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Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u 0 into a sum u + v . Here u ∈ BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H − s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer Science+Business Media, LLC
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-008-9047-8
Publisher site
See Article on Publisher Site

Abstract

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u 0 into a sum u + v . Here u ∈ BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H − s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2008

References