Why pixels on image edges are shifted when using Wiener filter?

Question

I implemented a Wiener filter using Matlab and i noticed that both functions(mine and matlabs) are creating an image with shifted pixels on the edges. Class teacher said it will happen. Is there a simple explanation to that?

Code:

%open image
I = imread('pout.tif');

%sigma for gaussian
sigma = 1.2;
%gaussian kernel max width/height 
N = 2*ceil(3*sigma)+1;

%gaussian PSF(point spread function) 
PSF = fspecial('gaussian',N,sigma);

%Noise
NSR = 0.05;

%Matlab's Wiener function.
I1 = deconvwnr(I,PSF,NSR);

%New PSF for my implementation
PSF1 = zeros(N,N);
%loop to create the 2D gaussian psf.
x = (N/2)-0.5;
for i = -x:x
    for j = -x:x
        %gaussian 
         PSF1(i+x+1,j+x+1) = exp( -( ( (i^2) + (j^2) ) / ( 2 * (sigma^2) ) ) );
    end
end

%normalization
PSF1=PSF1/sum(sum(PSF1));

%fourier transformation of PSF1.
fouri = zeros(size(I,1),size(I,2));
fouri(1:N,1:N)= PSF1;
H = fft2(fouri);

%uint8 to double of the original image.
I = double(I);
%fourier transformation of the image
Y = fft2(I);
%Wiener function.
X = conj(H)./ ( abs(H).^2 + NSR ).*  Y;

%exit from fourier transformation.
I2 = real(ifft2(X));
%go back to uint8 for both images.
I2 = uint8(I2); 
I = uint8(I);

%Results.
subplot(1,3,1); imshow(I); title('Original');
subplot(1,3,2); imshow(I1); title('Matlab function');
subplot(1,3,3); imshow(I2); title('My implementation');

Output:

Answer 1

The shift happens because of the way that the filter is defined in the spatial domain before applying the FFT. The kernel is centered on the pixel at index (N/2, N/2), instead of at (1, 1) (the top-left corner), where the FFT expects the origin to be.

Basically you convolve with a shifted kernel, which is identical to convolving with the non-shifted kernel and then shifting the output image.

This answer of mine on Stack Overflow shows how to properly pad and shift the kernel before applying the FFT. In short, pad the kernel equally on all sides, then apply ifftshift to shift the origin from the center of the array to the top-left corner.