A New Family Of Bivariate Orthogonal Nonseparable Wavelets

In this paper,the orthogonality and accuracy of a scaling function φ(x), which satisfies a scaling equation φ(x) = 2 ∑hkφ(Ax - A;) with special coefficients and a givendilation matrix A , is discussed . So a new family of bivariate orthogonal nonseparablewavelets is 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 {hk}k∈z~2 in the dilation equation are placed in two adjacent rows in the fourth quadrant , the dilation matrixand the symbolic function is given in the form of C(z,w) = A(z) +z~1w~-1B(z) in which / is odd . So an orthogonal scaling function with accuracy r + 1is acquired.In section 3, we get the conditons for A(z) and B(z) to make the symbolic function C(z,w) = A(z) + z~1w~-1 B(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(w) satisfies Cohen criterion , so a new family of bivariate orthogonal nonseparable wavelets 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 =andA =, no matter how placed the nonzero coefficients are, no scaling...