that minimizes any energy function in this class. Our results generalize several previous vision algorithms based on graph cuts, and also show how to minimize an interesting new class of energy functions.

No abstract