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# Orthogonal Quincunx Wavelets with Fractional Orders

## M. Feilner, M. Jacob, M. Unser

### Proceedings of the 2001 IEEE International Conference on Image Processing (ICIP'01), Θεσσαλονίκη (Thessaloniki), Ελληνική Δημοκρατία (Hellenic Republic), October 7-10, 2001, vol. I, pp. 606-609.

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 L2(ℜ2) for any α > 0. The wavelets are fractional in the sense that the approximation error at a given scale a decays like O(aα); 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.

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