Student Project
Semester Project, June
2001
Abstract
The nonseparable
wavelets may cost much more than the separable ones unless they are
implemented in the Fourier domain. We present here a new family of
such 2D orthogonal wavelets using a quincunx sampling and a fractional
filtering.
For the quincunx
downsampling we use a matrix D which turns the inital image by p/4 and reduce it by square root of 2. The upsampling consists
on filling the missing blanks by zeros so we keep only the wanted
pixels on the quincunx lattice.
The chosen matrix leads us to the separable case after each second iteration because D^{2}=2I.
The fractional
filtering offers more flexibility: In one hand, the refinement
filters are easily tunable thanks to the alpha parameter. In an other
hand, we can choose any size of filter since we use them in the
frequency domain (FIR, IIR).
The quincunx
transform has two advantages with respect to the separable wavelet
transform. First, it has less directionnality so the
privileged directions are no more the vertical and horizontal ones but
rather the diagonal ones. This can be more useful for some
applications such as edge detection. Then, it has one more iteration
before reducing the image size by 2 so the progression is slower and
finer. This can be more determinant for applications like fMRI and so
on.
The goal of the
project is to implement a fast and efficient algorithm of the
quincunx wavelet transform. We implemented this algorithm in Java in
order to provide a plugin on ImageJ which is a Java free software of
Image Processing. Some results of quincunx image transforms are shown.
Quincunx transform of a grayscale image
MRI 256x256 32bits 
order=p, 3 iterations 
order=0.6, 5 iterations 



Quincunx transform of a RGB image
Fluorescent Cells 256x256 
order=1.414, 2 iterations 
order=0.5, 4 iterations 



