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What energy functions can be minimized via graph cuts?
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2004
"... 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 co ..."
Abstract

Cited by 1047 (23 self)
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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 generalpurpose 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.
Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints. ..."
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Cited by 35 (6 self)
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We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints.
Convex Relaxations for Binary Image Partitioning and Perceptual Grouping
 In Pattern Recognition (23rd DAGM Symposium, Munich), Lecture Notes in Computer Science
, 2001
"... . We consider approaches to computer vision problems which require the minimization of a global energy functional over binary variables and take into account both local similarity and spatial context. The combinatorial nature of such problems has lead to the design of various approximation algorithm ..."
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Cited by 4 (0 self)
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. We consider approaches to computer vision problems which require the minimization of a global energy functional over binary variables and take into account both local similarity and spatial context. The combinatorial nature of such problems has lead to the design of various approximation algorithms in the past which often involve tuning parameters and tend to get trapped in local minima. In this context, we present a novel approach to the eld of computer vision that amounts to solving a convex relaxation of the original problem without introducing any additional parameters. Numerical ground truth experiments reveal a relative error of the convex minimizer with respect to the global optimum of below 2% on the average. We apply our approach by discussing two speci c problem instances related to image partitioning and perceptual grouping. Numerical experiments illustrate the quality of the approach which, in the partitioning case, compares favorably with established approaches like the ICMalgorithm. 1