The Jump Set under Geometric Regularization. Part 1: Basic Technique and First-Order Denoising
KAUST Grant NumberKUK-I1-007-43
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Abstract© 2015 Society for Industrial and Applied Mathematics. Let u ∈ BV(Ω) solve the total variation (TV) denoising problem with L<sup>2</sup>-squared fidelity and data f. Caselles, Chambolle, and Novaga [Multiscale Model. Simul., 6 (2008), pp. 879-894] have shown the containment H<sup>m-1</sup> (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.
CitationValkonen T (2015) The Jump Set under Geometric Regularization. Part 1: Basic Technique and First-Order Denoising. SIAM J Math Anal 47: 2587–2629. Available: http://dx.doi.org/10.1137/140976248.
SponsorsThis 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