Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Applying B-Spline Biorthogonal Wavelet Basis Functions to the Method of Moments in Solving Poisson equation

The aim of this paper is to introduce the application of B-spline biorthogonal wavelet with various orders (number of vanishing moments) in Electrostatic problems and making improvement in the moment method development. Due to the order of wavelet, the impedance matrix resulting in this problem is sparsified by wavelet, and consequently, the solution can be obtained rapidly. To illustrate these concepts, the two-body problem of parallel square conducting plates is presented. To demonstrate the effectiveness and accuracy of the proposed technique, numerical results for charge density potential, capacity, and error relative for different order of B-spline biorthogonal wavelets and dielectric constant are presented. Results are compared to the previous work done and published, excellent results are obtained

Pseudo-splines, wavelets and framelets

Master'sMASTER OF SCIENC