Correlation coefficient in 3D

I need to calculate the correlation between two 3D volumes. I've tried implementing Pearson's coefficient since it's well recognised in the literature but it only seems to spit out meaningless answers (see code snippet below):

    A = volume1 - mean(volume1, 'all');
    B = volume2 - mean(volume2, 'all');
    
    num = sum(A  .* B, 'all');
    den1 = sum(A.^2, 'all');
    den2 = sum(B.^2, 'all');
    r = num ./ (den1 .* den2)^0.5;

Is there something I'm doing wrong here? What alternative correlations metrics might work in this context?

Answer 1

[the cyclist]

Your formulas for den1 and den2 are not symmetric. It looks like den2 does not correspond to the formula in the screenshot, and should instead be

den2 = sum(B.^2, 'all');

[Alexander Collins]

Ah that is just a typo in my question - I don't think this is the issue.

[the cyclist]

OK. In that case, can you upload your data, or a sample that exhibits the problem? (Please upload the data in a MAT file, using the paper clip icon, not just an image.)

Also, please be more specific about what you mean by "meaningless" answers.

[Alexander Collins]

No problem. The correlation coefficient should =+1 for perfectly correlated data (i.e. identical volumes/images), 0 for completely uncorrelated data, and -1 for perfectly anti-correlated data.

I found that with this naive implementation I would get coefficients wildly outside this range.

Really my question was trying to get more at the theoretical side of things than the way I've written the code above - what's the best way to find a correlation coefficient between two 3D matrices?

[the cyclist]

Again, it would be useful if you uploaded your data, or small sample that shows a value outside the range [-1,1].

That is the correct formula for the Pearson correlation coefficient, and you have coded it correctly. So it must be something in the data. Did you convert the data to column vectors, so that volume1 and volume2 both column vectors? If not, then I'm guessing your formula is doing an implicit expansion that you are not expecting, which means the formula is not calculating what you want.