The Biorthogonal Features of the Ternary Vector-Valued Wavelet Wraps with Filter Functions and Pseudoframes - Advanced Research on Computer Education, Simulation and Modeling - page 66

The Biorthogonal Features of the Ternary Vector-Valued
Wavelet Wraps with Filter Functions and Pseudoframes

Honglin Guo 1,* and Zhihao Tang 2

1

Department of Fundamentals, Henan Polytechnic Institute, Nanyang, 473009
2 Dept. of Fundamentals, Henan Polytechnic Institute,
Nanyang, 473009, P.R. China
{sxxa11xauat,jhnsx123}@126.com

Abstract. The rise of wavelet analysis in applied mathematics is due to its ap-
plications and the flexibility. Vector-valued wavelet wraps with multi-scale di-
lation factor of space

v
L RC is introduced, which is the generalization of
multivariate wavelet packs. An approach for designing a sort of biorthogonal
vector-valued wavelet wraps in three-dimensional space is presented and their
biorthogonality traits are characterized by virtue of iteration method and time-
frequency analysis method. The biorthogonality formulas concerning these
wavelet wraps are established. Moreover, it is shown how to draw new Riesz
bases of space

23
(, )

v
L RC from these wavelet wraps. The pyramid decomposi-
tion scheme based on pseudoframes is established.

23
(,

)

Keywords: B-spline function; trivariate; vector-valued wavelet wraps; Riesz
bases; iteration method; time-frequency analysis representation.

1 Introduction

The frame theory has been one of powerful tools for researching into wavelets. Al-
though the Fourier transform has been a major tool in analysis for over a century, it
has a serious laking for signal analysis in that it hides in its phases information con-
cerning the moment of emission and duration of a signal. Every frame(or Bessel se-
quence) determines an analysis operator, the range of which is important for a lumber
of applications. Multiwavelets can simultaneously possess many desired properties
such as short support, orthogonality, symmetry, and vanishing moments, which a sin-
gle wavelet cannot possess simultaneously. This suggests that multiwavelet systems
can provide perfect reconstruction, good performance at the boundaries (symmetry),
and high approximation order (vanishing moments). Already they have led to exciting
applications in signal analysis [1], fractals [2] and image processing [3], and so on.
Vector-valued wavelets are a sort of special multiwavelets Chen [4] introduced the
notion of orthogonal vector-valued wavelets.However, vector-valued wavelets and
multiwavelets are different in the following sense. Pre-filtering is usually required for
discrete multiwavelet transforms [5] but not necessary for discrete vector-valued
* Corresponding author.