Fast and Exact Primal-Dual Iterations for Variational Problems in Computer Vision

Lellmann, Jan; Breitenreicher, Dirk; Schnörr, Christoph

doi:10.1007/978-3-642-15552-9_36

Cited by 18 publications

(15 citation statements)

References 19 publications

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“…Another algorithm was derived which was shown to converge faster than the approaches of Lellmann et al (2009), Zach et al (2008. Another variant of the Douglas-Rachford algorithm was derived in Lellmann et al (2010), related to the Split Bregman method (Goldstein et al 2009), where the subproblems are instead Laplace equations, which are easier to solve than TV minimization problems.…”

Section: Evaluation Of Efficiency and Convergencementioning

confidence: 99%

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

2010

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This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding dualitybased approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to ob- tain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.

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Section: Evaluation Of Efficiency and Convergencementioning

confidence: 99%

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

2010

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“…We plan to investigate this further and release another paper on fast implementations and more comparisons in the future. We also became aware of a simultaneous work [43] which gives another algorithm for minimizing the energy in the convex formulation of [52]. Comparison with this work will also be subject of future research.…”

Section: Conclusion and Future Topicsmentioning

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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“…In the MRF domain, algorithms such as α-expansion and α-β swap [1] were developed to solve the corresponding multilabel problem. In the continuous setting, to this end convex relaxation schemes were proposed by Chambolle et al [2], Pock et al [6] and Lellmann et al [3,7,8].…”

Section: Related Workmentioning

confidence: 99%

Nonmetric Priors for Continuous Multilabel Optimization

Strekalovskiy¹,

Nieuwenhuis²,

Cremers³

2012

Computer Vision – ECCV 2012

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Abstract. We propose a novel convex prior for multilabel optimization which allows to impose arbitrary distances between labels. Only symmetry, d(i, j) ≥ 0 and d(i, i) = 0 are required. In contrast to previous grid based approaches for the nonmetric case, the proposed prior is formulated in the continuous setting avoiding grid artifacts. In particular, the model is easy to implement, provides a convex relaxation for the Mumford-Shah functional and yields comparable or superior results on the MSRC segmentation database comparing to metric or grid based approaches.

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Fast and Exact Primal-Dual Iterations for Variational Problems in Computer Vision

Cited by 18 publications

References 19 publications

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

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

Nonmetric Priors for Continuous Multilabel Optimization

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