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

Abstract

Let $u \in \textup{BV}(\Omega)$ 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 $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ of the jump set $J_u$ 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 $\textup{TV}$ and, as a novelty, Huber-regularized $\textup{TV}$. We also demonstrate that the technique would apply to nonconvex $\textup{TV}$ models as well as the Perona--Malik anisotropic diffusion, if these approaches were well-posed to begin with.

Keywords

  1. total variation
  2. jump set
  3. regularization
  4. Lipschitz
  5. Huber
  6. Perona--Malik

MSC codes

  1. 26B30
  2. 49Q20
  3. 65J20

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 2587 - 2629
ISSN (online): 1095-7154

History

Submitted: 7 July 2014
Accepted: 13 April 2015
Published online: 2 July 2015

Keywords

  1. total variation
  2. jump set
  3. regularization
  4. Lipschitz
  5. Huber
  6. Perona--Malik

MSC codes

  1. 26B30
  2. 49Q20
  3. 65J20

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