Error in Generating Point Spread Function

After finding the Modulation Transfer Function, I used this method to rotate it about its vertical axis to obtain the 2D MTF. I then performed ifft2() on that to generate the PSF. However, my PSF looks off for some reason.

%restore fitted MTF properties

freq = q/T;

MTF_fit = abs(fft(yLSF_fit));

MTF_fit = fftshift(MTF_fit);

f = fit(freq.', MTF_fit.', 'gauss2');

% rotate MTF in fourier domain

n = length(freq);

x0 = freq;

y0 = f(x0).';

theta = linspace(0,2*pi,n).';

x = x0.*cos(theta);

y = x0.*sin(theta);

z = repmat(y0,[n,1]);

MTF2 = z;

%obtain 2D PSF using ifft2 on 2D MTF

PSF2 = abs(ifftshift(ifft2((MTF2)); % Calculate PSF

% or PSF2 = abs(fftshift(ifft2(fftshift(MTF2))));

PSF2_normalized = (PSF2-min(PSF2(:)))./(max(PSF2(:))-min(PSF2(:))); % Normalize PSF

%plot 2D MTF, PSF, and normalized PSF

subplot(3,2,4)

surf(x,y,z)

xlabel pixels

ylabel pixels

zlabel amplitude

title('2D MTF')

shading interp

axis tight

subplot(3,2,5)

surf(PSF2);

title('PSF')

shading interp

axis tight

xlabel pixels

ylabel pixels

zlabel amplitude

subplot(3,2,6)

surf(PSF2_normalized);

title('PSF (Normalized)')

shading interp

axis tight

rotate3d

xlabel pixels

ylabel pixels

zlabel amplitude

rotate3d('on');

What could be going wrong? I suspect it has something to do with the method I used for rotation because it generates exact copies of the same number that repeats throughout each column of the matrix, where each column holds a different value. See the MTF2 variable in the .mat workspace I have attached.

I've also attached another workspace (validMTF2.mat) with a correct MTF2 (i.e: one that has a viable shape and actually generates a 3D Gaussian PSF when ifft2() is applied) for reference. In this matrix, the columns aren't comprised of a single value, which is why I am suspicious of the rotation method I used, even though its graph looks perfect. Beware the values are very small though so they might show up as zeros. Thanks for the help!

Edit: here is what happens when n = 10 in the above code. The PSF has a more 3D shape in this case (higher n values make it thinner and thinner, which I don't understand). The tradeoff here is a less accurate 2D MTF shape as shown by the spikiness. How can I fix this issue and maintaing a high n value for accuracy?

Answer 1

[Debadipto]

Hi @Asser,

Replicating the 1D MTF using repmat actually creates a 2D MTF with repeated values, which is not rotationally symmetric. Instead, you should create a 2D grid and evaluate the 1D MTF at each point's radial distance from the center.

The following approach should give you a rotationally symmetric 2D MTF:

%restore fitted MTF properties

freq = q/T

MTF_fit = abs(fft(yLSF_fit));

MTF_fit = fftshift(MTF_fit);

f = fit(freq.', MTF_fit.', 'gauss2');

% Define the grid size

n_val = 100;

% Create a 2D grid of points

[x, y] = meshgrid(linspace(-max(freq), max(freq), n_val), linspace(-max(freq), max(freq), n_val));

% Calculate the radial distance from the center for each point

r = sqrt(x.^2 + y.^2);

% Evaluate the 1D MTF at each radial distance to get a 2D rotationally symmetric MTF

% Using arrayfun to apply 'f' to each element of 'r'

z = arrayfun(@(r_val) f(r_val), r);

MTF2 = z;

%obtain 2D PSF using ifft2 on 2D MTF

PSF2 = abs(ifftshift(ifft2((MTF2)))); % Calculate PSF

% or PSF2 = abs(fftshift(ifft2(fftshift(MTF2))));

PSF2_normalized = (PSF2-min(PSF2(:)))./(max(PSF2(:))-min(PSF2(:))); % Normalize PSF

%plot 2D MTF, PSF, and normalized PSF

subplot(3,2,4)

surf(x,y,z)

xlabel pixels

ylabel pixels

zlabel amplitude

title('2D MTF')

shading interp

axis tight

subplot(3,2,5)

surf(PSF2);

title('PSF')

shading interp

axis tight

xlabel pixels

ylabel pixels

zlabel amplitude

subplot(3,2,6)

surf(PSF2_normalized);

title('PSF (Normalized)')

shading interp

axis tight

rotate3d

xlabel pixels

ylabel pixels

zlabel amplitude

rotate3d('on');

Here is the plot generated on running the code: