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Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm

### Bin Han

## Abstract

In applications, it is well known that short support,
high vanishing moments and reasonable smoothness
are the three most important properties of a
biorthogonal wavelet. Based on our previous work on analysis and
construction of optimal fundamental refinable functions and optimal
biorthogonal wavelets, in this paper,
we shall discuss the mutual relations
among these three properties.
For example, we shall see that any orthogonal scaling function,
which is supported on $[0,2r-1]^s$ for some positive integer $r$
and has accuracy order $r$,
has $L_p$ $(1\le p \le \infty)$ smoothness not exceeding that of the univariate
Daubechies orthogonal
scaling function which is supported on $[0,2r-1]$.
Similar results hold true for fundamental refinable functions and
biorthogonal wavelets. Then, we shall discuss the relation between
symmetry and the smoothness of a refinable function.
Next, we discuss the construction by cosets (CBC) algorithm reported
in Han \cite{\Hanbw} to construct biorthogonal wavelets with arbitrary
order of vanishing moments.
We shall generalize this CBC algorithm to construct bivariate
biorthogonal wavelets. For any positive integer $k$ and a bivariate
primal mask $a$ such that $a$ is symmetric about the origin, such CBC
algorithm provides us a dual mask of $a$ such that the dual mask
satisfies the sum rules of order $2k$ and is also symmetric about the
origin. The resulting dual masks have certain optimal properties with
respect to their support. Finally, examples of bivariate biorthogonal wavelets
constructed by the CBC algorithm are provided to illustrate the
general theory. Advantages of the CBC algorithm in this paper over other
methods on constructing biorthogonal wavelets are also discussed.

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