A New Result Of Orthogonal Non-separable Wavelet

Wavelets Analysis, proposed in the middle of 1980's, is an interesting mathematical theory which has great value in theory research and application. Over the last decade, the theories for univariate wavelets have grown rapidly. Among them, multivariate wavelets received particular attention due to many applications involved in multi-dimension. Multivariate wavelets can be divided into separable wavelets and nonseparable wavelets. Compared to separable wavelets, noseparable wavelets have more design freedom and can offer more isotropic analysis on spacial structure, Moreover, nonseparable sampling can fit the human visual system better. Therefore, the research of nonseparable wavelets attracts much more attention and becomes the main focus of multivariate wavelets.There are two methods for nonseparable wavelets construction. One is to construct the polyphase matrixs and build the filters (coefficient mask) based on elements of the polyphase matrixs. The other is to construct wavelets with special formal through definition or properties directly, given a fixed sampling matrix. Compared with the former, the latter can get scaling functions with more flexible forms and be easier for properties analysis. In this paper, the latter is adopted.In this paper, we select a special sampling matrix and extend the othogonality and vanishing monments conditions to three-rows for the bivariate wavelets. Then the factorization of the coefficient mask is given by the othogonality conditions which are quadratic and the initial value conditions of the factors come from the vanishing moments conditions which are linear. For the complexity of the factorization and computation, we construct a kind of three-rows orthogonal nonseparable wavelets with the sufficient vanishing moments. Analysis of the proposed wavelets'smoothness and a plenty of examples are also presented.. The method proposed in this paper is based on the two-rows nonseparable bivariate wavelets construction methods proposed by Wang et al.(SIAM Journal on Mathematical Analysis, vol.30, pp. 678-697). Wang's method is difficult to extend to many rows. Our proposed method can not only construct a kind of new three-rows nonseparable wavelets by complex computation, but also hold many excellent properties such as orthogonality, high vanishing moments and arbitrarily smoothness.