%FILE: DigitalFilterDemo.m
%Creates and tests a digital filter for converting
% a cubic-phase PSF to a diffraction-limited PSF
%
%Create a digital representation of the cubic PSF:
% (We really don't care what the underlying optics
% are here -- the idea is to make a shape-changing filter)
N = 101; %Size of array;
rs = 25; %Size of digital filter (arbitrary)
Psize = .5; %Relative size of pupil in the array (radius)
x = linspace(-1,1,N);
[X,Y] = meshgrid(x,x);
R = sqrt(X.^2 + Y.^2);
%R is now an array with values equal to the distance
% from the center of the N x N array:
%Create a circular, uniform pupil, filling 1/5 the array:
P = zeros(N,N);
P(R<=Psize) = 1;
%Find the diffraction limited PSF and it's OTF:
DFpsf = fftshift(fft2(ifftshift(P)));
DFpsf = DFpsf.*conj(DFpsf);
%Normalize the psf:
DFpsf = DFpsf/sum(sum(DFpsf));
DFotf = fftshift(fft2(ifftshift(DFpsf)));
%Normalize the OTF:
DFotf = DFotf/max(max(abs(DFotf)));
%
figure(1)
subplot(1,2,1)
imagesc(DFpsf),axis image
set(gca,'XTickLabel',[],'YTickLabel',[])
%mesh(DFpsf)
colormap(gray)
title('Diffraction-Limited PSF')
subplot(1,2,2)
imagesc(abs(DFotf)),axis image
set(gca,'XTickLabel',[],'YTickLabel',[])
%mesh(abs(DFotf));
title('Diffraction-Limited MTF')
%
%Add cubic phase to the pupil:
Ac = 2/(Psize^3); %Waves of cubic phase at edge of pupil
C = Ac*(X.^3 + Y.^3);
Pc = P.*exp(i*2*pi*C);
Pc(R>Psize) = 0; %Clip amplitude and phase to actual pupil size.
%
%Find the PSF and OTF for the cubic-phase pupil:
Cpsf = fftshift(fft2(ifftshift(Pc)));
Cpsf = Cpsf.*conj(Cpsf);
%Normalize the PSF:
Cpsf = Cpsf/sum(sum(Cpsf));
Cotf = fftshift(fft2(ifftshift(Cpsf)));
%Normalize Cotf:
Cotf = Cotf/max(max(abs(Cotf)));
%
figure(2)
subplot(1,2,1)
imagesc(Cpsf),axis image
set(gca,'XTickLabel',[],'YTickLabel',[])
%mesh(Cpsf)
colormap(gray)
title('Cubic-Phase PSF')
subplot(1,2,2)
imagesc(abs(Cotf)),axis image
set(gca,'XTickLabel',[],'YTickLabel',[])
%mesh(abs(Cotf));
title('Cubic-Phase MTF')
%
%Make a digital filter to convert the Cubic PSF to a
% diffraction-limited PSF.
%
%************Make Cubic Filter***********************
%Make Filter for Cubic PSF:
%Calculate components for pseudo Weiner Filter:
%NOTE: Cotf can be a weighted sum of OTFs if the filter has to
%
reconstruct a number of similar PSFs.
H = conj(Cotf);
Hsq = Cotf .* conj(Cotf);
%Make Frequency Filter: (Note we are using the Diff-lim OTF
% as a limiting bandwith.)
W = DFotf; %Bandwidth limit of filter
wp = 0.001; %"Weiner parameter" => avoids divide by zero
F = (W .* H) ./ (Hsq + wp);
%F is now the frequency domain filter:
%Long Space Filter:
%IMPORTANT NOTE: The frequency arrays, W,H,Hsq,& F are shifted so
%
that the DC component is in the center. This is fine for
%
display purposes, and even for the filter arithmetic above,
%
but it is NOT OK for the FFT function, which insists on having
%
the DC component at the start of the array. Hence you MUST
%
make sure the array is ordered that way before using any FFT
%
algorithm:
F2 = fftshift(F);
f = real(ifftshift(ifft2(F2)));
%Adjust filter for unity gain:
f_long = f/sum(sum(f));
lng = sum(sum(f.^2)); %Noise gain of long filter
%Truncated Space Filter:
E = (N-rs)*0.5;
E1 = floor(E);
E2 = ceil(E);
rs = [E1:N-E2];
f_short =f(rs,rs);
%
%Display the filters:
figure(3)
subplot(1,2,1)
mesh(f_long)
colormap(gray)
title('Full-Size Space Filter')
subplot(1,2,2)
mesh(f_short)
title('Truncated Space Filter')
%
%Apply the filters to the cubic psf:
f_l_psf = conv2(Cpsf,f_long,'same');
f_s_psf = conv2(Cpsf,f_short,'same');
%
%Display the results:
figure(4)
subplot(1,2,1)
imagesc(f_l_psf),axis image, colormap(gray)
set(gca,'XTickLabel',[],'YTickLabel',[])
title('Filtered PSF (long filter')
subplot(1,2,2)
imagesc(f_s_psf),axis image
set(gca,'XTickLabel',[],'YTickLabel',[])
title('Filtered PSF (short filter')