Orthogonal Quincunx Wavelets with Fractional Orders

We present a new family of 2D 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 α, which may be non-integer. The wavelets have good isotropy properties. We can also prove that they yield wavelet bases of $L _{ 2 }$ $(\Re ^{ 2 }$ ) for any α > 0. The wavelets are fractional in the sense that the approximation error at a given scale a decays like $O(a ^{ \alpha }$ ); they also essentially behave like fractional derivative operators. To make our construction practical, we propose an FFT-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.