scholarly journals Regularization of linear inverse problems with total generalized variation

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Kristian Bredies ◽  
Martin Holler
Keyword(s):  
Inverse Problems ◽  
Minimization Problem ◽  
Regularization Parameter ◽  
Symmetric Tensor ◽  
Functions Of Bounded Variation ◽  
Well Posedness ◽  
Tensor Fields ◽  
Linear Inverse Problems ◽  
Total Generalized Variation ◽  
Generalized Variation

AbstractThe regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.

scholarly journals Variational regularization of inverse problems for manifold-valued data

2020 ◽  
Author(s):  
Martin Storath ◽  
Andreas Weinmann
Keyword(s):  
Inverse Problems ◽  
Total Variation ◽  
Real Data ◽  
Indirect Measurement ◽  
Numerical Realization ◽  
Experimental Results ◽  
Well Posedness ◽  
Total Generalized Variation ◽  
Generalized Variation ◽  
Variational Regularization

Abstract In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider total variation and total generalized variation regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results for synthetic and real data to show the potential of the proposed schemes for applications.

scholarly journals Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives

2021 ◽  
Vol 38 ◽  
pp. 36-53
Author(s):  
M. M. Turov ◽  
◽  
V. E. Fedorov ◽  
B. T. Kien ◽  
◽  
...  
Keyword(s):  
Inverse Problems ◽  
Fractional Derivative ◽  
Fractional Part ◽  
Well Posedness ◽  
Bounded Operators ◽  
Degenerate Problem ◽  
Linear Inverse Problems ◽  
Coefficient Problems ◽  
Degenerate Operator ◽  
Inverse Coefficient

The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

scholarly journals Total variation multiscale estimators for linear inverse problems

2020 ◽  
Vol 9 (4) ◽  
pp. 961-986 ◽  
Author(s):  
Miguel del Álamo ◽  
Axel Munk
Keyword(s):  
Inverse Problems ◽  
Convergence Result ◽  
Functions Of Bounded Variation ◽  
Linear Inverse Problems ◽  
Bv Functions ◽  
Regularization Techniques ◽  
Minimax Rate ◽  
Dimensional Domain ◽  
Problem Setting ◽  
Variational Regularization

Abstract Even though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on a $d$-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of $BV$ functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating $BV$ functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the $L^q$-risk, for any $q\in [1,\infty )$. This is to the best of our knowledge the first convergence result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101–126) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$ in which the minimax rate is slower than $n^{-1/(d+2\beta +2)}$, where $\beta$ is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of $BV$ functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

scholarly journals An Adaptive Total Generalized Variation Model with Augmented Lagrangian Method for Image Denoising

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Chuan He ◽  
Changhua Hu ◽  
Xiaogang Yang ◽  
Huafeng He ◽  
Qi Zhang
Keyword(s):  
Image Denoising ◽  
Augmented Lagrangian ◽  
Regularization Parameter ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Qualitative Assessment ◽  
Edge Preservation ◽  
Total Generalized Variation ◽  
Proposed Model ◽  
Generalized Variation

We propose an adaptive total generalized variation (TGV) based model, aiming at achieving a balance between edge preservation and region smoothness for image denoising. The variable splitting (VS) and the classical augmented Lagrangian method (ALM) are used to solve the proposed model. With the proposed adaptive model and ALM, the regularization parameter, which balances the data fidelity and the regularizer, is refreshed with a closed form in each iterate, and the image denoising can be accomplished without manual interference. Numerical results indicate that our method is effective in staircasing effect suppression and holds superiority over some other state-of-the-art methods both in quantitative and in qualitative assessment.

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