A New Augmented Lagrangian Approach for $L^1$-mean Curvature Image Denoising

Myllykoski, Mirko; Glowinski, Roland; Kärkkäinen, Tommi; Rossi, Tuomo

doi:10.1137/140962164

Cited by 40 publications

(29 citation statements)

References 106 publications

(129 reference statements)

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“…In this work we decided to use fixed point and Nesterov algorithms. Other techniques such as augmented Lagrangian methods [7,14], convexity splitting [2], non linear multigrid [3], homotopy methods [28] and so on may result in a more efficient solution and will be part of our future work.…”

Section: Numerical Solutionmentioning

confidence: 99%

A Mean Curvature Regularized Based Model for Demodulating Phase Maps from Fringe Patterns

Brito‐Loeza¹,

Legarda-Sáenz²,

Espinosa-Romero³

et al. 2018

CICP

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Section: Numerical Solutionmentioning

confidence: 99%

A Mean Curvature Regularized Based Model for Demodulating Phase Maps from Fringe Patterns

Brito‐Loeza¹,

Legarda-Sáenz²,

Espinosa-Romero³

et al. 2018

CICP

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“…In this paper, we use a unified characteristic function expression for all regions including redundant regions, but we add some simple area constraints of redundant regions to avoid estimations of redundant parameters, thus reducing costs. Although we transform the original model into a constrained optimization problem, we use ADMM [18][19][20] to solve it easily and systematically without additional constructions of characteristic functions for empty regions.…”

Section: Introductionmentioning

confidence: 99%

The Vese-Chan model without redundant parameter estimation for multiphase image segmentation

Wang

Pan

et al. 2020

J Image Video Proc.

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The Vese-Chan model for multiphase image segmentation uses m binary label functions to construct 2 m characteristic functions for different phases/regions systematically; the terms in this model have moderate degrees comparing with other schemes of multiphase segmentation. However, if the number of desired regions is less than 2 m , there exist some empty phases which need costly parameter estimation for segmentation purpose. In this paper, we propose an automatic construction method for characteristic functions via transformation between a natural number and its binary expression, and thus, the characteristic functions of empty phases can be written and recognized naturally. In order to avoid the redundant parameter estimations of these regions, we add area constraints in the original model to replace the corresponding region terms to preserve its systematic form and achieve high efficiency. Additionally, we design the alternating direction method of multipliers (ADMM) for the proposed modified model to decompose it into some simple sub-problems of optimization, which can be solved using Gauss-Seidel iterative method or generalized soft thresholding formulas. Some numerical examples for gray images and color images are presented finally to demonstrate that the proposed model has the same or better segmentation effects as the original one, and it reduces the estimation of redundant parameters and improves the segmentation efficiency.

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“…Variational methods allow easy integration of constraints and use of powerful modern optimisation techniques such as primal-dual [14][15][16], fast iterative shrinkagethresholding algorithm [17,18], and alternating direction method of multipliers [2][3][4][19][20][21][22][23][24]. Recent advances on how to automatically select parameters for different optimisation algorithms [16,18,25] dramatically boost performance of variational methods, leading to increased research interest in this field.…”

Section: Introductionmentioning

confidence: 99%

“…Among these is the second order total variation (SOTV) model [1,2,22,26,27]. Unlike the high order variational models, such as the Gaussian curvature [28], mean curvature [23,29], Euler's elastica [21] etc., the SOTV is a convex high order extension of the FOTV, which guarantees a global solution. The SOTV is also more efficient to implement [4] than the convex total generalised variation (TGV) [30,31].…”

Section: Introductionmentioning

confidence: 99%

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

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The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk JID:YDSPR AID:2153 /FLA [m5G; v1.219; Prn:17/07/2017; 11:02] Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1][2][3][4][5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.

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A New Augmented Lagrangian Approach for $L^1$-mean Curvature Image Denoising

Cited by 40 publications

References 106 publications

A Mean Curvature Regularized Based Model for Demodulating Phase Maps from Fringe Patterns

A Mean Curvature Regularized Based Model for Demodulating Phase Maps from Fringe Patterns

The Vese-Chan model without redundant parameter estimation for multiphase image segmentation

Introducing diffusion tensor to high order variational model for image reconstruction

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