High Balanced Biorthogonal Multiwavelets with Symmetry

Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. Yang et al. (2006) constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.

Generalized Refinable Function Vectors with Hermite Interpolating Property

Wavelet analysis has many applications in scientific areas such as computer graphics, image processing, numerical algorithms and signal denoising. In general, a wavelet is derived from a refinable function vector via a multiresolution analysis. In this paper, we presented a novel notion of generalized Hermite interpolating refinable function vector. In terms of its mask, several properties (such as interpolation property, symmetry property and approximation property) with respect to generalized Hermite interpolating refinable function vector. We shall present an example at the end of this paper.

Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

CONSTRUCTION OF COMPACTLY SUPPORTED CONJUGATE SYMMETRIC COMPLEX TIGHT WAVELET FRAMES

Two algorithms for constructing a class of compactly supported complex tight wavelet frames with conjugate symmetry are provided. Firstly, based on a given complex refinable function ϕ, an explicit formula for constructing complex tight wavelet frames is presented. If the given complex refinable function ϕ is compactly supported conjugate symmetric, then we prove that there exists a compactly supported conjugate symmetric/anti-symmetric complex tight wavelet frame Ψ = {ψ1, ψ2, ψ3} associated with ϕ. Secondly, under the conditions that both the low-pass filters and high-pass filters are unknown, we give a parametric formula for constructing a class of smooth conjugate symmetric/anti-symmetric complex tight wavelet frames. Free parameters in the algorithm are explicitly identified, and can be used to optimize the result with respect to other criteria. Finally, two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames.