The filters used to produce the figures in the paper are available in FgrFltr.zip.
The available software consists of three independent packages.
1) Rational DWT and DFT-Modulated FBs
2) Filter Design
3) 2D Directional Transforms Derived From DFT-Modulated FBs
The zip file TwoDim.zip contains the MATLAB programs that implement the forward and inverse 2D directional transforms derived from DFT-Modulated FBs along with a number of (almost) perfect reconstruction filter sets. See the file 'demo.m' for the usage.
First we load the filters for p=4, q=5 and 4 vanishing moments.
>> load P4Q5V4;
Notice that the file names give p, q and the number of vanishing moments.
>> who
Your variables are:
h p q
'h' is a symmetric filter that generates a {p,q} DFT-Modulated FB. Normally, we would set the synthesis filter 'g' to be the time reverse of 'h', but since 'h' is symmetric, we will use 'h' as also the synthesis filter. Let us check perfect reconstruction for 5 stages.
>> L = 5;
>> x = double(imread('house.bmp'));
>> coef = DIRDWT2D(x,h,p,q,L);
>> y = IDIRDWT2D(coef,h,p,q);
>> y = y(1:size(x,1), 1:size(x,2));
>> diff = x - y;
>> max(abs(diff(:)))
ans =
0.3022
Indeed, the frame operator is very close to identity as seen below.
'coef' is a length-L cell array where 'L' is the number of stages. Each element of the cell array holds another q by q cell array.
>> coef
coef =
{5x5 cell} {5x5 cell} {5x5 cell} {5x5 cell} {5x5 cell}
We claimed that the transform is directional. To verify this, let us look at a synthesis function utilized by the transform.
>> ze = zeros(size(x));
>> coefz = DIRDWT2D(ze,h,p,q,L);
>> coefz{3}{2,1}(round(end/2),round(end/2)) = 1;
>> di = IDIRDWT2D(coefz,h,p,q);
Repeating this (see 'demo.m'), we get the following synthesis functions.
