A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Tai, Xue–Cheng; Hahn, Jooyoung; Chung, Ginmo

doi:10.1137/100803730

Cited by 177 publications

(217 citation statements)

References 33 publications

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“…For more detailed description of the discretisation of other high order differential operators, we refer the reader to [3,4]. More advanced discretisation on a staggered grid can be found in [21,33,34]. Finally, we address the implementation problem of the first order derivatives of u σ in (2.7).…”

Section: Discretisation Of Differential Operatorsmentioning

confidence: 99%

“…In this section we test the capability of the TWSO model for image inpainting and compare it with several state-of-the-art inpainting methods such as the TV [36], TGV [37], SOTV [2,22], and Euler's elastica [21,38] models. We denote the inpainting domain as D in the following experiments.…”

Section: Image Inpaintingmentioning

confidence: 99%

“…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%

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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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Section: Discretisation Of Differential Operatorsmentioning

confidence: 99%

Section: Image Inpaintingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

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show abstract

“…Recently, it has also been tried to solve region-based curvature problems via non-convex constrained optimization [28]. While this is different from curve evolution, it is still dependent on initialization.…”

Section: Introductionmentioning

confidence: 99%

Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming

Schoenemann

Kuang

Kahl

2011

Lecture Notes in Computer Science

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Abstract. We follow recent work by Schoenemann et al. [25] for expressing curvature regularity as a linear program. While the original formulation focused on binary segmentation, we address several multi-label problems, including segmentation, denoising and inpainting, all cast as a single linear program. Our multi-label segmentation introduces a "curvature Potts model" and combines a well-known Potts model relaxation [14] with the above work. For inpainting, we improve on [25] by grouping intensities into bins. Finally, we address the problem of denoising with absolute differences in the data term. Furthermore, we explore alternative solving strategies, including higher order Markov Random Fields, min-sum diffusion and a combination of augmented Lagrangians and an accelerated first order scheme to solve the linear programs.

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Implementation of high‐order variational models made easy for image processing

Duan

Qiu

et al. 2016

Math Methods in App Sciences

115

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High‐order variational models are powerful methods for image processing and analysis, but they can lead to complicated high‐order nonlinear partial differential equations that are difficult to discretise to solve computationally. In this paper, we present some representative high‐order variational models and provide detailed descretisation of these models and numerical implementation of the split Bregman algorithm for solving these models using the fast Fourier transform. We demonstrate the advantages and disadvantages of these high‐order models in the context of image denoising through extensive experiments. The methods and techniques can also be used for other applications, such as image decomposition, inpainting and segmentation. Copyright © 2016 John Wiley & Sons, Ltd.

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A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Cited by 177 publications

References 33 publications

Introducing diffusion tensor to high order variational model for image reconstruction

Introducing diffusion tensor to high order variational model for image reconstruction

Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming

Implementation of high‐order variational models made easy for image processing

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