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Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage

Chambolle, Antonin; DeVore, Ron; Lee, Nam-Yong; Lucier, Bradley J. (1998), Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage, IEEE Transactions on Image Processing, 7, 3, p. 319-335. http://dx.doi.org/10.1109/83.661182

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Type
Article accepté pour publication ou publié
Date
1998
Journal name
IEEE Transactions on Image Processing
Volume
7
Number
3
Publisher
IEEE
Pages
319-335
Publication identifier
http://dx.doi.org/10.1109/83.661182
Metadata
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Author(s)
Chambolle, Antonin cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
DeVore, Ron

Lee, Nam-Yong

Lucier, Bradley J.
Abstract (EN)
This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem. Given an image F defined on a square I, minimize over all g in the Besov space B11(L1(I)) the functional |F-g|L2(I)2+λ|g|(B11(L1(I))). We use the theory of nonlinear wavelet image compression in L2(I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signal-to-noise ratio (SNR), which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest α for which F∈Bqα(Lq(I)),1/q=α/2+1/2, and the norm |F|Bqα(Lq(I)). Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Donoho and Johnstone's VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's (1994, 1995, 1996) SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves a lower error than our procedure.
Subjects / Keywords
variational problems; wavelet-based image processing algorithms

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