Convex Image Denoising via Non-convex Regularization with Parameter Selection

Lanza, Alessandro; Morigi, Serena; Sgallari, Fiorella

doi:10.1007/s10851-016-0655-7

Cited by 71 publications

(52 citation statements)

References 32 publications

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“…Compare with 1 norm, non-convex regularizers [27]- [29] can result in more accurate reconstruction but make the problem hard to solve. Recently, new non-convex regularizers are proposed [30]- [33]. With these regularizers, the large amplitude components can either be estimated accurately due to the non-convex penalties, and the corresponding problems can also be solved easily due to the convexity of the cost function.…”

Section: A Related Workmentioning

confidence: 99%

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Total Variation Denoising With Non-Convex Regularizers

et al. 2019

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Total variation (TV) denoising has attracted considerable attention in 1-D and 2-D signal processing. For image denoising, the convex cost function can be viewed as the regularized linear least squares problem (1 regularizer for anisotropic case and 2 regularizer for isotropic case). However, these convex regularizers often underestimate the high-amplitude components of the true image. In this paper, non-convex regularizers for 2-D TV denoising models are proposed. These regularizers are based on the Moreau envelope and minimax-concave penalty, which can maintain the convexity of the cost functions. Then, efficient algorithms based on forward-backward splitting are proposed to solve the new cost functions. The numerical results show the effectiveness of the proposed non-convex regularizers for both synthetic and real-world image. INDEX TERMS Total variation denoising, 1 norm, 2 norm, non-convex regularizer, forward-backward splitting.

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Section: A Related Workmentioning

confidence: 99%

“…Non-convex regularization for 2 norms, such as logarithmic, arctangent, first order rational and convex non-convex (CNC) penalties are proposed in [30] and [36] and used in image denoising. However, these penalties are separable and have strict convexity-preserving conditions.…”

Section: A Related Workmentioning

confidence: 99%

Total Variation Denoising With Non-Convex Regularizers

et al. 2019

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“…

{normalφ}_{exp_normalα} (|, x)

is defined as

{normalφ}_{exp_normalα} (|, x) = \frac{1 - e^{- normalα (|||, x)}}{normalα}

It was proposed by Lanza in Ref. very recently and applied to 2D image denoising. Compared with

g_{1} (|, x)

and

{normalφ}_{exp_normalα} (|, x)

{normalφ}_{M C_normalα} (|, x)

is closer to

g_{0} (|, x)

by selecting proper value of nonconvexity parameter

normalα

.…”

Section: Moreau Enhanced Function and Mctvmentioning

confidence: 99%

“…But

L_{1}

norm as a penalty tends to underestimate signal values, and it is not a very good proxy of

L_{0}

norm. Thus, a variety of non‐convex penalties were designed to outperform

L_{1}

norm regularization for sparse approximation, and some of them can maintain convexity of the cost function. For example, Selesnick defined Moreau‐enhanced TV in his paper, and applied it to onedimensional signal denoising problem.…”

Section: Introductionmentioning

confidence: 99%

Convex MR brain image reconstruction via non‐convex total variation minimization

Liu

Wang

et al. 2018

Int J Imaging Syst Tech

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Total variation (TV) regularization is a technique commonly utilized to promote sparsity of image in gradient domain. In this article, we address the problem of MR brain image reconstruction from highly undersampled Fourier measurements. We define the Moreau enhanced function of L 1 norm, and introduce the minmax-concave TV (MCTV) penalty as a regularization term for MR brain image reconstruction. MCTV strongly induces the sparsity in gradient domain, and fits the frame of fast algorithms (eg, ADMM) for solving optimization problems. Although MCTV is non-convex, the cost function in each iteration step can maintain convexity by specifying the relative nonconvexity parameter properly. Experimental results demonstrate the superior performance of the proposed method in comparison with standard TV as well as nonlocal TV minimization method, which suggests that MCTV may have promising applications in the field of neuroscience in the future. K E Y W O R D S brain image reconstruction, magnetic resonance imaging, non-convex regularization, total variation Int J Imaging Syst Technol. 2018;1-8. wileyonlinelibrary.com/journal/ima V C 2017 Wiley Periodicals, Inc. | 1

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“…A few recent papers consider the prescription of non-convex penalties that maintain the convexity of the TV denoising cost function [1], [30], [32], [48]. (The motivation for this is to leverage the benefits of both non-convex penalization and convex optimization, e.g., to accurately estimate the amplitude of jump discontinuities while guaranteeing the uniqueness of the solution.)…”

Section: Introductionmentioning

confidence: 99%

Total Variation Denoising Via the Moreau Envelope

Selesnick

2017

IEEE Signal Process. Lett.

102

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Abstract-Total variation denoising is a nonlinear filtering method well suited for the estimation of piecewise-constant signals observed in additive white Gaussian noise. The method is defined by the minimization of a particular non-differentiable convex cost function. This paper describes a generalization of this cost function that can yield more accurate estimation of piecewise constant signals. The new cost function involves a nonconvex penalty (regularizer) designed to maintain the convexity of the cost function. The new penalty is based on the Moreau envelope. The proposed total variation denoising method can be implemented using forward-backward splitting.

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Convex Image Denoising via Non-convex Regularization with Parameter Selection

Cited by 71 publications

References 32 publications

Total Variation Denoising With Non-Convex Regularizers

Total Variation Denoising With Non-Convex Regularizers

Convex MR brain image reconstruction via non‐convex total variation minimization

Total Variation Denoising Via the Moreau Envelope

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