Arbitrarily Smooth Orthogonal Nonseparable Wavelets in ²

For each r∈, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M: the coefficients are aligned along two adjacent rows, and . We prove that if ²=± 2I, e.\,g., %%= or =, M=(0\,\,2 1\,\,0) or M=(1\,\,-1 1\,\,-1), then the smoothness of the wavelets improves asymptotically by 1-1/2log₂3 ≈ 0.2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.