Properties of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mtext>-</mml:mtext><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>TGV</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>: The one-dimensional case

We revisit total variation denoising and study an augmented model where we assume that an estimate of the image gradient is available. We show that this increases the image reconstruction quality and derive that the resulting model resembles the total generalized variation denoising method, thus providing a new motivation for this model. Further, we propose to use a constraint denoising model and develop a variational denoising model that is basically parameter free, i.e. all model parameters are estimated directly from the noisy image.Moreover, we use Chambolle-Pock's primal dual method as well as the Douglas-Rachford method for the new models. For the latter one has to solve large discretizations of partial differential equations. We propose to do this in an inexact manner using the preconditioned conjugate gradients method and derive preconditioners for this. Numerical experiments show that the resulting method has good denoising properties and also that preconditioning does increase convergence speed significantly. Finally we analyze the duality gap of different formulations of the TGV denoising problem and derive a simple stopping criterion.

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“…Another approach to form a combined optimization problem out of DGTGV is to preserve both constraints, i.e. taking (4) and (5) to obtain…”

Section: Constrained and Morozov Total Generalized Variationmentioning

confidence: 99%

“…In section 2 we formulated a two-staged denoising method in two ways. First, as a constrained version (4) and (5). In this formulation we get the primal functionals for the first problem (4)…”

Section: A21 Dgtvmentioning

confidence: 99%

Denoising of Image Gradients and Total Generalized Variation Denoising

2018

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“…For a scalar field u ∈ L 1 (Ω), TGV 2 may, according to [12,13], be written as the "differentiation cascade"…”

Section: Second-order Total Generalized Variation (Tgv 2 ) For Tensormentioning

confidence: 99%

“…Remark 3.2 (dual-ball formulation). If we extended to the tensor case the equivalence proof [12,13] of the differentiation cascade formulation (3.1) of TGV 2 (β,α) , and the original dual-ball formulation [11], we could almost trivially obtain lower semicontinuity of TGV 2 (β,α) with respect to convergence in L p . This would imply weak lower semicontinuity in L 1 and could be used to replace Lemma 3.3 in the proof of Theorem 3.1.…”

Section: Existence Of Solutionsmentioning

confidence: 99%

Total Generalized Variation in Diffusion Tensor Imaging

Valkonen¹,

Bredies²,

Knoll³

2013

SIAM J. Imaging Sci.

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124

107

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We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalized variation (TGV 2 ) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational seminorms are rotation-invariant in a sense natural and well suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness constraint, we employ these novel seminorms as regularizers in Rudin-Osher-Fatemi (ROF) type denoising of medical in vivo brain images. For the numerical realization, we employ the ChambollePock algorithm, for which we develop a novel duality-based stopping criterion which guarantees error bounds with respect to the functional values. Our findings indicate that TD and TGV 2 , both of which employ the symmetrized differential, provide improved results compared to other evaluated approaches.

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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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Properties ofL1-TGV2: The one-dimensional case

Cited by 49 publications

References 13 publications

Denoising of Image Gradients and Total Generalized Variation Denoising

Denoising of Image Gradients and Total Generalized Variation Denoising

Total Generalized Variation in Diffusion Tensor Imaging

Mathematical Analysis of a Inf-Convolution Model for Image Processing

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