The structure of optimal parameters for image restoration problems

Reyes, Juan C.; Schönlieb, Carola-Bibiane; Valkonen, Tuomo

doi:10.1016/j.jmaa.2015.09.023

Cited by 47 publications

(68 citation statements)

References 32 publications

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“…2.2. Second, in [19], we indeed prove that even for the non-smooth problem (as μ → 0), under appropriate assumptions on the given data, the optimal α, β are guaranteed to be strictly positive.…”

Section: Definition 21mentioning

confidence: 68%

“…Remark 2.2 • Using the existence result in [19], in principle we could allow infinite values for α and β. This would include both TV 2 and TV as possible optimal regularisers in our learning problem.…”

Section: Theorem 21 We Consider the Learning Problem (23) For Tgv 2mentioning

confidence: 99%

“…In this setting and under the special assumption of box constraints on α and β, we provide a simple existence proof for an optimal solution. A more general existence result that holds also for the original non-smooth problem and does not require box constraints is derived in [19], and we refer the reader to this paper for a more sophisticated analysis on the structure of solutions. (a) Too low α, low β.…”

Section: Introductionmentioning

confidence: 99%

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Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

2016

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We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasiNewton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

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Section: Definition 21mentioning

confidence: 68%

Section: Theorem 21 We Consider the Learning Problem (23) For Tgv 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

2016

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“…This becomes particularity easy to perform in a computationally feasible manner for learned updating operators of this form. Furthermore, introducing the non-linear component in (12) allows the learned operator to approximate a large set of non-linear operators [18].…”

Section: Parametrizing the Learned Updating Operatorsmentioning

confidence: 99%

Solving ill-posed inverse problems using iterative deep neural networks

2017

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We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. The method results in a gradient-like iterative scheme, where the "gradient" component is learned using a convolutional network that includes the gradients of the data discrepancy and regularizer as input in each iteration.We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of 512 × 512 pixel images in about 0.4 s using a single graphics processing unit (GPU).

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“…Moreover, a set-valued analysis of the solvability of such programs with further symmetric cone structure is performed in [49]. In certain cases, with a finite-dimensional control u in an otherwise infinite-dimensional problem, it is also possible to do away with the regularizer G [32].…”

Section: V) − K(u)mentioning

confidence: 99%

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

Clason

Valkonen

2016

Set-Valued Var. Anal

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We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.

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The structure of optimal parameters for image restoration problems

Cited by 47 publications

References 32 publications

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Solving ill-posed inverse problems using iterative deep neural networks

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

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