Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage * Antonin Chambolle 1 , Ronald A. DeVore 2 , Nam-yong Lee 3 , and Bradley J. Lucier 4 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 prob- lems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image F de- fined on a square I , minimize over all g in the Besov space B 1 1 (L 1 (I )) the functional kF - gk 2 L 2 (I) + λkgk B 1 1 (L 1 (I)) . 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-to- noise ratio, which we claim more accurately reflects the visual per- ception of noise in images, arises in this derivation. We present ex- tensive 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 kF k 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, how- ever, that Donoho and Johnstone’s SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves lower error than our procedure. 1. Introduction This paper has several objectives. The first is to de- scribe several families of variational problems that can be solved quickly using wavelets. These variational problems take the form: given a positive parameter λ and an image, a signal, or noisy data f (x) defined for x in some finite do- main I , find a function ˜ f that minimizes over all possible functions g the functional (1) kf - gk 2 L2(I ) + λkgk Y , where kf - gk L2(I ) := Z I |f (x) - g(x)| 2 dx 1/2 * A shorter version of this paper appeared in the IEEE Trans- actions on Image Processing, v. 7, 1998, pp. 319–335. 1 CEREMADE (CNRS URA 749), Universit´ e de Paris– Dauphine, 75775 Paris CEDEX 16, France, Antonin.Chambolle@ ceremade.dauphine.fr. Supported by the CNRS. 2 Department of Mathematics, University of South Carolina, Columbia, SC 29208, [email protected]. Supported in part by the Office of Naval Research, Contract N00014-91-J-1076. 3 Department of Mathematics, Purdue University, West Lafay- ette, IN 47907-1395, [email protected]. Supported in part by the Purdue Research Foundation. 4 Department of Mathematics, Purdue University, West Lafay- ette, IN 47907-1395, [email protected]. Supported in part by the Office of Naval Research, Contract N00014-91-J-1152. Part of this work was done while the author was a visiting scholar at CEREMADE, Universit´ e de Paris–Dauphine, Paris, France. is the root-mean-square error (or more generally, differ- ence) between f and g, and kgk Y is the norm of the ap- proximation g in a smoothness space Y . The original image f could be noisy, or it could simply be “messy” (a medical image, for example), while ˜ f would be a denoised, seg- mented, or compressed version of f . The amount of noise removal, compression, or segmentation is determined by the parameter λ; if λ is large, then necessarily kgk Y must be smaller at the minimum, i.e., g must be smoother, while when λ is small, g can be rough, with kgk Y large, and one achieves a small error at the minimum. These types of variational problems have become fairly common in image processing and statistics; see, e.g., [32]. For example, Rudin-Osher-Fatemi [33] set Y to the space of functions of bounded variation BV(I ) for im- ages (see also [1]), and non-parametric estimation sets Y to be the Sobolev space W m (L 2 (I )) of functions all of whose mth derivatives are square-integrable; see the monograph by Wahba [34]. In fact, Y can be very general; one could, for example, let Y contain all piecewise constant func- tions, with kgk Y equal to the number of different pieces or segments of g; this would result in segmentation of the original image f . Indeed, Morel and Solimini [31] argue that almost any reasonable segmentation algorithm can be posed in this form. Techniques like this are also known as Tikhonov regularization; see [2]. In [12] we considered (1) in the context of interpolation of function spaces; in that theory, the infimum of (1) over all g is K(f,λ,L 2 (I ),Y ), the K-functional of f between L 2 (I ) and Y . A fast way of solving (1) is required for practical al- gorithms. In [12], we noted that the norms of g in many function spaces Y can be expressed in terms of the wavelet coefficients of g. In other words, if we choose an (orthogo- nal or biorthogonal) wavelet basis for L 2 (I ), and we expand g in terms of its wavelet coefficients, then the norm kgk Y is equivalent to a norm of the wavelet coefficients of g; see, for example, [30], [14], or [23]. In [12] we proposed that by choosing kgk Y to be one of these norms and by calculating approximate minimizers rather than exact minimizers, one can find efficient com- putational algorithms in terms of the wavelet coefficients of the data f . In particular, we showed how choosing Y = W m (L 2 (I )) and approximately minimizing (1) leads to wavelet algorithms that are analogous to well-known linear algorithms for compression and noise removal. In particular, we find that the wavelet coefficients of ˜ f are simply all wavelet coefficients of f with frequency below a fixed value, determined by λ. Additionally, we proposed choosing Y from the family
Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage
Jun 19, 2023
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