Signal recovery via TV-type energies

Abstract

Dedicated to the memory of Stefan Hildebrandt AMS Subject Classification: 26A45, 49J05, 49J45, 49M29, 34B15 Keywords: total variation, signal denoising, variational problems in one independent variable, linear growth, existence and regularity of solutions. AbstractWe consider one-dimensional variants of the classical first order total variation denoising model introduced by Rudin, Osher and Fatemi. This study is based on our previous work on various denoising and inpainting problems in image analysis, where we applied variational methods in arbitrary dimensions. More than being just a special case, the one-dimensional setting allows us to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, we obtain quite strong regularity results for a class of data functions that contains many of the standard examples from signal processing such as rectangle-or triangle signals as a special case. An analysis of the related Euler-Lagrange equation, which turns out to be a second order two-point boundary value problem with Neumann conditions, by ODE methods completes the picture of our investigation.Acknowledgement: The authors thank Michael Bildhauer for many stimulating discussions.

A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type

“…First, as it will become evident from the proof of Theorem 3, it plays the role of a technical restriction making it possible to manage certain quantities. Second, the results obtained in [1] suggest that there is some hope for the existence of regular weak solutions in case µ < 2 motivating our assumption u ∈ C 1 , whereas counterexamples taken from a slightly different setting (see [5]) indicate that for µ > 2 equations of the form (1) may fail to have solutions even on bounded domains, which can be found in some Sobolev space. Due to the linear growth of H stated in (17) and with respect to the ellipticity condition (16), the space of functions with bounded deformation (see, e.g.…”

“…holds with a given density h : [0, ∞) → [0, ∞) of class C 2 . Combining (3) and (4) we see that (5) T D (ε) = h ′ (|ε|) |ε| ε holds, and equation (5) includes as particular cases (i) power law models:…”

Convex regularization of multi-channel images based on variants of the TV-model

“…regularity results the bound on µ stated in (2.19) is optimal, since even in the case n = 1 = N there are counterexamples of singular solutions, if the case µ > 2 is considered. We refer the reader to [21].…”

On the local boundedness of generalized minimizers of variational problems with linear growth

“…in the monographs [1,2,3,4,5], where the reader interested in the subject will find a lot of further references as well as the definitions of the underlying spaces such as BV(Ω, R N ) and W 1,p (Ω, R N ) (and their local variants) consisting of all functions having finite total variation and the mappings with first order distributional derivatives located in the Lebesgue class L p (Ω, R N ), respectively. We will mainly concentrate on the case n ≥ 2 assuming that Ω is a bounded Lipschitz region, anyhow, the case n = 1 can be included but is accessible by much easier means as it is outlined for example in [6] and [7]. To begin with, we consider the minimization problem with boundary datum…”