Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

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

doi:10.1007/s10851-016-0662-8

Cited by 103 publications

(125 citation statements)

References 45 publications

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“…SSIM computed using on of them as the reference) is 0.98 for the "cameraman" image and 0.97 for the "owl" image. Although the TGV reconstructions depend on the parameter β any may differ more from TV pwL for other values of β, the one we chose here (β = 1.25) is reasonable and lies within the optimal range reported in [12].…”

Section: D Experimentsmentioning

confidence: 57%

“…We solve all problems in MATLAB using CVX [15]. For TGV we use the parameter β = 1.25, which is in the range [1, 1.5] recommended in [12]. A characteristic feature of the proposed regularizer TV pwL is its ability to efficiently encode the information about the gradient of the ground truth (away from jumps) if such information is available.…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

Burger

Korolev

Schönlieb

et al. 2019

Lecture Notes in Computer Science

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We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers TVL p and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.

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Section: D Experimentsmentioning

confidence: 57%

Section: Numerical Experimentsmentioning

confidence: 99%

A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

Burger

Korolev

Schönlieb

et al. 2019

Lecture Notes in Computer Science

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show abstract

“…As the regulariser we use R = TGV 2 (0.9α,α) , where the choice β = 0.9α was made somewhat arbitrarily, however yielding good results for all the models. This is slightly lower than the range [1, 1.5]α discovered in comprehensive experiments for other imaging modalities [7,38].…”

Section: Verification Of the Approach With Synthetic Datamentioning

confidence: 62%

Diffusion Tensor Imaging with Deterministic Error Bounds

2016

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Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to diffusion tensor imaging, where correct noise modelling is challenging: it involves the Rician distribution and the non-linear Stejskal-Tanner equation. Linearisation of the latter in the statistical framework would complicate the noise model even further. We avoid this using the error bounds approach, which preserves simple error structure under monotone transformations.

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“…So far these parameters are tuned manually. In the future, the parameters could be selected in an automatic fashion by treating the parameters as unknown variables in the proposed model Equation (13) and then solving the corresponding optimization problem using the bilevel approach (Kunisch and Pock, 2013; Calatroni et al, 2015; Reyes et al, 2016). …”

Section: Discussionmentioning

confidence: 99%

s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

et al. 2016

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EEG source imaging enables us to reconstruct current density in the brain from the electrical measurements with excellent temporal resolution (~ ms). The corresponding EEG inverse problem is an ill-posed one that has infinitely many solutions. This is due to the fact that the number of EEG sensors is usually much smaller than that of the potential dipole locations, as well as noise contamination in the recorded signals. To obtain a unique solution, regularizations can be incorporated to impose additional constraints on the solution. An appropriate choice of regularization is critically important for the reconstruction accuracy of a brain image. In this paper, we propose a novel Sparsity and SMOOthness enhanced brain TomograpHy (s-SMOOTH) method to improve the reconstruction accuracy by integrating two recently proposed regularization techniques: Total Generalized Variation (TGV) regularization and ℓ1−2 regularization. TGV is able to preserve the source edge and recover the spatial distribution of the source intensity with high accuracy. Compared to the relevant total variation (TV) regularization, TGV enhances the smoothness of the image and reduces staircasing artifacts. The traditional TGV defined on a 2D image has been widely used in the image processing field. In order to handle 3D EEG source images, we propose a voxel-based Total Generalized Variation (vTGV) regularization that extends the definition of second-order TGV from 2D planar images to 3D irregular surfaces such as cortex surface. In addition, the ℓ1−2 regularization is utilized to promote sparsity on the current density itself. We demonstrate that ℓ1−2 regularization is able to enhance sparsity and accelerate computations than ℓ1 regularization. The proposed model is solved by an efficient and robust algorithm based on the difference of convex functions algorithm (DCA) and the alternating direction method of multipliers (ADMM). Numerical experiments using synthetic data demonstrate the advantages of the proposed method over other state-of-the-art methods in terms of total reconstruction accuracy, localization accuracy and focalization degree. The application to the source localization of event-related potential data further demonstrates the performance of the proposed method in real-world scenarios.

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Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Cited by 103 publications

References 45 publications

A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

A Total Variation Based Regularizer Promoting Piecewise-Lipschitz Reconstructions

Diffusion Tensor Imaging with Deterministic Error Bounds

s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

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