Iteratively solving linear inverse problems under general convex constraints

By Ingrid Daubechies, Gerd Teschke, Luminita A. Vese. 2007

147 citations recorded by the source corpus.

We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algo- rithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively mod- erate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex con- straints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov- or BV-restoration for which we present also numerical experiments in the context of image processing.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a3d8970656b4dbe2ae5d3e8d1dd49a2d01bb7f9c

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In Computational Mechanics

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Citations in the corpus

Split Bregman Methods and Frame Based Image Restoration (2009, methodology): “While algorithms in [14, 31, 40, 43, 45] are the model of the synthesis based approach, algorithms in [7, 8, 10, 15, 26] balance the synthesis based and the analysis based approaches.”
Nonparametric sparsity and regularization (2012, methodology): “These methods have proved to be fast and accurate [10], both for `1-based regularization – see [18], [21], [30], [40] – and more general regularized learning methods – see for example [27], [43], [35] –.”
Dual Wavelet Frames and Riesz Bases in Sobolev Spaces (2009, unknown): “[5, 6, 7, 8, 9, 10, 22]).”
Inverse source problem for the hyperbolic equation with a time-dependent principal part (2015, unknown): “As an extension of classical gradient algorithms with regularization, the iterative thresholding algorithm and its updated versions have proved their feasibility mainly in the abundant applications to image processing due to their simplicity (see [2,4,8,10]).”
A proximal decomposition method for solving convex variational inverse problems (2008, background): “1) in which m = 2 and f2 is Lipschitz-differentiable on H is shown to cover a variety of seemingly unrelated inverse problem formulations such as Fourier regularization problems, constrained least-squares problems, split feasibility problems, multiresolution sparse regularization problems, geometry/”
A Singular Value Thresholding Algorithm for Matrix Completion (2008, background): “…have quite a bit of history in signal and image processing and we would like to mention the works [12, 39] for total-variation minimization, [24, 25, 31] for ℓ 1 minimization, and [5, 7, 8, 19, 20, 27, 28, 46] for some recent applications in the area of image inpainting and image restoration.”
Image restoration: Total variation, wavelet frames, and beyond (2012, methodology): “This is called the synthesis based approach, as the image is synthesized by the sparsest coefficient; see [46, 47, 48, 49, 50].”