Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction

Hintermüller, Michael; Rautenberg, Carlos N.; Hahn, Jooyoung

doi:10.1088/0266-5611/30/5/055014

Cited by 25 publications

(32 citation statements)

References 42 publications

(143 reference statements)

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“…Our definition of the discrete TV, using interpolation in the dual domain, is not new: it was proposed in [30] and called staggered grid discretization of the TV. With the isotropic TV, the projection of the image pair u onto the l ∞,2 norm ball, which amounts to simple pixelwise shrinkage, can be used.…”

Section: Classical Definitions Of the Discrete Tv And Their Propertiementioning

confidence: 99%

Discrete Total Variation: New Definition and Minimization

Condat¹

2017

SIAM J. Imaging Sci.

118

139

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International audienceWe propose a new definition for the gradient field of a discrete image, defined on a twice finer grid. The differentiation process from the image to its gradient field is viewed as the inverse operation of linear integration, and the proposed mapping is nonlinear. Then, we define the total variation of an image as the l1 norm of its gradient field amplitude. This new definition of the total variation yields sharp edges and has better isotropy than the classical definition

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Section: Classical Definitions Of the Discrete Tv And Their Propertiementioning

confidence: 99%

Discrete Total Variation: New Definition and Minimization

Condat¹

2017

SIAM J. Imaging Sci.

118

139

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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%

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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“…As in the split-split method, we use mass lumping for the variable s h to update it node-wise. A similar approach has been used by Hintermüller et al in [17] for the predual of the classical ROF problem and an alternate minimization technique has been employed for the numerical solution.…”

Section: Iterative Solutionmentioning

confidence: 99%

Iterative finite element solution of a constrained total variation regularized model problem

Bartels¹,

Milicevic²

2017

Discrete &Amp; Continuous Dynamical Systems - S

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The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the nondifferentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.

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Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction

Cited by 25 publications

References 42 publications

Discrete Total Variation: New Definition and Minimization

Discrete Total Variation: New Definition and Minimization

Introducing diffusion tensor to high order variational model for image reconstruction

Iterative finite element solution of a constrained total variation regularized model problem

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