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International Journal of Computer Vision

, Volume 63, Issue 1, pp 85–104

# Dual Norms and Image Decomposition Models

• Jean-François Aujol
• Antonin Chambolle
Article

## Abstract

Following a recent work by Y. Meyer, decomposition models into a geometrical component and a textured component have recently been proposed in image processing. In such approaches, negative Sobolev norms have seemed to be useful to modelize oscillating patterns. In this paper, we compare the properties of various norms that are dual of Sobolev or Besov norms. We then propose a decomposition model which splits an image into three components: a first one containing the structure of the image, a second one the texture of the image, and a third one the noise. Our decomposition model relies on the use of three different semi-norms: the total variation for the geometrical component, a negative Sobolev norm for the texture, and a negative Besov norm for the noise. We illustrate our study with numerical examples.

## Keywords

total variation minimization BV texture noise negative Sobolev spaces negative Besov spaces image decomposition

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## Copyright information

© Springer Science+Business Media, Inc. 2005

## Authors and Affiliations

1. 1.Laboratoire J.A. Dieudonné, UMR CNRS 6621 and ARIANAProjet Commun CNRS/INRIA/UNSAFrance
2. 2.CMAP (UMR 7641)Ecole PolytechniqueFrance