The Jump Set under Geometric Regularization. Part 1: Basic Technique and First-Order Denoising

Tuomo Valkonen

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

© 2015 Society for Industrial and Applied Mathematics. Let u ∈ BV(Ω) solve the total variation (TV) denoising problem with L$^{2}$-squared fidelity and data f. Caselles, Chambolle, and Novaga [Multiscale Model. Simul., 6 (2008), pp. 879-894] have shown the containment H$^{m-1}$ (Ju \Jf) = 0 of the jump set Ju of u in that of f. Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to higher-order, curvature-based, and other advanced geometric regularizers, such as total generalized variation and Euler's elastica. These have received increased attention in recent times due to their better practical regularization properties compared to conventional TV or wavelets. We prove analogous jump set containment properties for a general class of regularizers. We do this with novel Lipschitz transformation techniques and do not require the co-area formula. In the present Part 1 we demonstrate the general technique on first-order regularizers, while in Part 2 we will extend it to higher-order regularizers. In particular, we concentrate in this part on TV and, as a novelty, Huber-regularized TV. We also demonstrate that the technique would apply to nonconvex TV models as well as the Perona-Malik anisotropic diffusion, if these approaches were well-posed to begin with.
Original languageEnglish (US)
Pages (from-to)2587-2629
Number of pages43
JournalSIAM Journal on Mathematical Analysis
Volume47
Issue number4
DOIs
StatePublished - Jan 2015
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: This research was financially supported by SFB research program F32, “Mathematical Optimization and Applications in Biomedical Sciences,” of the Austrian Science Fund (FWF), by King Abdullah University of Science and Technology (KAUST) award KUK-I1-007-43, as well as EPSRC/Isaac Newton Trust small grant, “Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors,” and EPSRC first grant EP/J009539/1, “Sparse & Higher-order Image Restoration,” and by the Prometeo initiative of
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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