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.

A new theory is introduced for analyzing image compression methods that are based on compression of wavelet decompositions. This theory precisely relates a) the rate of decay in the error between the original image and the compressed image (measured in one of a family of so-called L p norms) as the size of the compressed image representation increases (i.e., as the amount of compression decreases) to b) the smoothness of the image in certain smoothness classes called Besov spaces. Within this theory, the error incurred by the quantization of wavelet transform coefficients is explained. Several compression algorithms based on piecewise constant approximations are analyzed in some detail. It is shown that if pictures can be characterized by their membership in the smoothness classes considered here, then wavelet-based methods are near optimal within a larger class of stable (in a particular mathematical sense) transform-based, nonlinear methods of image compression. Based on previous experimental research on the spatial-frequencyintensity response of the human visual system, it is argued that in most instances the error incurred in image compression should be measured in the integral ( L ' ) sense instead of the mean-square ( L~) sense. Index Terms-Image compression, wavelets, smoothness of images, quantization.

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We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in i-'(R) to within 0(N~2) by a piecewise linear function with O(N) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to 0(N~l). These numerical methods for conservation laws are the first to have proven convergence rates of greater than 0(N~l/2).

Abstract. In this paper we study finite-difference approximations to the variational problem using the BV smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an upwind finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by the first author for a simpler, less isotropic, finite-difference approximation to the (isotropic) BV seminorm. We introduce a multiscale method for speeding the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.AMS subject classifications. 15A15, 15A09, 15A231. Introduction. In an influential paper, Rudin, Osher, and Fatemi [28] suggested using the bounded variation seminorm to smooth images. The functional proposed in their work has since found use in a wide array of problems (see, e.g., [9]), both in image processing and other applications. The novelty of the work was to introduce a method that preserves discontinuities while removing noise and other artifacts.In the continuous setting, the behavior of the solutions of the model proposed in [28] is well understood (see, e.g., [11], [12], and [27]). The qualitative properties of solutions of its discrete versions are not, perhaps, as well known or understood. In this work we study the behavior of solutions of the discrete approach used in, e.g., [13] as well as a new "upwind" variant of this model that better preserves edges and "isotropic" features. We also introduce a multiscale method for improving the initial guess of certain iterative methods for solving discrete variational problems based on the BV variational model.We begin by giving some background on the work in [28]. Working on the unit square I = [0,1] 2 , where the (isotropic) bounded variation seminorm is defined as