Signal recovery via TV-type energies

One-dimensional variants are considered of the classical first order total variation denoising model introduced by Rudin, Osher, and Fatemi. This study is based on previous work of the authors on various denoising and inpainting problems in image analysis, where variational methods in arbitrary dimensions were applied. More than being just a special case, the one-dimensional setting makes it possible to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, quite strong regularity results are obtained 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. The 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 this investigation.