Fast Wavelet Techniques for Near-optimal Image Processing

In recent years many authors have introduced certain nonlinear algorithms for data compression, noise removal, and image reconstruction. These methods include wavelet and quadrature-mirror filters combined with quantization of filter coefficients for image compression, wavelet shrinkage estimation for Gaussian noise removal, and certain minimization techniques for image reconstruction. In many cases these algorithms have out-performed linear algorithms (such as convolution filters, or projections onto fixed linear subspaces) according to both objective and subjective measures. Very recently, several authors, e.g., [5], [7], [14], have developed mathematical theories that explain the improved performance in terms of how one measures the smoothness of images or signals. In this paper we present a unified mathematical approach that allows one to formulate both linear and nonlinear algorithms in terms of minimization problems related to the so-called K-functionals of harmonic analysis. We then summarize the previously developed mathematics that analyzes the image compression and Gaussian noise removal algorithms. We do not know of a specific formulation of the image reconstruction problem that supports an analysis or even definition of optimal solution. Although our framework and analysis can be applied to any d-dimensional signals (d = 2 for images, d = 1 for audio signals, etc.), we restrict our discussion in this paper to images. The outline of our paper is as follows. We want to find an approximation f̃ to a given image f on a square domain I, either to compress the image, remove noise from the image, etc. The size of the difference between f and f̃ is measured by a norm, which we shall take in this paper to be the L2(I) (mean-square) norm. (We emphasize that we do not believe that the L2(I) norm matches the spatialfrequency–contrast response of the human visual system— we use the L2(I) norm here only because the presentation is simpler; see, e.g., [5], where we develop a theory of image compression in Lp(I), with 0 < p < ∞.) We wish to balance the smoothness of f̃ with the goodness of fit ‖f − f̃‖L2(I); to this end we consider the problem of minimizing (1) ‖f − g‖L2(I) + λ‖g‖Y ,