A new approach for estimation of statistically matched wavelet

This paper presents a new approach for the estimation of wavelets that is matched to a given signal in the statistical sense. Based on this approach, a number of new methods to estimate statistically matched wavelets are proposed. The paper first proposes a new method for the estimation of statistically matched two-band compactly supported biorthogonal wavelet system. Second, a new method is proposed to estimate statistically matched semi-orthogonal two-band wavelet system that results in compactly supported or infinitely supported wavelet. Next, the proposed method of estimating two-band wavelet system is generalized to M-band wavelet system. Here, the key idea lies in the estimation of analysis wavelet filters from a given signal. This is similar to a sharpening filter used in image enhancement. The output of analysis highpass filter branch is viewed to be equivalent to an error in estimating the middle sample from the neighborhood. To minimize this error, a minimum mean square error (MMSE) criterion is employed. Since wavelet expansion acts like Karhunen-Loe`ve-type expansion for generalized 1/f^β processes, it is assumed that the given signal is a sample function of an mth-order fractional Brownian motion. Therefore, the autocorrelation structure of a generalized 1/f^β process is used in the estimation of analysis filters using the MMSE criterion. We then present methods to design a finite impulse response/infinite impulse response (FIR/IIR) biorthogonal perfect reconstruction filterbank, leading to the estimation of a compactly supported/infinitely supported statistically matched wavelet. The proposed methods are very simple. Simulation results to validate the proposed theory are presented for different synthetic self-similar signals as well as music and speech clips. Estimated wavelets for different signals are compared with standard biorthogonal 9/7 and 5/3 wavelets for the application of compression and are shown to have better results.