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Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 158 (36 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
Phase unwrapping via graph cuts
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2007
"... Phase unwrapping is the inference of absolute phase from modulo2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are firstorder Markov random fields. We provide an exact energy minimization algorithm, whenever the correspo ..."
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Cited by 42 (9 self)
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Phase unwrapping is the inference of absolute phase from modulo2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are firstorder Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical L p norm, with p ≥ 1. Its complexity is KT(n, 3n), where K is the length of the absolute phase domain measured in 2π units and T (n, m) is the complexity of a maxflow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NPhard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems, by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrapping maxflow/mincut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with stateoftheart phase unwrapping algorithms.
Primaldual algorithm for convex Markov random fields
, 2005
"... Computing maximum a posteriori configuration in a firstorder Markov Random Field has become a routinely used approach in computer vision. It is equivalent to minimizing an energy function of discrete variables. In this paper we consider a subclass of minimization problems in which unary and pairwis ..."
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Cited by 17 (1 self)
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Computing maximum a posteriori configuration in a firstorder Markov Random Field has become a routinely used approach in computer vision. It is equivalent to minimizing an energy function of discrete variables. In this paper we consider a subclass of minimization problems in which unary and pairwise terms of the energy function are convex. Such problems arise in many vision applications including image restoration, total variation minimization, phase unwrapping in SAR images and panoramic image stitching. We give a new algorithm for computing an exact solution. Its complexity is K · T (n, m) where K is the number of labels and T (n, m) is the time needed to compute a maximum flow in a graph with n nodes and m edges. This is the fastest maxflowbased algorithm for this problem: previously best known technique takes T (nK, mK 2) time for general convex functions. Our approach also needs much less memory (O(n + m) instead of O(nK + mK 2)). Experimental results show for the panoramic stitching problem our method outperforms other techniques.
New algorithms for convex cost tension problem with application to computer vision
 a) hal00530369, version 1  28 Oct 2010 (b) (c
"... Motivated by various applications to computer vision, we consider the convex cost tension problem, which is the dual of the convex cost
ow problem. In this paper, we rst propose a primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal vari ..."
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Cited by 15 (3 self)
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Motivated by various applications to computer vision, we consider the convex cost tension problem, which is the dual of the convex cost
ow problem. In this paper, we rst propose a primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. We show that the time complexity of the primal algorithm is O(K T (n; m)), where K is the range of primal variables and T (n; m) is the time needed to compute a minimum cut in a graph with n nodes and m edges. We then develop an improved version of the primal algorithm, called the primaldual algorithm, by making good use of dual variables in addition to primal variables. Although its time complexity is the same as that of the primal algorithm, we can expect a better performance in practice. We nally consider an application to a computer vision problem called the panoramic image stitching. Key words: minimum cost tension, minimum cost
ow, discrete convex function, submodular function 1.
A steepest descent algorithm for Mconvex functions on jump system
 IEICE Trans. Fundamentals of Electr., Commun., Comput. Sci
"... The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have o®ered their works here electro ..."
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Cited by 7 (1 self)
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The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have o®ered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder.
CAPE: combinatorial absolute phase estimation
"... An absolute phase estimation algorithm for interferometric applications is introduced. The approach is Bayesian. Besides coping with the 2�periodic sinusoidal nonlinearity in the observations, the proposed methodology assumes a firstorder Markov random field prior and a maximum a posteriori probab ..."
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An absolute phase estimation algorithm for interferometric applications is introduced. The approach is Bayesian. Besides coping with the 2�periodic sinusoidal nonlinearity in the observations, the proposed methodology assumes a firstorder Markov random field prior and a maximum a posteriori probability (MAP) viewpoint. For computing the MAP solution, we provide a combinatorial suboptimal algorithm that involves a multiprecision sequence. In the coarser precision, it unwraps the phase by using, essentially, the previously introduced PUMA algorithm [IEEE Trans. Image Proc. 16, 698 (2007)], which blindly detects discontinuities and yields a piecewise smooth unwrapped phase. In the subsequent increasing precision iterations, the proposed algorithm denoises each piecewise smooth region, thanks to the previously detected location of the discontinuities. For each precision, we map the problem into a sequence of binary optimizations, which we tackle by computing mincuts on appropriate graphs. This unified rationale for both phase unwrapping and denoising inherits the fast performance of the graph mincuts algorithms. In a set of experimental results, we illustrate the effectiveness of the proposed approach. © 2009 Optical Society of America OCIS codes: 100.5088, 100.3020, 100.3175. 1.
A Fast Solver for TruncatedConvex Priors: QuantizedConvex Split Moves
, 2011
"... This paper addresses the problem of minimizing multilabel energies with truncated convex priors. Such priors are known to be useful but difficult and slow to optimize because they are not convex. We propose two novel classes of binary GraphCuts (GC) moves, namely the convex move and the quantized m ..."
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This paper addresses the problem of minimizing multilabel energies with truncated convex priors. Such priors are known to be useful but difficult and slow to optimize because they are not convex. We propose two novel classes of binary GraphCuts (GC) moves, namely the convex move and the quantized move. The moves are complementary. To significantly improve efficiency, the label range is divided into even intervals. The quantized move tends to efficiently put pixel labels into the correct intervals for the energy with truncated convex prior. Then the convex move assigns the labels more precisely within these intervals for the same energy. The quantized move is a modified αexpansion move, adapted to handle a generalized Potts prior, which assigns a constant penalty to arguments above some threshold. Our convex move is a GC representation of the efficient Murota’s algorithm. We assume that the data terms are convex, since this is a requirement for Murota’s algorithm. We introduce QuantizedConvex Split Moves algorithm which minimizes energies with truncated priors by alternating both moves. This algorithm is a fast solver for labeling problems with a high number of labels and convex data terms. We illustrate its performance on image restoration.