Analysis of wavelet image denoising model in Besov spaces

Abstract

ABSTRACT In this paper, we discuss the image denoising model which DeV ore et al. had established, in which both distance and smoothness can be measured by the objective function, and anal ysis the model for wavelet image denoising in the Besov spaces with pq . In addition, we give the exact thresholds for the model, and prove that for 01 p the effect of noise removal using our methods is in between hard wavelet shrinkage and soft wavelet shrinkage. For the case 01 p and 1 ddfp , which refers to the problems on the converge nce of the iteration of the equations and on the complexity of computation, we give the simp lified algorithms. Comparing the thresh old given by this paper with Lorenz threshold, we conclude that the former is more meticulous than the latter for the model. Keywords: Besov SpacesE Bounded Variation, Wavelet ShrinkageE Hard ThresholdingE Soft Thresholding 1. INTRODUCTION Nonlinear approximation has recently played an important role in several problems of image processing including noise removal, compression, and feature extraction ([1]). The Besov space, consisting of functions with finite Besov semi-norms, is of particular interest for applications to noise removal and data compression. It is often chosen as a model for piecewise smooth signals suc h as images. Ronald A. DeVore proved that Besov spaces are more suitable for image processing than other smoothness spaces, such as Sobolev spaces. The traditional image denoising principle is based on the fact that after wavelet transformation the smooth pa rts of the images mainly conc entrated in the low frequency, and the noise and the details of the image information containe d in the high frequency components. The shrinkage of wavelet coefficients has been used widely since Donoho and Johnstone firs tly proposed the method in 1994. This method makes use of the characteristic that after wavelet decomposition the amplitude of noise is relatively small to remove minor coefficients and get directly the denoising image. When the noisy images are seen as a stochastic model in Besov spaces, Donoho and Johnstone have already proved wavelet shrinkage is approximate optimal noise removal method ([8] [9]). In image Denoising, we must not only consider the degree of approach between the denoising images and the original images, but also consider the smoothness of images. We should balance between them to determine the objective function. Since Besov spaces was introduced to the field of nonlinear approximation, some good properties of the Besov spaces have been used in image processing gradually. DeVore et al. successfully apply Besov space to the wavelet shrinkage ([2]) . Chambolle et al. not only prove the module in [2] has good result ([3]) for image denoising, but also prove that when letting Besov space be