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URN: urn:nbn:de:bsz:291-scidok-44404
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4440/


On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs

Steidl, Gabriele ; Weickert, Joachim ; Brox, Thomas ; Mrázek, Pavel ; Welk, Martin

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Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer: 94
Sprache: Englisch
Erstellungsjahr: 2003
Publikationsdatum: 04.01.2012
Kurzfassung auf Englisch: Soft wavelet shrinkage, total variation (TV) diffusion, total variation regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the 1-D case. First we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV difusion and TV regularization are identical, and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyse possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thesholds. Finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE / variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds.
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