# 2D Gaussian Noise

• Oct 12th 2012, 10:03 PM
peeyman
2D Gaussian Noise
I generated a 2d Gaussian noise then modulated it with a 2d cosine (f(x,y)=cos(2*pi/N*a*x)) witch oscillate only with the x direction and "a" is the frequency.

In a nutshell
N: Gaussian Noise size N*N
P: f(x,y)=cos(2*pi/N*a*x) a: frequency
I=P.N

after i take the Fourier transform of I, frequency pairs at offset (u,v) and (u+2a,v) have correlated phase.

If i do the experiment with another noise for example uniform noise (in Matlab: rand(N,N)) and then modulate it with the same cosine there
is no such correlations.

I want to what property of Gaussian noise cause this correlations?

Thank u all for help in advance.
• Oct 12th 2012, 11:51 PM
chiro
Re: 2D Gaussian Noise
Hey peeyman.

First question is your f(x,y) is only a function of x: what is the y? Second question is the Noise determined by P in any way (i.e. is P a function of N or N a function of P in any way)?

If the second question is no, then the correlation should be 0 or at least if you test with a simulation, it should get close to zero is your sample size increases.
• Oct 13th 2012, 12:06 AM
peeyman
Re: 2D Gaussian Noise
but first question i wrote f(x,y)=cos(2*pi/N*a*x) means f(x,y)=cos(2*pi/n*a*x+2*pi/n*0*y) witch means that frequency of y is zero so it is 2D.

but the second question is about my mistake. I correct it as follow.
N: Gaussian Noise size n*n
P: f(x,y)=cos(2*pi/n*a*x) a: frequency
I=P.N

in the about "n" is size but N is noise witch means that P and N are not functions of each other.

sorry for the mistake.
I will be appreciate if have an answer for that.
• Oct 13th 2012, 12:09 AM
chiro
Re: 2D Gaussian Noise
There should be no correlation between the variables if they are independent.

Cov(X,Y) = 0 if X is independent from Y and it looks like there is a mistake if you get some kind of correlation theoretically when you derive it.

Maybe you are talking about a different kind of correlation though: what kind of correlation are you talking about?
• Oct 13th 2012, 12:13 AM
peeyman
Re: 2D Gaussian Noise
i don't talk about correlation between P and N.

I=P.N
I take the Fourier transform of I
FI=Fourier of I

then i say that frequency pair at offset (u,v) and (u+2a,v) in FI are correlated.
• Oct 13th 2012, 12:16 AM
chiro
Re: 2D Gaussian Noise
Can you show the calculation? Whether it's hand-written or computer generated (like in Maple/Mathematica)?
• Oct 13th 2012, 12:20 AM
peeyman
Re: 2D Gaussian Noise
hear is the code in matlab

%------------------------------------------------------------
clear all
clc
len=1000;
x=[0:len];
y=[0:len];
[X,Y]=meshgrid(x,y);
P=cos(4*2*pi/(len+1)*X);

N=wgn(len+1,len+1,0);

IF=fft2(I);
figure
imshow(angle(IF),[]);

ang1=angle(IF);

offset=8;

for i=1:len-offset
phase1(1,i)=ang1(1,i);
phase1(2,i)=ang1(1,i+offset);
end
plot(phase1(1,:),phase1(2,:),'ro');
corrcoef(phase1(1,:)',phase1(2,:)')
• Oct 13th 2012, 12:21 AM
peeyman
Re: 2D Gaussian Noise
%------------------------------------------------------------
clear all
clc
len=1000;
x=[0:len];
y=[0:len];
[X,Y]=meshgrid(x,y);
P=cos(4*2*pi/(len+1)*X);

N=wgn(len+1,len+1,0);

I=P.*N

IF=fft2(I);
figure
imshow(angle(IF),[]);

ang1=angle(IF);

offset=8;

for i=1:len-offset
phase1(1,i)=ang1(1,i);
phase1(2,i)=ang1(1,i+offset);
end
plot(phase1(1,,phase1(2,,'ro');
corrcoef(phase1(1,',phase1(2,')
• Oct 13th 2012, 02:07 AM
chiro
Re: 2D Gaussian Noise
I'm afraid I can't really give you a suggestion for why this is the case.

The only thing I could suggest is to look at the spectral attributes of the noise and see if there are any similarities or things that stick out with regard to your function.

Since you are modulating the two functions, the first thing that comes to mind is to look at some kind of convolution properties of your modulated result in a way to see how the spectral attributes of the noise are related to the that of the function and look for how this would affect the correlation component you observe.
• Oct 13th 2012, 02:18 AM
peeyman
Re: 2D Gaussian Noise