Nonstationary Biorthogonal Wavelet Filters

Nonstationary multiresolution analyses have gained a great interest in recent years. In fact, in the nonstationary framework it is possible to construct nonstationary biorthogonal bases whose approximation properties are better than those achieved in the stationary case. Nonstationary biorthogonal bases are associated with nonstationary biorthogonal filters, defined by two pairs of filter sequences - the low-pass pair and the high-pass pair - whose taps change at each step $ of the decomposition/reconstruction scheme. Nonstationary filters can be used to realize nonstationary wavelet transforms through a classical subband coding (analysis/synthesis) scheme, which permits to change the properties of the analysis/synthesis filters at each decomposition/reconstruction level. This property makes them a flexible and adaptive tool to be successfully applied in signal and image processing, while preserving all the computational advantages of the discrete wavelet transform. In this work we will present some families of nonstationary biorthogonal wavelet filters and analyze their main properties - compact support, number of vanishing moments, regularity, etc. - that contribute to the good compaction property of the nonstationary wavelet transform in both time and frequency domains. The performances of such filters in some classical problems, such as compression, denoising and edge detection, will be also evaluated.