A New Approach for Estimation of Statistically
Anubha Gupta, Shashank Joshi, Surendra Prasad · 2005
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 pro- poses 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 sup- ported or infinitely supported wavelet. Next, the proposed method of estimating two-band wavelet system is generalized to -band wavelet system. Here, the key idea lies in the estimation of anal- ysis wavelet filters from a given signal. This is similar to a sharp- ening 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 mini- mize this error, a minimum mean square error (MMSE) criterion is employed. Since wavelet expansion acts like Karhunen-Loeve- type expansion for generalized processes, it is assumed that the given signal is a sample function of an th-order fractional Brownian motion. Therefore, the autocorrelation structure of a generalized process is used in the estimation of analysis fil- ters using the MMSE criterion. We then present methods to de- sign a finite impulse response/infinite impulse response (FIR/IIR) biorthogonal perfect reconstruction filterbank, leading to the esti- mation of a compactly supported/infinitely supported statistically matched wavelet. The proposed methods are very simple. Simula- tion results to validate the proposed theory are presented for dif- ferent 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.