Cross-correlation in frequency domain and xcorr2 in MATLAB

What are the reasons for the differences between my frequency domain cross-correlation results and those obtained using the xcorr2() function in MATLAB? Is it possible that xcorr2() performs spatial domain cross-correlation, and how might this affect the results?

img1 = imread('1.jpg');
img2 = imread('2.jpg');

img1_gray = rgb2gray(img1);
img2_gray = rgb2gray(img2);

[m, n] = size(img1_gray);
[p, q] = size(img2_gray);

fft_img1 = fft2(img1_gray, m + p - 1, n + q - 1);
fft_img2 = fft2(img2_gray, m + p - 1, n + q - 1);

cross_correlation = (fft_img1 .* conj(fft_img2));
cross_correlation = ifft2(cross_correlation);
cross_correlation = fftshift(cross_correlation);

cross_correlation_xcorr2 = xcorr2(img2_gray, img1_gray);

figure;
subplot(2, 1, 1);
imshow(cross_correlation, []);
title('Custom Cross-Correlation');
subplot(2, 1, 2);
imshow(cross_correlation_xcorr2, []);
title('xcorr2');

Results:

[Paul]

Can you add 1.jpg and 2.jpg to your post using the Paperclip icon on the Insert menu?

[Shy]

Sure.

Answer 1

[Paul]

Hi Shy,

Assuming that the output of the xcorr2 is the expected result, it can be obtained with the changes below

img1 = imread('1.jpg');
img2 = imread('2.jpg');
img1_gray = rgb2gray(img1);
img2_gray = rgb2gray(img2);

[m, n] = size(img1_gray);
[p, q] = size(img2_gray);

%fft_img1 = fft2(img1_gray, m + p - 1, n + q - 1);
fft_img1 = fft2(flipud(fliplr(img1_gray)), m + p - 1, n + q - 1);
fft_img2 = fft2(img2_gray, m + p - 1, n + q - 1);

%cross_correlation = (fft_img1 .* conj(fft_img2));
cross_correlation = (fft_img1 .* fft_img2);
cross_correlation = ifft2(cross_correlation);
%cross_correlation = fftshift(cross_correlation);

cross_correlation_xcorr2 = xcorr2(img2_gray, img1_gray);


figure;
subplot(2, 1, 1);
imshow(cross_correlation, []);
title('Custom Cross-Correlation');
subplot(2, 1, 2);
imshow(cross_correlation_xcorr2, []);
title('xcorr2');

[Shy]

It turns out that we are not using a complex conjugate matrix here? Then how does the algorithm work? I thought it was necessary to multiply the matrix of the first image and the complex conjugate matrix of the second image.

[Paul]

As I undersand it, xcorr2(x,y) is the the same as conv2 of x and the 2D-reversed conjugate of y. But in this case, img1_gray is real, so its conjugate is itself. With that understanding, I just made the fft stuff implement the convolution of img2 and the 2D-reverse of img1 (because img1 is the second argument to xcorr2).

[Shy]

Okay, I get it. Thanks