An adaptive bayesian wavelet thresholding approach to multifractal signal denoising

Abstract

Multifractal functions are widely used to model irregular signals, while thresholding of the empirical wavelet coefficients is an effective tool for signal denoising. This paper outlines a Bayesian thresholding approach for multifractal functions observed in a white noise model. To do that, lacunary wavelet series are used to approximate the functions. These random functions are statistically characterized by two parameters. The first parameter governs the intensity of the wavelet coefficients while the second one governs its lacunarity. The estimation is obtained by placing priors on the wavelet coefficients that consists of a mixture of two normal distributions with different standard deviations. These variances are chosen adaptively according to the resolution level of the coefficients and depend on the multifractal function parameters. Estimators of these parameters are constructed and a closed form expressions for the posterior means of the unknown wavelets coefficients are obtained. An example is used to illustrate the method, and a comparison is made with other thresholding methods.