Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage

Abstract

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. We use the theory of nonlinear wavelet image compression in L 2 (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-tonoise ratio, 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 ∈ B α q (Lq(I)), 1/q = α/2 + 1/2, and the norm F B α q (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 SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves lower error than our procedure.

Adjoint-based optimization of PDE systems with alternative gradients

“…Our computational examples illustrating application of this strategy to solve optimal control problems for two different PDE systems show advantages of the proposed approach with respect to traditional techniques. As argued in [18] and as is also well-known in the image processing literature (see, e.g., [20]), extraction of gradients in different functional spaces is in fact equivalent to applying different filters to the adjoint field. Gradients extracted in Hilbert spaces can be regarded as obtained via an application of a linear filter to the adjoint state and, for example, the Sobolev gradients can be viewed as obtained via application of suitable low-pass filters (defined in wavenumber space) to the adjoint field [18].…”

“…Relation (20) can now be transformed to a form consistent with Riesz identity (5) by introducing an adjoint operator L * : X → X * and the corresponding adjoint state q * ∈ X via the following identity…”

Spatiotemporal wavelet analysis for functional MRI

“…While robust, such approaches tend to degrade edges and other potentially informative irregularities in the recovered image. ' 1 constraints can facilitate a larger range of candidate fits by including those belonging to a minimally smooth subspace (see Chambolle et al, 1998 of all possible functions). In the current application, use of an ' 1 penalty enriches detail in the activation maps while accounting for low-amplitude, smooth connected structures.…”

“…These have included applications in signal and image processing (Chambolle et al, 1998;Coifman et al, 1992;Devore and Lucier, 1992;Vetterli and Kovacević, 1995), fractals (Abry et al), vision (Mallat, 1996), meteorology (Fournier, 1996), time series (Nason and von Sachs, 1999;von Sachs, 1998), and statistics (Benjamini and Hochberg, 1995;Donoho and Johnstone, 1995;Johnstone and Silverman, 1997).…”

Wavelet thresholding via MDL for natural images

“…13,12,14,11,5 . In a series of papers, Donoho and Johnstone 13, 12, 14 assume the following additive, white noise model in the wavelet domain: y i = i + i ; i = 1 ; : : : ; n 1 where the i are wavelet coe cients of the signal and the i are iid N 0; 2 .…”