Please explain difference in results between these variance algorithms which should be null?

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

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

conv1=zeros(nrows,ncols,nzs);
conv2=zeros(nrows,ncols,nzs);
varIm=zeros(nrows,ncols,nzs);
En=zeros(nrows,ncols,nzs);
Mn=zeros(nrows,ncols,nzs);
Vn=zeros(nrows,ncols,nzs);

gImgSq=mat2gray(Img.^2);
gImg=mat2gray(Img);

%Algorithm #1

conv1 = imfilter(gImgSq, h/n1 , 'symmetric');
conv2 = imfilter(gImg, h, 'symmetric').^2 / (n*n1);
VarIm = (max((conv1 - conv2),0));

%Algorithm #2

for i=1:1:nzs;
En(:,:,i)=filter2(h, gImgSq(:,:,i)) / n;
end

for i=1:1:nzs;
Mn(:,:,i)=filter2(h, gImg(:,:,i)) / n;
end

Vn = n/n1.*(En-Mn.^2);

%Test the difference

Test= VarIm - Vn;

Answer 1

[Image Analyst]

Look up what mat2gray() does and I think you'll see the answer. If not, ask.

[Eric Diaz]

Not sure. Are you trying to say that it has something to do with the double type? Want to throw me another bone here... I would prefer just an explanation as opposed to this vagueness

[Image Analyst]

mat2gray says "The returned matrix I contains values in the range 0.0 (black) to 1.0 (full intensity or white)" So if you put both the image and the image squared through that function, it will normalize both to 0-1, which destroys the relationship. Let's say the max value was 200, and so the max squared value will be 4000. But you put them both through mat2gray, and now that location has a value of 1 for both the image and the image squared. Here, run this illustration to see how the relative relationship changes:

workspace
x=1:20;
y1 = x;
y2 = x.^2;
subplot(2,1,1);
plot(x,y1, x,y2, 'LineWidth', 3);
grid on;
title('No mat2gray');

ym1 = mat2gray(x);
ym2 = mat2gray(x.^2);
subplot(2,1,2);
plot(x,ym1, x,ym2, 'LineWidth', 3);
grid on;
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Give a name to the title bar.
set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off') 
title('With mat2gray');

[Eric Diaz]

Oh, I see what you are saying. Thank you for the clarification.

I'm wondering though whether in the end that would cause the problem I am getting. The reason that I'm wondering that is because both algorithms square the same variables, convert to grayscale and take the differences. Does this come down to a significant digits and/or catastrophic cancellation problem?