Non-stationary biorthogonal wavelet system based on Gaussian function

Abstract

We discuss a non-stationary biorthogonal wavelet system. An ordinary(i.e., stationary) biorthogonal wavelet system consists of a set of filters: a lowpass filter, a highpass filter, and their respective duals (a dual lowpass filter, and a dual highpass filter). If the system is non-stationary, then each filter becomes a sequence of filters which work in a predetermined order. As the lowpass filters, we employ masks from a Gaussian based subdivision scheme. Being a non-stationary scheme, a Gaussian based subdivision has a sequence of masks instead of a single one. By the virtue of the refinable property which is shared by both of subdivision and wavelets, the mask of a subdivision scheme can serve as a lowpass filter for a wavelet system. In our case, the refinable relation in the form of non-stationary version guarantees the same end. The given lowpass filters define a non-stationary Riesz Multiresolution Analysis(MRA) whose structure is exactly the same as an ordinary(i.e., stationary) MRA except for one point: it has a sequence of generators while the ordinary has a single. Hence we can also define a non-stationary biorthogonal MRA in a similar manner. Then we have a condition for the dual lowpass filters. We present an algorithm to construct dual lowpass filters, especially the symmetric choice. Once the lowpass filter and its dual are determined, there is an immediate choice for highpass filter and its dual. As the construction of Gaussian based subdivision scheme reveals, the lowpass filters are symmetric and compactly supported. We construct its dual to preserve these properties. Hence the resulted wavelet system is symmetric and compactly supported. Different choice for the shape parameter (λ from the Gaussian function) leads to a different set of filters. As a result, we take two different choice for λ to compute the filters numerically. In an ordinary case, a lowpass/highpass filter defines a scaling/wavelet function and a respecti...