A Combined First and Second Order Variational Approach for Image Reconstruction

Papafitsoros, Kostas; Schönlieb, Carola-Bibiane

doi:10.1007/s10851-013-0445-4

Cited by 255 publications

(256 citation statements)

References 57 publications

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“…By using such norm, (2.1) has several capabilities: 1) allowing discontinuity of gradients of u; 2) imposing smoothness on u; 3) satisfying the rotation-invariant property. Thought this high order model (2.1) is able to reduce the staircase artefact associated with the FOTV for image denoising, it can blur object edges in the image [1][2][3][4]. For inpainting, as investigated in [2,22], though the SOTV has the ability to connect large gaps in the image, such ability depends on the geometry of the inpainting region, and it can blur the inpainted image.…”

Section: The Twso Modelmentioning

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%

“…High order variational models thus can be applied to remedy these side effects. 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.…”

Section: Introductionmentioning

confidence: 99%

“…The SOTV is also more efficient to implement [4] than the convex total generalised variation (TGV) [30,31]. However, the inpainting results of the model highly depend on the geometry of the inpainting region, and it also tends to blur the inpainted area [2,22]. Further, according to the numerical experimental results displayed in [1][2][3][4][5] the SOTV model tends to blur object edges for image denoising.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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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: The Twso Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

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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A magnetic resonance image reconstruction method using support of first-second order variation

Gao

Jia

et al. 2015

Int. J. Imaging Syst. Technol.

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This article addresses the problem of reconstructing a magnetic resonance image from highly undersampled data, which frequently arises in accelerated magnetic resonance imaging. We propose to impose sparsity of first and second order difference sparse coefficients within the complement of the known support. Second order variation is involved to overcome blocky effects and support information is used to reduce the sampling rate further. The resulting optimization problem consists of a data fidelity term and first-second order variation terms penalizing entries within the complement of the known support. The efficient split Bregman algorithm is used to solve the problem. Reconstruction results from magnetic resonance imaging data corresponding to different sampling rates are shown to illustrate the performance of the proposed method. Then, we also assess the tolerance of the new method to noise briefly. (2014) proposed a combined first and second order variation approach successively. The reconstruction quality of the latter method is not far off from that of TGV, and computational burden caused by numerical solution shows that TGV is, in general, about 10 times slower than the latter one.Methods mentioned above only exploit the sparsity which is implicit in MR images. Beyond utilizing sparsity, researchers pro-

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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 Combined First and Second Order Variational Approach for Image Reconstruction

Cited by 255 publications

References 57 publications

Introducing diffusion tensor to high order variational model for image reconstruction

Introducing diffusion tensor to high order variational model for image reconstruction

A magnetic resonance image reconstruction method using support of first-second order variation

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

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