Need help with implementing a 2D elliptical Gaussian function

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Ronni
Ronni 2011-7-5
评论： MJ HL 2017-8-22
I'm trying to implement a 2D gaussian function, which has an elliptical shape rather than circular. For example if I have a standard Gaussian fuction such as: f=a1*exp(-(((x-b1)/c1).^2+((y-b1)/c1).^2))
where [x,y]=meshgrid(xmin:spacing:xmax,ymin:spacing,yman); I can just f=a1*exp(-((r-b1)/c).^2) where r=sqrt(x^2+y^2);
however if my c1 is not the same, I will have an ellipse. I'm not sure how to implement this. Any help will be highly appreciated.

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回答（2 个）

David Young
David Young 2011-7-5
You can make an elliptical filter aligned with the axes by replacing the second occurrence of c1 by a different variable, say c2, and giving c1 and c2 different values. Here's some code that demonstrates what I mean; I've taken the opportunity to change b1 to b2 as well, so the centre of the filter can be moved to an arbitrary point:
% Set up mesh
xmin = -100;
xmax = 100;
ymin = -100;
ymax = 100;
spacing = 1;
xvals = xmin:spacing:xmax+spacing/2;
yvals = ymin:spacing:ymax+spacing/2;
[x,y] = meshgrid(xvals, yvals);
% parameters for the gaussian
a1 = 1;
b1 = 20;
b2 = 40;
c1 = 10;
c2 = 40;
% Compute the filter and display it
f=a1.*exp(-(((x-b1)./c1).^2+((y-b2)./c2).^2));
contour(x, y, f);
If you want the ellipse to be oriented in an arbitrary direction, you need to rotate the axes before the computation. This involves multiplying x and y by a rotation matrix. Please say if you also need help with this.
  1 个评论
MJ HL
MJ HL 2017-8-22
Hello David, ... As you guessed I'm from that kind of people that need more help in rotating this filter :) . How should I do this? I need to rotate this filter in an arbitrary direction... thanks

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Ronni
Ronni 2011-7-5
Thanks David. I tried that and it seems to be working. I also had something similar what the problem that I had was that my function was sum of multiple functions. That I still can't get it working. Hopefully I can explain here what I mean.
I have this 2D data, which looks like a combination of gaussians. So since it was centered around zero, to fit this 2D data, I just took 1D profile across the center and fitted it with just using x variable. I assumed I can use the same parameters for y since for my initial test it was just a circular distribution. Later, I will be tweeking it so the FWHM of the added of function of one side is longer than the other. Hence, it will turn into an elliptical multi-gaussian function rather than just a circular mult-gaussian function.
This is what I have written, but the contour looks weird:
xgrid=-2:0.05:2;
ygrid=-2:0.05:2;
[x,y]=meshgrid(xgrid,ygrid);
f=zeros(size(x));
a1 = 57.27;
b1 = 0.5494;
c1 = 0.3432;
a2 = 58.35;
b2 = -0.5444;
c2 = 0.3461;
a3 = 33.19;
b3 = 0.8118;
c3 = 0.2174 ;
a4 = 33.38 ;
b4 = -0.8114;
c4 = 0.2188;
a5 = 90.86;
b5 = 0.005504;
c5 = 0.5079;
b_r = -10.81;
a = 9.4e+005;
b_l = 10.81;
f_l_35= a*exp(b_l*x+b_l*y);
f_m_35= a1*exp(-(((x-b1)/c1).^2+((y-b1)/c1).^2)) ...
+ a2*exp(-(((x-b2)/c2).^2+((y-b2)/c2).^2)) ...
+ a3*exp(-(((x-b3)/c3).^2+((y-b3)/c3).^2)) ...
+ a4*exp(-(((x-b4)/c4).^2+((y-b4)/c4).^2))...
+ a5*exp(-(((x-b5)/c5).^2+((y-b5)/c5).^2));
f_r_35= a*exp(b_r*x+b_r*y);
%I'm fitting the tails to the exponential function.
f(find(x<=-1))= f_l_35(find(x<=-1));
f(find(y<=-1))= f_l_35(find(y<=-1));
%Middle part to mult-gaussian
f(find(x<1 & x>-1))= f_m_35(find(x<1 & x>-1));
f(find(y<1 & y>-1))= f_m_35(find(y<1 & y>-1));
%again my right tail to exponential
f(find(x>=1))= f_r_35(find(x>=1));
f(find(y>=1))= f_r_35(find(y>=1));
Again, I got these coefficients by fitting central 1D profile. I kind of know why the distribution looks weird. I think when I did 1D profile, I did it at y=0 and so I also assumed b1_y (written as b1) to be zero. When I plot in 2D, it's shifting the centeral profile I fitted by b1/b2/b3 etc.
The distribution should really be looking like
f_m_35_revised= a1*exp(-(((r-b1)/c1).^2)) ...
+ a2*exp(-(((r-b2)/c2).^2))...
+ a3*exp(-(((r-b3)/c3).^2))...
+ a4*exp(-(((r-b4)/c4).^2))...
+ a5*exp(-(((r-b5)/c5).^2));
where r = sqrt(x.^2+y.^2);
Since I know my data is center about zero. May be I will change my fitting model and change the b parameters to 0. so more like f_m_35(x)= a1*exp(-(((x)/c1).^2)) ... + a2*exp(-(((x)/c2).^2))... + a3*exp(-(((x)/c3).^2))... + a4*exp(-(((x)/c4).^2))... + a5*exp(-(((x)/c5).^2)); I'll see how that goes...

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