最小二乘法拟合散点组成的椭圆曲线

一次图像处理的作业：

用最小二乘法拟合散点集所组成的椭圆：

function ellipse = fitellipse(X,Y)
if nargin == 0
  % Read the original picture
  x_ori = [];
  y_ori = [];
  f=imread('test2.png');
  % Convert the picture to gray scale
  f = rgb2gray(f);
  [m,n]=size(f);
  for i=1:m
	for j=1:n
		if(f(i,j)>200)
			x_ori=[x_ori,j];
			y_ori=[y_ori,i];
		end
	end
  end
  % Plot
  clf
  plot(x_ori,y_ori,'o');set(gca,'YDir','reverse');axis([1,n,1,m])
  figure(gcf)
  params = fitellipse(x_ori,y_ori);
  
  Returned = round(params.*[1 1 1 1 180])
  % Show the ellipse
  t = linspace(0,pi*2);
  x = params(3) * cos(t);
  y = params(4) * sin(t);
  x_ori = x*cos(params(5))-y*sin(params(5)) + params(1); 
  y_ori = x*sin(params(5))+y*cos(params(5)) + params(2);
  hold on
  plot(x_ori,y_ori,'r-')
  
  return
end
% Normalize the matrix
mx = mean(X);
my = mean(Y);
sx = (max(X)-min(X))/2;
sy = (max(Y)-min(Y))/2; 

x = (X-mx)/sx;
y = (Y-my)/sy;
x = x(:);
y = y(:);
% Create matrix D
D = [ x.*x  x.*y  y.*y  x  y  ones(size(x)) ];
% Create matrix S = Trans(D)*D
S = D'*D;
% Scatter coefficient matrix
C(6,6) = 0; C(1,3) = -2; C(2,2) = 1; C(3,1) = -2;

% Calculate eigenvalue
[gevec, geval] = eig(S,C);

% Find the eigenValue
invValue = find(real(diag(geval)) < 1e-8 & ~isinf(diag(geval)));
  
% Calculate the EigenVector
A = real(gevec(:,invValue));

% Revert the normalization
par = [
  A(1)*sy*sy,   ...
      A(2)*sx*sy,   ...
      A(3)*sx*sx,   ...
      -2*A(1)*sy*sy*mx - A(2)*sx*sy*my + A(4)*sx*sy*sy,   ...
      -A(2)*sx*sy*mx - 2*A(3)*sx*sx*my + A(5)*sx*sx*sy,   ...
      A(1)*sy*sy*mx*mx + A(2)*sx*sy*mx*my + A(3)*sx*sx*my*my   ...
      - A(4)*sx*sy*sy*mx - A(5)*sx*sx*sy*my   ...
      + A(6)*sx*sx*sy*sy   ...
      ]';
	  
% Calculate the parameters of the ellipse
thetarad = 0.5*atan2(par(2),par(1) - par(3));
a=par(1)/par(6);
b=par(2)/par(6);
c=par(3)/par(6);
d=par(4)/par(6);
e=par(5)/par(6);
f=1;
valueTemp = 4*a*c-b*b;
xCentre = (b*e-2*c*d)/valueTemp;
yCentre = (b*d-2*a*e)/valueTemp;

temp1 = 2*(a*xCentre*xCentre + c*yCentre*yCentre + b*xCentre*yCentre - 1);
temp2 = sqrt((a-c)*(a-c) + b*b);
Ru = sqrt(temp1/(a+c-temp2));
Rv = sqrt(temp1/(a+c+temp2));
ellipse = [xCentre, yCentre, Ru, Rv, thetarad];