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34
A comparative study of energy minimization methods for Markov random fields
 In ECCV
, 2006
"... Abstract. One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Ma ..."
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Cited by 241 (24 self)
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Abstract. One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Random Fields (MRF’s), the resulting energy minimization problems were 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 topperforming stereo methods. Unfortunately, most papers define their own energy function, which is minimized with a specific algorithm of their choice. As a result, the tradeoffs among different energy minimization algorithms are not well understood. In this paper we describe a set of energy minimization benchmarks, which we use to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods—graph cuts, LBP, and treereweighted message passing—as well as the wellknown older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching and interactive segmentation. We also provide a generalpurpose software interface that allows vision researchers to easily switch between optimization methods with minimal overhead. We expect that the availability of our benchmarks and interface will make it significantly easier for vision researchers to adopt the best method for their specific problems. Benchmarks, code, results and images are available at
M.: Fast gradientbased algorithms for constrained total variation image denoising and deblurring problems
 IEEE Transaction on Image Processing
, 2009
"... This paper studies gradientbased schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TVbased image deburring problem. To achieve this task we combine an acceleration of ..."
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Cited by 67 (1 self)
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This paper studies gradientbased schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TVbased image deburring problem. To achieve this task we combine an acceleration of the well known dual approach to the denoising problem with a novel monotone version of a fast iterative shrinkage/thresholding algorithm (FISTA) we have recently introduced. The resulting gradientbased algorithm shares a remarkable simplicity together with a proven global rate of convergence which is significantly better than currently known gradient projectionsbased methods. Our results are applicable to both the anisotropic and isotropic discretized TV functionals. Initial numerical results demonstrate the viability and efficiency of the proposed algorithms on image deblurring problems with box constraints. 1
Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization
, 2006
"... This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps ..."
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Cited by 58 (6 self)
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This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under L1 data fidelity term yields a selfdual contrast invariant filter. Finally we present some results.
Applications of parametric maxflow in computer vision
"... The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for ..."
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Cited by 39 (7 self)
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The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for different λ’s. An example is a weighting between data and regularization terms in image segmentation or stereo: it is desirable to vary it both during training (to learn λ from ground truth data) and testing (to select best λ using highknowledge constraints, e.g. user input). We review algorithmic aspects of this parametric maximum flow problem previously unknown in vision, such as the ability to compute all breakpoints of λ and corresponding optimal configurations in finite time. These results allow, in particular, to minimize the ratio of some geometric functionals, such as flux of a vector field over length (or area). Previously, such functionals were tackled with shortest path techniques applicable only in 2D. We give theoretical improvements for “PDE cuts ” [5]. We present experimental results for image segmentation, 3D reconstruction, and the cosegmentation problem. 1.
Solving The Convex Cost Integer Dual Network Flow Problem
 MANAGEMENT SCIENCE
, 1999
"... In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form (i,j)Q ij ij F(w)+ iP ii B( ) ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the f ..."
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Cited by 29 (5 self)
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In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form (i,j)Q ij ij F(w)+ iP ii B( ) ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form i  j w ij , (i, j) Q), with lower and upper bounds on variables. Let n = P, m = Q, and U be the largest magnitude in the lower and upper bounds of variables. We call this problem the convex cost integer dual network flow problem. In this paper, we describe several applications of the convex cost integer dual network flow problem arising in dialaride transit problems, inverse spanning tree problem, project management, and regression analysis. We develop network flow based algorithms to solve the convex cost integer dual network flow problem. We show that using the Lagrangian relaxation technique, the convex cost integer dual network flow problem can be transformed to a convex cost primal network flow problem where each cost function is a piecewise linear convex function with integer slopes. Its special structure allows the convex cost primal network flow problem to be solved in O(nm log n log(nU)) time using a costscaling algorithm, which is the best available time bound to solve the convex cost integer dual network flow problem.
An efficient algorithm for Cosegmentation
"... This paper is focused on the Cosegmentation problem [1] – where the objective is to segment a similar object from a pair of images. The background in the two images may be arbitrary; therefore, simultaneous segmentation of both images must be performed with a requirement that the appearance of the ..."
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Cited by 25 (1 self)
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This paper is focused on the Cosegmentation problem [1] – where the objective is to segment a similar object from a pair of images. The background in the two images may be arbitrary; therefore, simultaneous segmentation of both images must be performed with a requirement that the appearance of the two sets of foreground pixels in the respective images are consistent. Existing approaches [1, 2] cast this problem as a Markov Random Field (MRF) based segmentation of the image pair with a regularized difference of the two histograms – assuming a Gaussian prior on the foreground appearance [1] or by calculating the sum of squared differences [2]. Both are interesting formulations but lead to difficult optimization problems, due to the presence of the second (histogram difference) term. The model proposed here bypasses measurement of the histogram differences in a direct fashion; we show that this enables obtaining efficient solutions to the underlying optimization model. Our new algorithm is similar to the existing methods in spirit, but differs substantially in that it can be solved to optimality in polynomial time using a maximum flow procedure on an appropriately constructed graph. We discuss our ideas and present promising experimental results. 1.
PARAMETRIC MAXIMUM FLOW ALGORITHMS FOR FAST TOTAL VARIATION MINIMIZATION
"... Abstract. This report studies the global minimization of discretized total variation (TV) energies with an L p (in particular, L 1 and L 2) fidelity term using parametric maximum flow algorithms to minimize st cut representations of these energies. The TV/L 2 model, also known as the RudinOsherFa ..."
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Cited by 20 (4 self)
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Abstract. This report studies the global minimization of discretized total variation (TV) energies with an L p (in particular, L 1 and L 2) fidelity term using parametric maximum flow algorithms to minimize st cut representations of these energies. The TV/L 2 model, also known as the RudinOsherFatemi (ROF) model is suitable for restoring images contaminated by Gaussian noise, while the TV/L 1 model is able to remove impulsive noise from greyscale images, and perform multiscale decompositions of them. Preliminary numerical results on largescale twodimensional CT and threedimensional Brain MR images are presented to illustrate the effectiveness of these approaches.
Phase unwrapping via graph cuts
 IEEE Transactions on Image Processing
, 2007
"... Abstract — 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 th ..."
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Cited by 18 (6 self)
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Abstract — 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. Index Terms — Phase unwrapping, energy minimization, integer optimization, submodularity, graph cuts, image
An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 15 (2 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
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 11 (0 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.