Difference Wavelet - Theory and A Comparison Study
I‐Liang Chern, Chien-Chang Yen · Methods and Applications of Analysis · 2002
Abstract. Wavelet methods with polynomial filters are usually favored in applications for their fast wavelet transforms and compact support. However, wavelet methods with rational filters have more freedom to achieve smaller condition numbers, more regularity and better efficiency. Such methods can be attractive if they also possess fast algorithms and have fast decay (as if the corresponding wavelets had compact support). In the first part of this paper, we propose a new wavelet method with rational filters which do have these properties. We call it the difference wavelet method. It is a generalization of Butterworth wavelets. The analysis part is simply averaging and finite differencing. The wavelet coefficients measure the finite differences of the averages of an input data sequence. Its synthesis part involves rational filters, which can be performed with linear computational complexity by the cyclic reduction method. Their Riesz basis property, biorthogonality, decay and regularity are investigated. In the second part of this paper, we perform comparison studies of the difference wavelet method (Diff) with three other popular wavelet methods: the Cohen-Daubechies-Feauveau biorthogonal wavelets (CDF), the Daubechies orthogonal wavelets (Daub) and the Chui-Wang semi-orthogonal wavelets (CW). Natural criteria in designing good wavelet methods for representing functions and