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Constrained and SNR-Based Solutions for TV-Hilbert Space Image Denoising

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Abstract

We examine the general regularization model which is based on total-variation for the structural part and a Hilbert-space norm for the oscillatory part. This framework generalizes the Rudin-Osher-Fatemi and the Osher-Sole-Vese models and opens way for new denoising or decomposition methods with tunable norms, which are adapted to the nature of the noise or textures of the image. We give sufficient conditions and prove the convergence of an iterative numerical implementation, following Chambolle’s projection algorithm.

In this paper we focus on the denoising problem. In order to provide an automatic solution, a systematic method for choosing the weight between the energies is imperative. The classical method for selecting the weight parameter according to the noise variance is reformulated in a Hilbert space sense. Moreover, we generalize a recent study of Gilboa-Sochen-Zeevi where the weight parameter is selected such that the denoised result is close to optimal, in the SNR sense. A broader definition of SNR, which is frequency weighted, is formulated in the context of inner products. A necessary condition for maximal SNR is provided. Lower and upper bounds on the SNR performance of the classical and optimal strategies are established, under quite general assumptions.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, California, 90095, USA

    Jean-François Aujol & Guy Gilboa

  2. CMLA, UMR CNRS 8536, ENS Cachan, France

    Jean-François Aujol

Authors
  1. Jean-François Aujol
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  2. Guy Gilboa
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Additional information

Jean-François Aujol graduated from “1’ Ecole Normale Supérieure de Cachan” in 2001. He was a PH’D student in Mathematics at the University of Nice-Sophia-Antipolis (France). He was a member of the J.A. Dieudonné Laboratory at Nice, and also a member of the Ariana research group (CNRS/INRIA/UNSA) at Sophia-Antipolis (France). His research interests are calculus of variations, nonlinear partial differential equations, numerical analysis and mathematical image processing (and in particular classification, texture, decomposition model, restoration). He is Assistant Researcher at UCLA (Math Department).

Guy Gilboa is currently an Assistant Researcher (postdoctoral position) at the Department of Mathematics, UCLA, hosted by Prof. Stanley Osher. He received his Ph.D. in Electrical Engineering from the Technion—Israel Institute of Technology, in 2004. He previously worked for three years at Intel Development Center, Haifa, Israel, in the design of processors.

His main research interests are related to variational and PDE-based processes applied to image enhancement, denoising and decomposition.

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Aujol, JF., Gilboa, G. Constrained and SNR-Based Solutions for TV-Hilbert Space Image Denoising. J Math Imaging Vis 26, 217–237 (2006). https://doi.org/10.1007/s10851-006-7801-6

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