Globally minimal surfaces by continuous maximal flows

Appleton, Ben; Talbot, Hugues

doi:10.1109/tpami.2006.12

Cited by 128 publications

(151 citation statements)

References 26 publications

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“…On the other hand, it is possible to apply global optimization techniques to make the segmentation more robust to poor initialization, e.g. [8], [27], [2]. These methods have great optimizability, but often the model is less sophisticated and impoverished.…”

Section: A Optimizability and Fidelitymentioning

confidence: 99%

An Efficient Optimization Framework for Multi-Region Segmentation Based on Lagrangian Duality

Ulén

Strandmark

Kahl

2013

IEEE Trans. Med. Imaging

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Abstract-We introduce a multi-region model for simultaneous segmentation of medical images. In contrast to many other models, geometric constraints such as inclusion and exclusion between the regions are enforced, which makes it possible to correctly segment different regions even if the intensity distributions are identical. We efficiently optimize the model using a combination of graph cuts and Lagrangian duality which is faster and more memory efficient than current state of the art. As the method is based on global optimization techniques, the resulting segmentations are independent of initialization. We apply our framework to the segmentation of the left and right ventricles, myocardium and the left ventricular papillary muscles in MRI and to lung segmentation in full-body X-ray CT. We evaluate our approach on a publicly available benchmark with competitive results.

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Section: A Optimizability and Fidelitymentioning

confidence: 99%

An Efficient Optimization Framework for Multi-Region Segmentation Based on Lagrangian Duality

Ulén

Strandmark

Kahl

2013

IEEE Trans. Med. Imaging

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“…More generally, such problems can be formulated in terms of a labeling function. Examples include image denoising [41,57] where gray-scale values are directly regarded as labels, image segmentation [13,2,14] for which each label represents a region, two-view stereo reconstruction [40,41] where discrete-valued depths are used as labels, multi-view reconstruction [45] where inside and outside are simply indicated by two labels (see [49] for a good reference to more applications).…”

mentioning

confidence: 99%

A Fast Continuous Max-Flow Approach to Non-convex Multi-labeling Problems

Bae

Yuan

Tai

et al. 2014

Efficient Algorithms for Global Optimization Methods in Computer Vision

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Abstract. This work addresses a class of multilabeling problems over a spatially continuous image domain, where the data fidelity term can be any bounded function, not necessarily convex. Two total variation based regularization terms are considered, the first favoring a linear relationship between the labels and the second independent of the label values (Pott's model). In the spatially discrete setting, Ishikawa [33] showed that the first of these labeling problems can be solved exactly by standard max-flow and min-cut algorithms over specially designed graphs. We will propose a continuous analogue of Ishikawa's graph construction [33] by formulating continuous max-flow and min-cut models over a specially designed domain. These max-flow and min-cut models are equivalent under a primal-dual perspective. They can be seen as exact convex relaxations of the original problem and can be used to compute global solutions. Fast continuous max-flow based algorithms are proposed based on the max-flow models whose efficiency and reliability can be validated by both standard optimization theories and experiments. In comparison to previous work [53,52] on continuous generalization of Ishikawa's construction, our approach differs in the max-flow dual treatment which leads to the following main advantages: A new theoretical framework which embeds the label order constraints implicitly and naturally results in optimal labeling functions taking values in any predefined finite label set; A more general thresholding theorem which, under some conditions, allows to produce a larger set of non-unique solutions to the original problem; Numerical experiments show the new max-flow algorithms converge faster than the fast primal-dual algorithm of [53,52]. The speedup factor is especially significant at high precisions. In the end, our dual formulation and algorithms are extended to a recently proposed convex relaxation of Pott's model [50], thereby avoiding expensive iterative computations of projections without closed form solution.Key words. image processing and segmentation, continuous max-flow / min-cut, optimization AMS subject classifications. . . . Introduction.Many problems in image processing and computer vision can be modeled as energy minimization problems. In image restoration, such minimization problems may be defined over a set of functions which indicate the gray value of the restored image at each pixel. In image segmentation, the minimization problem can be defined over a set of partitions of the image domain. More generally, such problems can be formulated in terms of a labeling function. Examples include image denoising [41,57] where gray-scale values are directly regarded as labels, image segmentation [13, 2, 14] for which each label represents a region, two-view stereo reconstruction [40,41] where discrete-valued depths are used as labels, multi-view reconstruction [45] where inside and outside are simply indicated by two labels (see [49] for a good reference to more applications).Such optimization problems can be a...

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“…The minimum cut should separate sources and sinks and have the smallest possible cost γ c which can be interpreted as a length of hypersurface γ in isotropic metric defined by a scalar function c. Strang also establishes duality between continuous versions of minimum cut and maximum flow problems that is analogous to the discrete version established by Ford and Fulkerson [12]. On a practical note, a recent work by Appleton&Talbot [2] proposed a finite differences approach that, in the limit, converges to a globally optimal solution of continuous mincut/max-flow problem defined by Strang. Note, however, that they use graph cuts algorithms to "greatly increase the speed of convergence".…”

Section: Theories Connecting Graph-cuts and Hypersurfaces In R Nmentioning

confidence: 86%

“…Besides finding new applications, in the last years researchers discovered and rediscovered interesting links connecting graph cuts with other combinatorial algorithms (dynamic programming, shortest paths [6,22]), Markov random fields, statistical physics, simulated annealing and other regularization techniques [17,10,19], sub-modular functions [25], random walks and electric circuit theory [15,16], Bayesian networks and belief propagation [37], integral/differential geometry, anisotropic diffusion, level sets and other variational methods [36,7,2,22].…”

Section: Introductionmentioning

confidence: 99%

Graph Cuts in Vision and Graphics: Theories and Applications

Boykov¹,

Veksler²

Handbook of Mathematical Models in Computer Vision

143

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Abstract. Combinatorial min-cut algorithms on graphs emerged as an increasingly useful tool for problems in vision. Typically, the use of graphcuts is motivated by one of the following two reasons. Firstly, graph-cuts allow geometric interpretation; under certain conditions a cut on a graph can be seen as a hypersurface in N-D space embedding the corresponding graph. Thus, many applications in vision and graphics use min-cut algorithms as a tool for computing optimal hypersurfaces. Secondly, graphcuts also work as a powerful energy minimization tool for a fairly wide class of binary and non-binary energies that frequently occur in early vision. In some cases graph cuts produce globally optimal solutions. More generally, there are iterative graph-cut based techniques that produce provably good approximations which (were empirically shown to) correspond to high-quality solutions in practice. Thus, another large group of applications use graph-cuts as an optimization technique for low-level vision problems based on global energy formulations. This chapter is intended as a tutorial illustrating these two aspects of graph-cuts in the context of problems in computer vision and graphics. We explain general theoretical properties that motivate the use of graph cuts, as well as, show their limitations.

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Globally minimal surfaces by continuous maximal flows

Cited by 128 publications

References 26 publications

An Efficient Optimization Framework for Multi-Region Segmentation Based on Lagrangian Duality

An Efficient Optimization Framework for Multi-Region Segmentation Based on Lagrangian Duality

A Fast Continuous Max-Flow Approach to Non-convex Multi-labeling Problems

Graph Cuts in Vision and Graphics: Theories and Applications

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