The Nice Characters of Orthogonal Bidimensional Vector Wavelet Packs and Orthogonal Binary Framelets - Advanced Research on Computer Education, Simulation and Modeling - page 73

The Nice Characters of Orthogonal Bidimensional Vector
Wavelet Packs and Orthogonal Binary Framelets

Jinfan Yang * and Sanhua Song

Department of Computer Science, Huanghuai University, Zhumadian, 463000, China
[e-mail]

Abstract. Wavelet analysis has become a developing branch of mathematics for
over twenty years. It has been a powerful tool for exploring and solving many
complicated problems in natural science and engineering computation. In this
work, the notion of orthogonal vector bivariate wavelet packs and wavelet
frame packs, which are generalization of uni-wavelet packets, is introduced. A
new procedure for designing these vector bivariate wavelet packs is presented.
Their characteristics are studied by using time-frequency analysis method, Ba-
nach space theory and finite group theory. Orthogonal formulas concerning the
wavelet packs are established. The biorthogonality formulas concerning these
wavelet wraps are established. Moreover, it is shown how to draw new Riesz
bases of space

v
L RC from these wavelet wraps.

22
(,

)

Keywords: Banach space theorem, Bivariate, vector wavelet wraps, Riesz
bases, Bessel sequence, time-frequency analysis representation.

1 Introduction and Notations

The wavelet 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 concerning
the moment of emission and duration of a signal. Since 1986, wavelet analysis has be-
come a developing branch of mathematics. Wavelet packets have been applied to sig-
nal processing [1], image compression [2] so on. Coifman and Meyer are those who
firstly introduced the notion of univariate orthogonal wavelet packets. Shen[3] con-
structed the multivariate orthogonal wavelet packets. Wavelet packets include multiple
orthonormal basis , which means that a signal could be represented in many different
ways by using wavelet packets. But the performances in presenting the specified signal
are different using different bases. The one which could provide the best performance
according to some criterion will be the best basis. Nowadays, most of the related stud-
ies use the algorithm proposed by Coisman and Wickerhanser to select the best basis.
Vector-valued wavelets are a class of generalized multiwavelets. Xia [4] introduced
the notion of orthogonal vector-valued wavelets. Moreover, he studied the existence
* Corresponding author.