Construction Of Compactly Supported Symmetric Orthonormal Wavelets With Dilation4

Wavelets with some good properties such as symmetric and orthonormal are desired in application, while it’s well known that2-band compactly supported symmetric orthonormal wavelets do not exist. In order to overcome this shortage, r-band wavelets are proposed. The object of this paper is to exhibit the construction of compactly supported o.n. symmetric scaling functions with dilation4and regularity m, and the three corresponding compactly supported o.n. wavelets, one being symmetric and the other two being antisymmetric. Firstly, we study the relationship between4-band orthonormal scaling function and scaling symbols, secondly, an explicit formula is obtained for m-regular scaling filters with dilation4, thirdly, by adding the condition of symmetric, the property of the factor of scaling function is studied and an approach to construct the wavelet filters from the scaling filter is described. Several examples are given to illustrate our approach and the corresponding images are depicted.