Total Variation Denoising in $l^1$ Anisotropy

Abstract

We aim at constructing solutions to the minimizing problem for the variant of RudinOsher-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 (P CR). 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 P CR function, then solutions to both considered problems also have this property. For P CR 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.MSC: 68U10, 35K67, 35C05, 49N60, 35B65

Numerical computations of split Bregman method for fourth order total variation flow

“…Figure 9b presents the evolution of fourth order anisotropic total variation flow for u 0 (x, y) = x(x − 1)y(y − 1) − 1/36, N x = N y = 40, λ = 5h −4 and µ = 20h −2 . For second order anisotropic total variation flow, Lasica, Moll and Mucha [28] have considered rectangular domain Ω ⊂ R 2 or Ω = R 2 and rigorously proved that if the initial profile is piecewise constant, then the exact solution is piecewise constant. We can infer from our numerical experiment 9b that their theoretical result is true also for fourth order anisotropic total variation flow.…”

Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow

“…In [31], we proved the following comparison principle for the quantity ƒ. Note that Proposition 4.1 implies a comparison for ƒ of facets: if .F 1 ; 1 /, .F 2 ; 2 / are two W ı -.L 2 / Cahn-Hoffman facets that are ordered, For n D 2 and W .p/ D jp 1 j C jp 2 j, a rather thorough characterization of ƒOE.A; / for axes-aligned polygons A is available in [35]. In particular, if ƒOE.A; / is a constant, then…”

Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy

“…In this section we introduce several notions related to functions which are piecewise constant on rectilinear subsets of R n . Some of these are n-dimensional analogues of notions from [11,Sec. 2.3].…”