Please explain difference in image variance algorithms ... E-10 for all tests

Can someone please explain why these different algorithms are giving slightly different results for the variance of an image?

%References:

• http://en.wikipedia.org/wiki/Variance#Formulae_for_the_variance
• http://www.mathworks.com/help/images/ref/stdfilt.html
• http://mathworld.wolfram.com/Variance.html

% Load images and get basic parameters

% You may need to load your own image or one of Matlab's demo images, like cameraman

load('img.mat')

nrows = size(img,1);
ncols = size(img,2);
nzs   = size(img,3);

h  = ones(7);
n  = sum(h(:));
n1 = n - 1;

%Preinitialize

M2a    = zeros(nrows,ncols,nzs);
M1a    = zeros(nrows,ncols,nzs);
varImA = zeros(nrows,ncols,nzs);

M2b    = zeros(nrows,ncols,nzs);
M1b    = zeros(nrows,ncols,nzs);
varImB = zeros(nrows,ncols,nzs);

M2c    = zeros(nrows,ncols,nzs);
M1c    = zeros(nrows,ncols,nzs);
varImC = zeros(nrows,ncols,nzs);

imgSq  = img.^2;

%Algorithm #1

M2a    = imfilter(imgSq, h/n1 , 'symmetric');
M1a    = imfilter(img, h, 'symmetric').^2 / (n*n1);
varImA = (max((M2a - M1a),0));

%Algorithm #2

M2b    = imfilter(imgSq, h/n1 , 'symmetric');
M1b    = imfilter(img, h/n, 'symmetric');
M1S    = (M1b.^2)*n/n1;
varImB = max((M2b - M1S),0);

%Algorithm #3

for i=1:1:nzs;
    M2c(:,:,i) = filter2(h, imgSq(:,:,i)) / n;
end

for i=1:1:nzs;
    M1c(:,:,i) = filter2(h, img(:,:,i)) / n;
end

varImC = n/n1.*(M2c-M1c.^2);

% Test the difference (need to crop image borders to get relevant results)

Test1 = varImA(20:end-20,20:end-20,:) - varImB(20:end-20,20:end-20,:);
Test2 = varImA(20:end-20,20:end-20,:) - varImC(20:end-20,20:end-20,:);
Test3 = varImB(20:end-20,20:end-20,:) - varImC(20:end-20,20:end-20,:);

% Display results;

max(Test1(:))
min(Test1(:))
max(Test2(:))
min(Test2(:))
max(Test3(:))
min(Test3(:))

[Jan]

Loading cameraman.tif replies an integer matrix, such that it sufficient as test data. There are some typos in "VarImB" and "VarImC", which both need a lowercase "v".

Answer 1

[Jan]

When I use img = rand(256, 256, 3) as input data and fix the typos, I get differences of the magnitude 6.6613e-16. This is 3*eps, which means, that the differences are very tiny. Such differences are caused by the limited precision of the representation of numbers. See FAQ: Why is 0.3-0.2-0.1 not equal to 0.

[Eric Diaz]

That's kind of what I was thinking.

I'm wondering why I am getting differences on the order of E-10 with my image data, which is imported as UINT12 --> double.