Quincunx Wavelets 
Investigators: Dimitri Van De Ville, Manuela Feilner 

Summary: We have constructed several families of quincunx wavelet bases with an adjustable order parameter, which may be noninteger. They are implemented very efficiently by means of an FFTbased algorithm. 

Separable wavelets transforms are easy to implement and have become quite popular for image processing. Unfortunately, they tend to privilege the vertical and horizontal directions; they also produce a socalled “diagonal” wavelet component, which does not have a straightforward directional interpretation.
Wavelets defined on a quincunx lattice offer an interesting alternative. Their main advantages over the separable ones are: (1) there is one single wavelet instead of three, (2) the scaling function and wavelet can be designed to be nearly isotropic, and (3) the scale reduction is more progressive. 

We have introduced two families of quincunx wavelet transforms with some remarkable properties.
The first one is orthogonal and based on filterbank design techniques. The starting point of the construction is a fractional version of the Butterworth filter that is then mapped to 2D using the McClellan transformation.
The second construction is explicit and uses fractional polyharmonic spline functions, which are localized on the quincunx lattice. The advantage is that the basis functions are defined analytically and that one has full control of their mathematical properties such as regularity or decay. The corresponding wavelets are semiorthogonal and come in a variety of flavors: orthogonal, Bspline, and dual.
The key point is that both wavelet families are indexed by a fractional order parameter and that they are tunable in a continuous fashion. They essentially behave like fractional iterates of the Laplacian operator. They are also quite efficient computationally, thanks to our FFTbased algoritm.
In fact, our method is almost as efficient as the standard Mallat algorithm for separable wavelets. This can be checked on this demo. 

Collaborations: Prof. Michael Unser 




