Total variation denoising in $l^1$ anisotropy

Abstract

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.

34 pages, 9 figures

Cited by in corpus (7)

• Solution Paths of Variational Regularization Methods for Inverse Problems
• Numerical computations of split Bregman method for fourth order total variation flow
• Invariant $\phi$-minimal sets and total variation denoising on graphs
• Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy
• Weak solutions for the Stokes system for compressible non-Newtonian fluids with unbounded divergence
• Anisotropic tempered diffusion equations
• Sparse Signal Reconstruction for Overdispersed Low-photon Count Biomedical Imaging Using ${\ell }_p$ Total Variation