Bayesian Denoising of Visual Images in the Wavelet Domain

Abstract

The use of multi-scale decompositions has led to significant advances in representation, compression, restoration, analysis, and synthesis of signals. The fundamental reason for these advances is that the statistics of many natural signals, when decomposed in such bases, are substantially simplified. Choosing a basis that is adapted to statistical properties of the input signal is a classical problem. The traditional solution is principal components analysis (PCA), in which a linear decomposition is chosen to diagonalize the covariance structure of the input. The most well-known description of image statistics is that their Fourier spectra take the form of a power law [e.g., 1, 2, 3]. Coupled with a constraint of translation-invariance, this suggests that the Fourier transform is an appropriate PCA representation. Fourier and related representations are widely used in image processing applications. For example, the classical solution to the noise removal problem is the Wiener filter, which can be derived by assuming a signal model of decorrelated Gaussian-distributed coefficients in the Fourier domain.

Recently a number of authors have noted that statistics of order greater than two can be utilized in choosing a basis for images. Field [2, 4] noted that the coefficients of frequency subbands of natural scenes have much higher kurtosis than a Gaussian density. Recent work on so-called “independent components analysis” (ICA) has sought linear bases that optimize higher-order statistical measures [e.g., 5, 6]. Several authors have constructed optimal bases for images by optimizing such information-theoretic criterion [7, 8]. The resulting basis functions are oriented and have roughly octave bandwidth, similar to many of the most common multi-scale decompositions. A number of authors have explored the optimal choice of a basis from a library of functions based on entropy or other statistical criterion [e.g. 9, 10, 11, 12, 13].

In this chapter, we examine the empirical statistical properties of visual images within two fixed multi-scale bases, and describe two statistical models for the coefficients in these bases. The first is a non-Gaussian marginal model, previously described in [14]. The second is a joint non-Gaussian Markov model for wavelet subbands, previous versions of which have been described in [15, 16]. We demonstrate the use of each of these models in Bayesian estimation of an image contaminated by additive Gaussian white noise.