Standard pairs and existence of symmetric multiscaling functions

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

BALANCED MULTIWAVELETS ON THE INTERVAL [0,1] WITH ARBITRARY INTEGER DILATION FACTOR a

In this paper, orthogonal multiwavelets on interval [0, 1] with arbitrary compact support γ and integer dilation factor a are studied. Firstly, the concept of multiwavelets on the interval [0, 1] is generalized to the arbitrary integer dilation factor a, a ≥ 2, a ∈ ℤ, and an algorithm for constructing orthogonal multiscaling function and multiwavelets on interval [0, 1] with dilation factor a is presented. Secondly, the decomposition and reconstruction formulas of multiwavelets on interval [0, 1] are deduced. Finally, the "balancing" concept of multiwavelets on the interval [0, 1] with dilation factor a is defined, and the algorithm of balancing the unbalanced multiwavelets on the interval [0, 1] is also given.

HIGH APPROXIMATION ORDER BIORTHOGONAL MULTISCALING FUNCTIONS WITH DILATION FACTOR a

An algorithm is presented for constructing a pair of high approximation order biorthogonal multiscaling function with dilation factor a in terms of any given pair of biorthogonal multiscaling function. The special case that a = 2 is discussed. If the dilation factor a = 2, then a biorthogonal multiwavelet pair is constructed explicitly. Finally, examples are given.

MULTIWAVELETS AND APPLICATIONS TO IMAGE DENOISING WITH EFFECTIVE ITERATION METHOD

The multiscaling function of Alpert multiwavelet system and Chui–Lian multiwavelet system consist of one symmetric and one antisymmetric scaling functions. In its single level decomposition, only one of 16 subband blocks may be considered as a core part because the antisymmetric scaling function may play the high pass filter. We complete the Alpert multiwavelet system by determining the high pass filter using the algorithm of paraunitary extension principle. Taking the advantage of the symmetric/antisymmetric (SA) feature of Alpert and Chui–Lian multiscaling functions, our iteration (SA iteration) of multiwavelet decomposition continues to the core part (1/16 of subbands) for the image denoising problem while the traditional iteration applies to the one fourth of 16 subbands as core parts. The numerical results demonstrate that our method of symmetric/antisymmetric iteration exhibits performance superior to the traditional ways.