New wavelet-based algorithms for signal decomposition and reconstruction via the theory of circular stationary vector sequences and the Zak transform with applications to image compression

In this thesis we develop a new track to the theory of the time-frequency decomposition and apply the results to image compression.; First we develop the theory of circular stationary sequences and apply it to the theory of the Gabor decomposition. Using our method, we show how to circumvent the problem of zero in the Zak transform domain under fast Gabor decomposition that caused many researches to look for some more computationally intensive methods of computing the Gabor transform, such as resorting to oversampling. We also show how to compute the Gabor transform and obtain the best possible reconstruction of the signal in the case of undersampling.; We derive some interesting and important theorems and algorithms for the filter bank theory based on the Zak transform properties. These algorithms allow decomposition and reconstruction of signals on virtually any filter bank. A fast procedure for filter bank orthogonalization is derived. Finally the theory is applied to the derivation of the procedure for the wavelet transform in the frequency domain.; The filter bank theory is then applied to the derivation of the wavelet filtering algorithm for decomposition and reconstruction of signals (images), allowing unusual flexibility for choosing the filters. According to this algorithm, one can choose practically any filter to be the low pass filter of the filter bank consisting of two filters. The high pass filter can be the mirror filter of the low pass one. During the reconstruction some short recursive filters are applied to the signal before the upsampling. The reconstruction filters are the same as those used for decomposition. Using the flexibility of this method, we found the filters producing the best compression performance for different filter support sizes under SPIHT compression algorithm. Our 7 tap filter with recursive post filtering turned out to compress better than the widely acclaimed 9/7 biorthogonal filter of Antonini et al. (31) and approximately the same as the 10/18 filter of Tsai et al. (29). While because of the recursive post filtering, our filter is slower than the 9/7 filter, it is faster than the 10/18 filter. We apply our method together with the lifting procedure to find good wavelet filters mapping integers into integers.