The calibration method for the Mumford-Shah functional and free-discontinuity problems

Alberti, Giovanni; Bouchitté, Guy; Maso, Gianni Dal

doi:10.1007/s005260100152

Cited by 98 publications

(181 citation statements)

References 29 publications

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“…Because of the empirical nature of the coupling, it is not clear how the computed solutions u mathematically relate to the original model (3), or even to (1). In fact, the computed solutions are actually not piecewise constant, but vary smoothly over large areas.…”

Section: Related Workmentioning

confidence: 99%

“…In the recent past, several authors have overcome the issue of non-convexity by suggesting convex relaxations for respective functionals [1,10]. Convex relaxations for the piecewise constant MumfordShah functional were proposed in [13,7,22].…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we propose a novel algorithm to e ciently compute approximate minimizers of the full Mumford-Shah functional (1). It combines several important advantages:…”

Section: Contributionmentioning

confidence: 99%

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Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional

Strekalovskiy¹,

Cremers²

2014

Computer Vision – ECCV 2014

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Abstract. We propose an algorithm for e ciently minimizing the piecewise smooth Mumford-Shah functional. The algorithm is based on an extension of a recent primal-dual algorithm from convex to non-convex optimization problems. The key idea is to rewrite the proximal operator in the primal-dual algorithm using Moreau's identity. The resulting algorithm computes piecewise smooth approximations of color images at 15-20 frames per second at VGA resolution using GPU acceleration. Compared to convex relaxation approaches [18], it is orders of magnitude faster and does not require a discretization of color values. In contrast to the popular Ambrosio-Tortorelli approach [2], it naturally combines piecewise smooth and piecewise constant approximations, it does not require an epsilon-approximation and it is not based on an alternation scheme. The achieved energies are in practice at most 5% o↵ the optimal value for one-dimensional problems. Numerous experiments demonstrate that the proposed algorithm is well-suited to perform discontinuity-preserving smoothing and real-time video cartooning.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional

Strekalovskiy¹,

Cremers²

2014

Computer Vision – ECCV 2014

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“…The continuous variant of the Potts model [54] describes a partition of the continuous domain Ω into n disjoint subdomains {Ω i } n i=1 as the minimum of a weighted sum of data fidelity and the length of the partition boundaries It was shown in [18] that the non-convex binary constraint u(x) ∈ {0, 1} could be relaxed and replaced by the convex constraint u(x) ∈ [0, 1]. Global and exact binary optimums could be obtained by thresholding the result of the convex relaxed problem at almost any level in the interval [0,1]. In [60], such a convex relaxation scheme was redefined under a novel continuous max-flow/min-cut perspective and studied by an elegant variational theory.…”

Section: Convex Relaxation Approaches For Pott's Modelmentioning

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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“…Then, motivated by the "calibration" idea [1], it was shown in [18] that f ′ can be expressed in terms of the {0, 1}-indicator function χ H(u ′ ) of the hypograph…”

Section: Bilinear Saddle-point Problems In Computer Visionmentioning

confidence: 99%

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

Lellmann

Breitenreicher

Schnörr

2010

Computer Vision – ECCV 2010

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Abstract. The saddle point framework provides a convenient way to formulate many convex variational problems that occur in computer vision. The framework unifies a broad range of data and regularization terms, and is particularly suited for nonsmooth problems such as Total Variation-based approaches to image labeling. However, for many interesting problems the constraint sets involved are difficult to handle numerically. State-of-the-art methods rely on using nested iterative projections, which induces both theoretical and practical convergence issues. We present a dual multiple-constraint Douglas-Rachford splitting approach that is globally convergent, avoids inner iterative loops, enforces the constraints exactly, and requires only basic operations that can be easily parallelized. The method outperforms existing methods by a factor of 4 − 20 while considerably increasing the numerical robustness.

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The calibration method for the Mumford-Shah functional and free-discontinuity problems

Cited by 98 publications

References 29 publications

Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional

Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional

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

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

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