An Infeasible Primal-Dual Algorithm for Total Bounded Variation--Based Inf-Convolution-Type Image Restoration

Hintermüller, Michael; Stadler, Georg

doi:10.1137/040613263

Cited by 147 publications

(212 citation statements)

References 31 publications

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“…We briefly mention several motivating applications where (1) emerges from Fenchel dualization (Ekeland and Temam, 1976) and semismooth Newton solvers (Hintermüller et al, 2003). In a rather abstract setting, regularized total variation type image restoration (Hintermüller and Stadler, 2006), energies related to Bingham fluids (Hintermüller and de los Reyes, 2011), simplified friction problems or elastoplastic problems in material science (Duvaut and Lions, 1976;Johnson, 1976;Carstensen, 1997;Stadler, 2004;Hintermüller and Rösel, 2013) can be associated with the following problem:…”

Section: One Readily Observes Thatmentioning

confidence: 99%

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

Hintermüller

Rautenberg

2015

Journal of Mathematical Analysis and Applications

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For a Banach space X of R M -valued functions on a Lipschitz domain, let K(X) ⊂ X be a closed convex set arising from pointwise constraints on the value of the function, its gradient or its divergence, respectively. The main result of the paper establishes, under certain conditions, the density of K(X 0 ) in K(X 1 ) where X 0 is densely and continuously embedded in X 1 . The proof is constructive, utilizes the theory of mollifiers and can be applied to Sobolev spaces such as H 0 (div, Ω) and W 1,p 0 (Ω), in particular. It is also shown that such a density result cannot be expected in general.

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Section: One Readily Observes Thatmentioning

confidence: 99%

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

Hintermüller

Rautenberg

2015

Journal of Mathematical Analysis and Applications

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“…methods, for instance, the dual algorithm [10], the split-Bregman algorithm [9,23,43,50], the primal-dual algorithm [11,18], the infeasible primal-dual algorithm of semi-smooth Newtontype [25], the ADMM algorithm [14,52], as well as the max-flow algorithm [22]. Here, we apply the dual algorithm proposed in [10].…”

Section: The Admm Algorithm For Solving (5)mentioning

confidence: 99%

Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees

et al. 2017

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Image restoration is one of the essential tasks in image processing. In order to restore images from blurs and noise while also preserving their edges, one often applies total variation (TV) minimization. Cauchy noise, which frequently appears in engineering applications, is a kind of impulsive and non-Gaussian noise. Removing Cauchy noise can be achieved by solving a nonconvex TV minimization problem, which is difficult due to its nonconvexity and nonsmoothness. In this paper, we adapt recent results in the literature and develop a specific alternating direction method of multiplier to solve this problem. Theoretically, we establish the convergence of our method to a stationary point. Experimental results demonstrate that the proposed method is competitive with other methods in visual and quantitative measures. In particular, our method achieves higher PSNRs for 0.5 dB on average.

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“…Hintermüller and Stadler [14] (HS) discuss an infeasible-interior-point method for a modification of (2) in which an term µ Ω |∇u| 2 dx is added, for some small but positive µ. By perturbing the dual of their problem with a regularization term, then applying a semismooth Newton method to the primal-dual formulation of the resulting problem, they obtain superlinear convergence to a solution of the modified problem.…”

Section: Previous Algorithmsmentioning

confidence: 99%

Duality-based algorithms for total-variation-regularized image restoration

Zhu¹,

Wright

Chan³

2008

Comput Optim Appl

158

122

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Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies, including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required.

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An Infeasible Primal-Dual Algorithm for Total Bounded Variation--Based Inf-Convolution-Type Image Restoration

Cited by 147 publications

References 31 publications

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees

Duality-based algorithms for total-variation-regularized image restoration

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