Replicate MATLAB's `conv2()` Using Fourier Domain Convolution

Question

I would like to take two images and convolve them together in Matlab using the 2D FFT without recourse to the conv2 function. However, I am uncertain with respect to how the matrices should be properly padded and prepared for the convolution.

The mathematical operation is the following:

A * B = C

In the above, * is the convolution operator (Wikipedia link).

The following Matlab program shows the difference between padding and not padding the matrices. I suspect that not padding the matrices results in a circular convolution, but I would like to perform a linear convolution without aliasing.

If I do pad the two matrices, then how do I truncate the output of the convolution so that C is the same size as A and B?

A = rgb2gray(im2double(imread('1.png'))); % input A
B = rgb2gray(im2double(imread('2.png'))); % kernel B

figure;
imagesc(A); colormap gray;
title ('A')

figure;
imagesc(B); colormap gray;
title ('B')

[m,n] = size(A);
mm = 2*m - 1;
nn = 2*n - 1;

C = (ifft2(fft2(A,mm,nn).* fft2(B,mm,nn)));

figure;
imagesc(C); colormap gray;
title ('C with padding')

C0 = (ifft2(fft2(A).* fft2(B)));

figure;
imagesc(C0); colormap gray;
title ('C without padding')

Here is the output of the program:

Answer 1

Without padding the result will be equivalent to circular convolution as you point out. For linear convolution, in convolving 2 images (2D signals) A*B the full output will be of size Ma+Mb-1 x Na+Nb-1, where Ma x Na, Mb x Nb the sizes of images A and B resp.

After padding to the expected size, multiplying and transforming back, via ifft2, you can keep the central part of the resulting image (usually corresponding to the largest one of A and B).

A = double(imread('cameraman.tif'))./255; % image
B = fspecial('gaussian', [15 15], 2); % some 2D filter function

[m,n] = size(A);
[mb,nb] = size(B); 
% output size 
mm = m + mb - 1;
nn = n + nb - 1;

% pad, multiply and transform back
C = ifft2(fft2(A,mm,nn).* fft2(B,mm,nn));

% padding constants (for output of size == size(A))
padC_m = ceil((mb-1)./2);
padC_n = ceil((nb-1)./2);

% frequency-domain convolution result
D = C(padC_m+1:m+padC_m, padC_n+1:n+padC_n); 
figure; imshow(D,[]);

Now, compare the above with doing spatial-domain convolution, using conv2D

 % space-domain convolution result
 F = conv2(A,B,'same');
 figure; imshow(F,[]);

Results are visually the same, and total error between the two (due to rounding) on the order of e-10.

Answer 2

I created a MATLAB function which is basically conv2() in Frequency Domain:

function [ mO ] = ImageConvFrequencyDomain( mI, mH, convShape )
% ----------------------------------------------------------------------------------------------- %
% [ mO ] = ImageConvFrequencyDomain( mI, mH, convShape )
% Applies Image Convolution in the Frequency Domain.
% Input:
%   - mI                -   Input Image.
%                           Structure: Matrix.
%                           Type: 'Single' / 'Double' (Single Channel).
%                           Range: (-inf, inf).
%   - mH                -   Filtering Kernel.
%                           Structure: Matrix.
%                           Type: 'Single' / 'Double'.
%                           Range: (-inf, inf).
%   - convShape         -   Convolution Shape.
%                           Sets the convolution shape.
%                           Structure: Scalar.
%                           Type: 'Single' / 'Double'.
%                           Range: {1, 2, 3}.
% Output:
%   - mI                -   Output Image.
%                           Structure: Matrix (Single Channel).
%                           Type: 'Single' / 'Double'.
%                           Range: (-inf, inf).
% References:
%   1.  MATLAB's 'conv2()' - https://www.mathworks.com/help/matlab/ref/conv2.html.
% Remarks:
%   1.  A
% TODO:
%   1.  
%   Release Notes:
%   -   1.0.000     29/04/2021  Royi Avital     [email protected]
%       *   First release version.
% ----------------------------------------------------------------------------------------------- %

CONV_SHAPE_FULL     = 1;
CONV_SHAPE_SAME     = 2;
CONV_SHAPE_VALID    = 3;

numRows     = size(mI, 1);
numCols     = size(mI, 2);

numRowsKernel = size(mH, 1);
numColsKernel = size(mH, 2);

switch(convShape)
    case(CONV_SHAPE_FULL)
        numRowsFft  = numRows + numRowsKernel - 1;
        numColsFft  = numCols + numColsKernel - 1;
        firstRowIdx = 1;
        firstColIdx = 1;
        lastRowIdx  = numRowsFft;
        lastColdIdx = numColsFft;
    case(CONV_SHAPE_SAME)
        numRowsFft  = numRows + numRowsKernel;
        numColsFft  = numCols + numColsKernel;
        firstRowIdx = ceil((numRowsKernel + 1) / 2);
        firstColIdx = ceil((numColsKernel + 1) / 2);
        lastRowIdx  = firstRowIdx + numRows - 1;
        lastColdIdx = firstColIdx + numCols - 1;
    case(CONV_SHAPE_VALID)
        numRowsFft = numRows;
        numColsFft = numCols;
        firstRowIdx = numRowsKernel;
        firstColIdx = numColsKernel;
        % The Kernel when transformed is shifted (Namely its (0, 0) is top
        % left not middle).
        lastRowIdx  = numRowsFft;
        lastColdIdx = numColsFft;
end

mO = ifft2(fft2(mI, numRowsFft, numColsFft) .* fft2(mH, numRowsFft, numColsFft), 'symmetric');
mO = mO(firstRowIdx:lastRowIdx, firstColIdx:lastColdIdx);


end

It is fully compatible and validated.
The full code is available on my StackExchange Signal Processing Q74803 GitHub Repository.