Construction Of A Family Of Bivariate Orthogonal Nonseparable Wavelets

In this paper.the orthogonality and accuracy of a scaling function φ(x), which satisfies a scaling equation with special coefficients and a given dilation matrix A , is discussed . So a family of bivariate orthogonal nonseparable wavelets in R~2is constructed . Meanwhile, the smoothness of φ(x) is investigated.In section 1 , we introduce some basic notions and related conclusions .In section 2 , we give the main result of the paper. We find the expression of A(z) and B(z) on the assumption that nonzero coefficients {h_k}k∈z~2 in the dilation equation are placed in two arbitrary rows in the first,fourth quadrant , the dilationmatrix A is and the symbolic function is given in the form of C(z,ω) =ω~mA(z)+z~lω~nB(z) in which l is odd . So an orthogonal scaling function with accuracy r + 1 is acquired.In section 3, we get the conditons for A(z) and B{z) to make the symbolic function C(z,ω) =ω~mA(z) + z~lω~nB(z) has accuracy r + 1, which guarantees that the scaling function to be constructed is r +1 accurate. On the basis of this, we find the concrete expression of A(z) and B{z) satisfing the orthogonal condition of bidimensional filters. Furthermore , we verify that such filter function h(u) satisfies Cohen criterion , so a family of bivariate orthogonal nonseparable wavelets in R~2 with accuracy r + 1 is acquired.In section 4, the smoothness of the scaling function φ(x) is investigated tentatively. As we all know, it is usually difficult for the smoothness of a scaling function φ(x) to be investigated , and the sufficient and necessary condition of smoothness is not easy to get, the paper draws a conclusion that the scaling function acquired does not satisfy the sufficient condition given in [6],which is not enough to illuminate the non-smoothness of φ(x), and we will make further study in the future.In section 5, we give an example with our method of constructing bivariate orthogonal wavelets.At last, we raise a conjecture, that is , for the dilation matrix A =and , no matter how placed the nonzero coefficients are, no scalingfunctions are both smooth and orthogonal.