Two-dimensional orthogonal wavelets with vanishing moments

We investigate a very general subset of 2-D, orthogonal, compactly supported wavelets. This subset includes all the wavelets with a corresponding wavelet (polyphase) matrix that can be factored as a product of factors of degree-1 in one variable. In this paper, we consider, in particular, wavelets with vanishing moments. The number of vanishing moments that can be achieved increases with the increase in the McMillan degrees of the wavelet matrix. We design wavelets with the maximal number of vanishing moments for given McMillan degrees by solving a set of nonlinear constraints on the free parameters defining the wavelet matrix and discuss their relation to regular, smooth wavelets. Design examples are given for two fundamental sampling schemes: the quincunx and the four-band separable sampling. The relation of the wavelets to the well-known 1-D Daubechies wavelets with vanishing moments is discussed