MATLAB: Least Squares Fitting Method with a Circle

circlefittingImage Processing Toolboxleast squaresMATLABpolar plotrandom number generator

Hi everyone,
Forgive me, I am no expert at MATLAB. I would appreciate it greatly if someone could explain to me the method of nonlinear least squares and how to fit it with a circle of random points. Here's what I have so far:
% radius information-------------------------------------------------------
r = 10;                 % constant radius
Rc = @(x,y,z) (x.^2 + y.^2 + z.^2).^(1/2);      % radius as a function
n = rand(1000,3);       % generate random points on nominal circle
xc = n(:,1);
yc = n(:,2);            % separate into coordinates
zc = n(:,3);
%--------------------------------------------------------------------------



% angular information------------------------------------------------------
theta = linspace(0,2*pi,1000);      % plot bounds
theta = theta';
%--------------------------------------------------------------------------
% generate random radii----------------------------------------------------
R = Rc(xc,yc,zc);
R(:,2) = R(:,1);       % same radii in all directions
R(:,3) = R(:,1);
p = n + (r/100)*R;     % generate plot points around the circle
figure(1)
polar(theta,Rc(xc,yc,zc))
%--------------------------------------------------------------------------
% Least Squares fitting----------------------------------------------------
e = dist(p,R');
perf = sse(e);
lsqrfit = lsqr(e,Rc(xc,yc,zc),[],1000);
%--------------------------------------------------------------------------
So from what I understand, the residuals (errors) need to be found and then the sum of squares of the residuals need to be found. After that I am not sure where to go. Thanks for all the help.
Ian

Best Answer

Maybe this will help, it is adapted from "measuring the radius of a roll of tape" demo in the image processing toolbox:
r = 5+.5*rand(100,1);         %100 random radii
ang = linspace(0,2*pi,100)';  % angles
x = r.*cos(ang)+2; y = r.*sin(ang)+3; % x,y coordinates centered [2,3]
plot(x,y,'+')
axis equal; hold on;
c = [x y ones(length(x),1)]\-(x.^2+y.^2); %least squares fit
xhat = -c(1)/2;
yhat = -c(2)/2;
rhat = sqrt(xhat^2+yhat^2-c(3));
plot(rhat*cos(ang)+xhat,rhat*sin(ang)+yhat,...
    'g','linewidth',2) %best fit circle
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