# Normalized cross-correlation function

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Manolis Michailidis on 18 Sep 2015
Commented: devi sowjanya on 6 Mar 2021
Hello, i am trying to write a normilized cross-correlation method function , but i can't complete it. Here are the details of the formula :
My code is
x=rand(1,1000);
N=length(x); %// length of signal
n1=128; %// length of window
xf=framing(x,n1,n1/2,rectwin(n1)); %this function frames the signal i will get xf(128,14)
win_num=size(xf,2);
for col=1:win_num
for m=1:n1+1
for n=1:n1-m
ccor(m,col)=sum(xf(n+m,col)*(xf(n,col)))/ sqrt(sum(xf(n)^2)*sum(xf(n+m)^2) );
end
end
end
Note that i want to see the correlation values betwen -1 and 1 like in the figure but instead i get
.
devi sowjanya on 6 Mar 2021
Can you please provide the framing function code

Kirby Fears on 21 Sep 2015
Edited: Kirby Fears on 21 Sep 2015
Hi Manolis,
To directly replicate what the equations say (shifting indices with +1), the calculation would look more like this:
x=rand(1,1000);
N=length(x); %// length of signal
n1=128; %// length of window
xf=framing(x,n1,n1/2,rectwin(n1)); %this function frames the signal i will get xf(128,14)
win_num=size(xf,2);
% M0=???
ccor=NaN(M0+1,win_num);
for col=1:win_num
for m=1:M0+1,
[term1,term2,term3]=deal(0);
for n=1:N-m,
term1=term1+xf(n,col)*xf(n+m,col);
term2=term2+xf(n,col)^2;
term3=term3+xf(n+m,col)^2;
end
ccor(m,col)=term1/sqrt(term2*term3);
end
end
You would just need to explicitly specify M0. This calculation can be simplified as follows:
x=rand(1,1000);
N=length(x); %// length of signal
n1=128; %// length of window
xf=framing(x,n1,n1/2,rectwin(n1)); %this function frames the signal i will get xf(128,14)
win_num=size(xf,2);
% M0=???
ccor=NaN(M0+1,win_num);
for col=1:win_num
for m=1:M0+1,
term1=sum(xf(1:N-m,col).*xf(1+m:N,col));
term2=sum(xf(1:N-m,col).^2);
term3=sum(xf(1+m:N,col).^2);
ccor(m,col)=term1/sqrt(term2*term3);
end
end
I don't have the toolbox required to run this code, so it may have bugs. Hope this helps.
Manolis Michailidis on 29 Sep 2015
Thank you i finally solved it, for M0=n1-1 shifts i had to replace in the loops N with n1 and it works. I get the results i want , from the start i wanted the correlation to be biased it the region [-1 1] and that's what i get. So thanks a lot for your effort.

Kirby Fears on 18 Sep 2015
Edited: Kirby Fears on 18 Sep 2015
Hi Manolis,
You are looping n and m by starting at 1, but in the formula you posted the sum starts at 0. You will need to correct your calculations to include the 0 terms in the summation. You should also consider pre-allocating "ccor" to save yourself some computational time.
You may have access to the xcorr function which calculates cross correlations. It has several options to control normalization.
Hope this helps.
##### 2 CommentsShowHide 1 older comment
Kirby Fears on 21 Sep 2015
Indeed the loop variables are only used as indices, so you are correct. My mistake. I assumed the loop variables were also used as numeric values in the summand. I'm writing another answer below.

Image Analyst on 29 Sep 2015
Do you have the Image Processing Toolbox? If so, why not just use the built-in function normxcorr2()?
Manolis Michailidis on 29 Sep 2015
Yes i have it, i was just curious and wanted to confirm the result with the matlabs, anyway thank you i will take a look at the function you mentioned. It's always nice to learn something new.