Collaborative Total Variation: A General Framework for Vectorial TV Models

Duran, Joan; Moeller, Michael; Sbert, Catalina; Cremers, Daniel

doi:10.1137/15m102873x

Cited by 61 publications

(61 citation statements)

References 50 publications

(92 reference statements)

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“…It makes sense to look at vectorial TV as applying collaborative norms to the discrete gradient of the image as we define in [17] and reproduce below for the sake of completeness. Definition 1.…”

Section: Collaborative Total Variationmentioning

confidence: 99%

“…Interestingly, there are cases where one can imagine different versions due to an ambiguity of the permutation of the dimensions. We refer the reader to [17] for a more detailed discussion on the literature of vectorial total variation methods.…”

Section: Collaborative Total Variationmentioning

confidence: 99%

“…The case of an inner 2 coupling is also well known in the image processing literature and its proximity operator is often called (generalized) shrinkage. By a short computation or via the application of [17,Theorem 7] we find…”

Section: Proximity Operators Of Pqr Normsmentioning

confidence: 99%

“…If the order of norms of the previous case is switched such that the inner norm is 2 , we can use [17,Theorem 7] to obtain the proximity operator as follows…”

Section: Thementioning

confidence: 99%

“…We proposed in [16,17] to take a generalized viewpoint that unifies most vectorial TV based models proposed in the literature. By considering the derivatives of a color image as a linear operator, one obtains a 3D data structure of the gradient: one dimension corresponding to the pixels, one dimension corresponding to the derivatives, and one dimension corresponding to the color channels.…”

mentioning

confidence: 99%

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On the Implementation of Collaborative TV Regularization: Application to Cartoon+Texture Decomposition

Duran¹,

Moeller²,

Sbert³

et al. 2016

Image Processing on Line

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This paper deals with the analysis, implementation, and comparison of several vector-valued total variation (TV) methods that extend the Rudin-Osher-Fatemi variational model to color images. By considering the discrete gradient of a multichannel image as a 3D structure with dimensions corresponding to the spatial extent, the differences to other pixels, and the color channels, we introduce in [J. Duran, M. Moeller, C. Sbert, and D. Cremers, "Collaborative Total Variation: A General Framework for Vectorial TV Models", SIAM Journal on Imaging Sciences, 9(1), pp. 116-151, 2016] collaborative sparsity enforcing norms for penalizing the resulting tensor. We call this class of regularizations collaborative total variation (CTV). We first analyze the denoising properties of each collaborative norm for suppressing color artifacts while preserving image features and aligning edges. We then describe the primal-dual hybrid gradient method for solving the minimization problem in detail. The resulting CTV-L2 variational model can successfully be applied to many image processing tasks. On the one hand, an extensive performance comparison of several collaborative norms for color image denoising is provided. On the other hand, we analyze the ability of different CTV methods for decomposing a multichannel image into a cartoon and a textural part. Finally, we also include a short discussion on alternative minimization methods and compare their computational efficiency.Source Code ANSI C source code to produce the same results as the demo is accessible at the IPOL web part of this article 1 .

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Section: Collaborative Total Variationmentioning

confidence: 99%

Section: Collaborative Total Variationmentioning

confidence: 99%

Section: Proximity Operators Of Pqr Normsmentioning

confidence: 99%

“…If the order of norms of the previous case is switched such that the inner norm is 2 , we can use [17,Theorem 7] to obtain the proximity operator as follows…”

Section: Thementioning

confidence: 99%

mentioning

confidence: 99%

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On the Implementation of Collaborative TV Regularization: Application to Cartoon+Texture Decomposition

Duran¹,

Moeller²,

Sbert³

et al. 2016

Image Processing on Line

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Non-smooth Variational Regularization for Processing Manifold-Valued Data

Höller

Weinmann

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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Many methods for processing scalar and vector valued images, volumes and other data in the context of inverse problems are based on variational formulations. Such formulations require appropriate regularization functionals that model expected properties of the object to reconstruct. Prominent examples of regularization functionals in a vector-space context are the total variation (TV) and the Mumford-Shah functional, as well as higher-order schemes such as total generalized variation models. Driven by applications where the signals or data live in nonlinear manifolds, there has been quite some interest in developing analogous methods for nonlinear, manifold-valued data recently. In this chapter, we consider various variational regularization methods for manifold-valued data. In particular, we consider TV minimization as well as higher order models such as total generalized variation (TGV). Also, we discuss (discrete) Mumford-Shah models and related methods for piecewise constant data. We develop discrete energies for denoising and report on algorithmic approaches to minimize them. Further, we also deal with the extension of such methods to incorporate indirect measurement terms, thus addressing the inverse problem setup. Finally, we discuss wavelet sparse regularization for manifold-valued data. arXiv:1909.08921v1 [math.NA]

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