Verify the convolution theorem

Question

My main goal is to show that the convolution theorem works (just a reminder: the convolution theorem means that idft(dft(im) .* dft(mask)) = conv(im, mask)). I'm trying to program that.

Here is my code:

function displayTransform( im )
% This routine displays the Fourier spectrum of an image.
% 
% Input:      im - a grayscale image (values in [0,255]) 
% 
% Method:  Computes the Fourier transform of im and displays its spectrum,
%                    (if F(u,v) = a+ib, displays sqrt(a^2+b^2)).
%                    Uses display techniques for visualization: log, and stretch values to full range,
%                    cyclic shift DC to center (use fftshift).
%                    Use showImage to display and fft2 to apply transform.  

%displays the image in grayscale in the Frequency domain
imfft = fft2(im);
imagesc(log(abs(fftshift(imfft))+1)), colormap(gray);

% building mask and padding it with Zeros in order to create same size mask
b = 1/16*[1 1 1 1;1 1 1 1; 1 1 1 1; 1 1 1 1];
paddedB = padarray(b, [floor(size(im,1)/2)-2 floor(size(im,2)/2)-2]);
paddedB = fft2(paddedB);
C = imfft.*paddedB;
resIFFT = ifft2(C);

%reguler convolution
resConv = conv2(im,b);
showImage(resConv);

end

I want to compare resIFFT and resConv. I think I'm missing some casting because I am getting numbers in the matrix closer one to another if I'm using casting to double. Maybe I have some mistake in the place of the casting or the padding?

I don't see a question here. All I see is a code dump and a statement that you're "missing something". Well, yes, you probably are, and so are we -- namely a clear description of your problem. — Ilmari Karonen
@IlmariKaronen i have tried to clearify my question. please review it again. — Gilad
@Androidy - still unclear. Convert your im and b to doubles. — Shai
@Shai i want to program in matlab a simple demo to show that the convolution theorem works. my idea was to take an image make a convolution with the mask b. and on the other side ifft2(fft2(im).*fft2(b)). and then to compare the values of the two results. however my problem is that i'm getting two different matrices as a result. — Gilad

Eitan T Eitan T · Accepted Answer · 2012-12-25T16:42:25

In order to compute the linear convolution using DFT, you need to post-pad both signals with zeros, otherwise the result would be the circular convolution. You don't have to manually pad a signal though, fft2 can do it for you if you add additional parameters to the function call, like so:
```
fft2(X, M, N)
```
This pads (or truncates) signal X to create an M-by-N signal before doing the transform.
Pad each signal in each dimension to a length that equals the sum of the lengths of both signals, that is:
```
M = size(im, 1) + size(mask, 1);
N = size(im, 2) + size(mask, 2);
```

Just for good practice, instead of:

b = 1 / 16 * [1 1 1 1; 1 1 1 1; 1 1 1 1; 1 1 1 1];

you can write:

b = ones(4) / 16;

Anyway, here's the fixed code (I've generated a random image just for the sake of the example):

im = fix(255 * rand(500));            % # Generate a random image
mask = ones(4) / 16;                  % # Mask

% # Circular convolution
resConv = conv2(im, mask);

% # Discrete Fourier transform
M = size(im, 1) + size(mask, 1);
N = size(im, 2) + size(mask, 2);
resIFFT = ifft2(fft2(im, M, N) .* fft2(mask, M, N));
resIFFT = resIFFT(1:end-1, 1:end-1);  % # Adjust dimensions

% # Check the difference
max(abs(resConv(:) - resIFFT(:)))

The result you should get is supposed to be zero:

ans =

    8.5265e-014

Close enough.

Verify the convolution theorem

1 Answers