Exact solutions for the denoising problem of piecewise constant images in dimension one

Cristoferi, Riccardo

doi:10.48550/arxiv.1612.05508

Cited by 2 publications

(2 citation statements)

References 35 publications

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“…Let I ⊂ R be an interval, X = BV(I), H = L 2 (I), and J = TV. If f is piecewise constant, then according to [42] the solution v τ is piecewise affine with v τ = f − τ p for τ ∈ [0, τ ] and p ∈ ∂J( f ). In [43] the authors prove similar results in two dimensions, using anisotropic total variation as regularization and assuming the data to be piecewise constant on rectangles.…”

Section: Affine Solution Paths Of the Quadratic Problemmentioning

confidence: 99%

Solution paths of variational regularization methods for inverse problems

Bungert

Burger

2019

Inverse Problems

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We consider a family of variational regularization functionals for a generic inverse problem, where the data fidelity and regularization term are given by powers of a Hilbert norm and an absolutely one-homogeneous functional, respectively. We investigate the small and large time behavior of the associated solution paths and, in particular, prove finite extinction time for a large class of functionals. Depending on the powers, we also show that the solution paths are of bounded variation or even Lipschitz continuous. In addition, it will turn out that the models are "almost" mutually equivalent in terms of the minimizers they admit. Finally, we apply our results to define and compare two different nonlinear spectral representations of data and show that only one of it is able to decompose a linear combination of nonlinear eigenvectors into the individual eigenvectors. Finally, we also briefly address piecewise affine solution paths.

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Section: Affine Solution Paths Of the Quadratic Problemmentioning

confidence: 99%

Solution paths of variational regularization methods for inverse problems

Bungert

Burger

2019

Inverse Problems

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show abstract

“…As a result, the differences between the effects that these terms have on the solutions of the corresponding denoising models are well understood. See for instance [1,2,3,16,17,18,20,28,36,43,51,53] for TV regularisation and [10,48,49,50,54] for TGV. For example, in the case of TV regularisation, it is known that the use of L 2 fidelity does not introduce new discontinuities in the solution, which is not the case for the L 1 fidelity.…”

Section: Introductionmentioning

confidence: 99%

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

Calatroni¹,

Papafitsoros²

2019

Inverse Problems

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We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines L 1 and L 2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with L 1 and L 2 data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.

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Exact solutions for the denoising problem of piecewise constant images in dimension one

Abstract: In this paper we propose a method to determine explicitly the solution of the total variation denoising problem with an L p fidelity term, where p > 1, for piecewise constant initial data in dimension one.

Cited by 2 publications

References 35 publications

Solution paths of variational regularization methods for inverse problems

Solution paths of variational regularization methods for inverse problems

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

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