Physics 114: Lecture 19
Least Squares Fit to
2D Data
Dale E. Gary
NJIT Physics Department
Nonlinear Least Squares Fitting
We saw that for a function y(x) the chi-square is
2


y

y
(
x
)
2   i


i


which can be used to fit a curve to data points yi.
 In just the same way, we can easily extend this idea to fitting a function
z(x,y), that is, a surface, to a 2D data set of measurements zi(x,y).
2
 z  z ( x, y ) 
2   i

2
i
 x b 




c
 You know that it takes three parameters to fit a Gaussian curve y ( x)  ae   .
 To extend this to a 2D Gaussian is straightforward. It is:

2
z ( x, y)  ae

 x b1   y b2 

 

 c1   c2 
2
.
Note that this has 5 parameters, but the most general way to describe a
2D Gaussian actually has 6 parameters, which you will do for homework.
Apr 19, 2010
Plotting a 2D Gaussian in MatLAB
Let’s introduce some useful MatLAB commands for creating a 2D Gaussian.
2D Gaussian with 5 Parameters
 One can use a brute force method such as
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z = zeros(101,101);
a = 2.5; bx = 2.3; by = -1.2; cx = 1.5; cy = 0.8;
i = 0;
for x = -5:0.1:5
i = i + 1;
j = 0;
for y = -5:0.1:5
j = j + 1;
z(i,j) = a*exp(-((x-bx)/cx)^2 – ((y-by)/cy)^2);
end
end
imagesc(-5:0.1:5,-5:0.1:5,z)
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y
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Or, we can plot it as a surface to provide x and y coordinates
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x
surf(-5:0.1:5,-5:0.1:5,z)
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Better Way of Creating 2D Functions

That way of using for loops is a very slow, so here is another way of doing
it for “round” Gaussians:
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x = -5:0.1:5;
y = -5:0.1:5;
siz = size(x); siz = siz(2);
a = 2.5; bx = 2.3; by = -1.2; c = 0.8;
dist = zeros(siz,siz);
for i=1:siz
for j=1:siz
dist(i,j) = x(i)^2 + y(j)^2; % Create “distance” array once
end
end
% use the next two statements multiple times for different Gaussians
sdist = circshift(dist,round([bx/0.1,by/0.1])); % Shifted “distance” array
z = a*exp(-sdist/c);
imagesc(x,y,z);
Apr 19, 2010
Best Way of Creating 2D Functions

The best way, however, is to use the meshgrid() MatLAB function:
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[x, y] = meshgrid(-5:.1:5, -5:.1:5);
a = 2.5; bx = 2.3; by = -1.2; cx = 1.5; cy = 0.8;
z = a*exp( - (((x-bx)/cx).^2 + ((y-by)/cy).^2) ;
surf(x,y,z);
This is completely general, and allows vector calculations which are very
fast.
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 Here is another example, the sinc() function:
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z = a*sinc(sqrt(((x-bx)/cx).^2 + ((y-by)/cy).^2));
2
You can see that the meshgrid() function is 0
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very powerful as a shortcut for 2D function 5
calculation.
 To add noise, just add sig*randn(size(z)),
where sig is the standard deviation of the noise.
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0
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Apr 19, 2010
Searching Parameter Space in 2D
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Just as we did in the 1D case, searching parameter space is just a trial and
error exercise.
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% Initial values around which to search
a0 = max(max(z));
[ix iy] = find(z == a0);
bx0 = x(ix,iy); by0 = y(ix,iy); cx0 = 1; cy0 = 1;
astep = 0.05; bstep = 0.05; cstep = 0.05;
chisqmin = 1e20;
for a = a0-5*astep:astep:a0+5*astep
for bx = bx0-5*bstep:bstep:bx0+5*bstep
for by = by0-5*bstep:bstep:by0+5*bstep
for cx = cx0-5*cstep:cstep:cx0+5*cstep
for cy = cy0-5*cstep:cstep:cy0+5*cstep
ztest = a*exp(-(((x-bx)/cx).^2 + ((y-by)/cy).^2));
chisq = sum(sum(((ztest-z)/sig).^2));
if (chisq < chisqmin); chisqmin = chisq; params = [a,bx,by,cx,cy]; end
end
end
end
params = 2.1465 2.3000 -1.2000 1.2500
end
end
0.8000
Apr 19, 2010
The Result Looks Good, but…
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To calculate the result of the fit, just type
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ztest = params(1)*exp(-(((x-params(2))/params(4)).^2 + ((y-params(3))/params(5)).^2));
imagesc(x(1,:),y(:,1),ztest)
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This gives a reasonable result, but compare
the fit parameters with the (known) params:
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a = 2.1; bx = 2.3; by = -1.2; cx = 1.3; cy = 0.8;
params = 2.15 2.3 -1.2 1.25 0.8
The value of cx is especially far off, because
our search window was limited to 0.75-1.25,
so it could not reach 1.3.
 Look at the residuals:
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imagesc(x(1,:),y(:,1),ztest-z)
Can you see a pattern in the noise?
 To bring out the pattern better, let’s introduce convolution.
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Apr 19, 2010
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1D Convolution and Smoothing
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Let’s create a noisy sine wave:
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u = -5:.1:5;
w = sin(u*pi)+0.5*randn(size(u));
plot(u,w)
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We can now smooth the data by convolving
it with a vector [1,1,1], which does a 3-point
running sum.
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wsm = conv(w,[1,1,1]);
whos wsm
Name
wsm
Size
1x103
Bytes Class
Attributes
824 double
sin(u)
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w

w
wsm
1.5
00
-0.5
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-1.5
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
0
u0
u
conv(w, k )  (w * k )(t )   w( )k (t   )d

Notice wsm is now of length 103. That means we cannot plot(u,wsm),
but we can plot(u,wsm(2:102)). Now we see another problem.
 Try this:
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plot(u,w)
hold on; plot(u,wsm(2:102)/3,’r’)
Apr 19, 2010
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2D Convolution

To do 2D smoothing, we will use a 2D kernel k = ones(3,3), and use the
conv2() function. So to smooth the residuals of our fit, we can use
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zsm = conv2(ztest-z,k)/9.;
imagesc(x,y,zsm(2:102,2:102))
Now we can see the effect of missing the
value of cx by 0.05 due to our limited
search range.
 There are other uses for convolution, such
as edge detection. For example, we can
convolve with a kernel k = [1,1,1,1,1,1].
 Or a kernel k = [1,1,1,1,1,1]’. Or even
a kernel k = eye(6). Or k = rot90(eye(6)).
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Apr 19, 2010
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