A two-level domain decomposition method for image restoration

Xu, Jing; Tai, Xue–Cheng; Wang, Li-Lian

doi:10.3934/ipi.2010.4.523

Cited by 34 publications

(24 citation statements)

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“…We reiterate the relevance of our result, especially in light of the classical counterexamples mentioned above, which may hold for alternating algorithms on orthogonal decompositions of the space. Nevertheless, it is important to mention that there are also other very recent attempts of addressing domain decomposition methods for total variation minimization with successful numerical results [30,40], although lacking of a rigorous theoretical analysis.…”

Section: Our Approach Results and Technical Issuesmentioning

confidence: 99%

A convergent overlapping domain decomposition method for total variation minimization

2010

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In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles. Mathematics Subject Classification (2000)

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Section: Our Approach Results and Technical Issuesmentioning

confidence: 99%

A convergent overlapping domain decomposition method for total variation minimization

2010

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“…Since the structure of BD-S (2) is sequential, it requires elaborated effort to be implemented efficiently in a parallel computing machine. For instance, the coloring technique [29] is used for the block decomposition method of TV. To overcome this drawback, the following parallel nonoverlapping block decomposition method (or nonlinear block Jacobi) is commonly used.…”

Section: Block Decomposition Methodsmentioning

confidence: 99%

Block Decomposition Methods for Total Variation by Primal–Dual Stitching

Lee

Woo

et al. 2015

J Sci Comput

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Due to the advance of image capturing devices, huge size of images are available in our daily life. As a consequence the processing of large scale image data is highly demanded. Since the total variation (TV) is kind of de facto standard in image processing, we consider block decomposition methods for TV based variational models to handle large scale images. Unfortunately, TV is non-separable and non-smooth and it thus is challenging to solve TV based variational models in a block decomposition. In this paper, we introduce a primaldual stitching (PDS) method to efficiently process the TV based variational models in the block decomposition framework. To characterize TV in the block decomposition framework, we only focus on the proximal map of TV function. Empirically, we have observed that the proposed PDS based block decomposition framework outperforms other state-of-art methods such as Bregman operator splitting based approach in terms of computational speed.

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“…[15,36]), graph cuts method, additive operator splitting (AOS), and multigrid method. A concise outline of these approaches can be found in [37]. The second one is the dual approach.…”

Section: Introductionmentioning

confidence: 99%

“…Overlapping DDMs were used there based on a primal-dual formulation for the anisotropic total variation problem [20]. The PSC and SSC were applied to variational image restoration and segmentation in [9,28,37]. In these applications, the original problem was successfully decomposed into smaller-size subproblems, which were solved in parallel.…”

Section: Introductionmentioning

confidence: 99%

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Domain Decomposition Methods for Total Variation Minimization

Chang

Tai

Yang

2015

Lecture Notes in Computer Science

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Abstract. In this paper, overlapping domain decomposition methods (DDMs) are used for solving the Rudin-Osher-Fatemi (ROF) model in image restoration. It is known that this problem is nonlinear and the minimization functional is non-strictly convex and non-differentiable. Therefore, it is difficult to analyze the convergence rate for this problem. In this work, we use the dual formulation of the ROF model in connection with proper subspace correction. With this approach, we overcome the problems caused by the non-strict-convexity and non-differentiability of the ROF model. However, the dual problem has a global constraint for the dual variable which is difficult to handle for subspace correction methods. We propose a stable unit decomposition, which allows us to construct the successive subspace correction method (SSC) and parallel subspace correction method (PSC) based domain decomposition. Numerical experiments are supplied to demonstrate the efficiency of our proposed methods.

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A two-level domain decomposition method for image restoration

Cited by 34 publications

References 50 publications

A convergent overlapping domain decomposition method for total variation minimization

A convergent overlapping domain decomposition method for total variation minimization

Block Decomposition Methods for Total Variation by Primal–Dual Stitching

Domain Decomposition Methods for Total Variation Minimization

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