Regularization of linear inverse problems with
total generalized variation

Bredies, Kristian; Höller, Martin

doi:10.1515/jip-2013-0068

Cited by 96 publications

(134 citation statements)

References 27 publications

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“…The remaining assertions follow immediately from the equivalence of ICTV n β to TGV k α with k = md(β), α ∈ R k + , and the corresponding assertions on TGV k α as in [10]. Below, for any orthogonal matrix O ∈ R d×d and ξ ∈ Sym k (R d ), k ∈ N, we define the right multiplication ξO ∈ Sym k (R d ) by…”

Section: Properties Of Ictvmentioning

confidence: 99%

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On Infimal Convolution of TV-Type Functionals and Applications to Video and Image Reconstruction

Höller¹,

Kunisch

2014

SIAM J. Imaging Sci.

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The infimal convolution of total (generalized) variation-type functionals and its application as regularization for video and image reconstruction is considered. The definition of this particular type of regularization functional is motivated by the need of suitably combining spatial and temporal regularity requirements for video processing. The proposed functional is defined in an infinite dimensional setting, and important analytical properties are established. As applications, the reconstruction of compressed video data and of noisy still images is considered. The resulting problem settings are posed in function space, and suitable numerical solution schemes are established. Experiments confirm a significant improvement compared to standard total variation-type methods, which originates from the introduction of spatio-temporal and spatial anisotropies. Introduction.Motivated by the aim of defining a suitable spatio-temporal regularization for image sequences we consider the infimal convolution of an arbitrary number of modified total (generalized) variation functionals. We discuss analytical properties and propose a numerical realization for two concrete applications.The combination of different regularization functionals by infimal convolution was mentioned early in [15] and is discussed in [31] in a more general, discrete setting, both motivated by combining first and higher order derivatives in order to reduce staircasing effects for still images. Given two functionals J 1 and J 2 , their infimal convolution is defined by

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Section: Properties Of Ictvmentioning

confidence: 99%

“…In [10] it has been shown that TGV k β , with | · | β i = α −1 i | · |, can equivalently be written as…”

Section: Introductionmentioning

confidence: 99%

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On Infimal Convolution of TV-Type Functionals and Applications to Video and Image Reconstruction

Höller¹,

Kunisch

2014

SIAM J. Imaging Sci.

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“…The use of such a model allows to get rid of the staircasing effect that appears with the ROF model in denoising processes. To achieve this goal, Bredies et al [15][16][17] have recently introduced a second-order generalized total variation definition that is a nice compromise/mixture between the firstand second-order derivatives. It is, in some sense, an extension of the inf-convolution (we recall the definition later) of the first-and second-order derivatives.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of a Inf-Convolution Model for Image Processing

Bergounioux

2015

J Optim Theory Appl

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We deal with a second-order image decomposition model to perform denoising and texture extraction that was previously presented. We look for the decomposition of an image as the summation of three different order terms. For highly textured images, the model gives a two-scale texture decomposition: The first-order term can be viewed as a macro-texture (larger scale) which oscillations are not too large, and the zero-order term is the micro-texture (very oscillating) that contains the noise. Here, we perform mathematical analysis of the model and give qualitative properties of solutions using the dual problem and inf-convolution formulation.

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“…Variational methods were rapidly applied along with data fidelity terms Ψ. The use of differential operators D k of various orders k 2 in the prior Φ has been recently investigated, see, e.g., [22,23]. More details on variational methods for image processing can be found in several textbooks like [3,5,91].…”

Section: F(u V) = − Ln π(V|u) − Ln π(U)mentioning

confidence: 99%

Energy Minimization Methods

Nikolova

SpringerReference

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Regularization of linear inverse problems with total generalized variation

Cited by 96 publications

References 27 publications

On Infimal Convolution of TV-Type Functionals and Applications to Video and Image Reconstruction

On Infimal Convolution of TV-Type Functionals and Applications to Video and Image Reconstruction

Mathematical Analysis of a Inf-Convolution Model for Image Processing

Energy Minimization Methods

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