2D Deconvolution using FFT in Matlab Problems

Question

I have convoluted an image I created in matlab with a 2D Gaussian function which I have also defined in matlab and now I am trying to deconvolve the resultant matrix to see if I get the 2D Gaussian function back using the fft2 and ifft2 commands. However the matrix I get as a result is incorrect (to my knowledge). Here is the code for what I have done thus far:

% Code for input image (img) [300x300 array]

N = 100;
t = linspace(0,2*pi,50);
r = (N-10)/2;
circle = poly2mask(r*cos(t)+N/2+0.5, r*sin(t)+N/2+0.5,N,N);
img = repmat(circle,3,3);

% Code for 2D Gaussian Function with c = 0 sig = 1/64 (Z) [300x300 array]

x = linspace(-3,3,300);
y = x';
[X Y] = meshgrid(x,y);
Z = exp(-((X.^2)+(Y.^2))/(2*1/64));

% Code for 2D Convolution of img with Z (C) [599x599 array]

C = conv2(img,Z);

% I have tested that this convolution is correct using cross section profile vectors for img and C and the resulting x-y plots are what i expect from the convolution.

% From my knowledge of convolution, the algorithm works as a multiplier in Fourier space, therefore by dividing the Fourier transform of my output (convoluted image) by my input (img) I should get back the point spread function (Z - 2D Gaussian function) after the inverse Fourier transform is applied to this result by division.

% Code for attempted 2D deconvolution

Fimg = fft2(img,599,599);

% zero padding added to increase result to 599x599 array

FC = fft2(C);
R = FC/Fimg;

% I now get this error prompt: Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.551432e-22

iFR = ifft2(R);

I'm expecting iFR to be close to Z but I'm getting something completely different. It may be an approximation of Z with complex values but I can't seem to check it since I don't know how to plot a 3D complex matrix in matlab. So if anyone can tell me whether my answer is correct or incorrect and how to get this deconvolution to work? I'd be much appreciated.

Answer 1

R = FC/Fimg needs to be R = FC./Fimg; You need to do division element-wise.

Answer 2

Here are some Octave (version 3.6.2) plots of that deconvolved Gaussian.

% deconvolve in frequency domain
Fimg = fft2(img,599,599);
FC = fft2(C);
R = FC ./ Fimg;
r = ifft2(R);

% show deconvolved Gaussian
figure(1);
subplot(2,3,1), imshow(img), title('image');
subplot(2,3,2), imshow(Z), title('Gaussian');
subplot(2,3,3), imshow(C), title('image blurred by Gaussian');
subplot(2,3,4), mesh(X,Y,Z), title('initial Gaussian');
subplot(2,3,5), imagesc(real(r(1:300,1:300))), colormap gray, title('deconvolved Gaussian');
subplot(2,3,6), mesh(X,Y,real(r(1:300,1:300))), title('deconvolved Gaussian');

% show difference between Gaussian and deconvolved Gaussian
figure(2);
gdiff = Z - real(r(1:300,1:300));
imagesc(gdiff), colorbar, colormap gray, title('difference between initial Gaussian and deconvolved Guassian');