An Orthogonal Family of Quincunx Wavelets with Continuously Adjustable Order

We present a new family of two-dimensional and three-dimensional orthogonal wavelets which uses quincunx sampling. The orthogonal refinement filters have a simple analytical expression in the Fourier domain as a function of the order λ, which may be noninteger. We can also prove that they yield wavelet bases of $L _{ 2 } (R ^{ 2 })$ for any λ>0. The wavelets are fractional in the sense that the approximation error at a given scale a decays like $O(a ^{ \lambda } )$; they also essentially behave like fractional derivative operators. To make our construction practical, we propose an fast Fourier transform-based implementation that turns out to be surprisingly fast. In fact, our method is almost as efficient as the standard Mallat algorithm for separable wavelets.