最小二乘法拟合椭圆设平面任意位置椭圆方程为：x2+Axy+By2+Cx+Dy+E=0设Pi(xi,yi)(i=1,2,…,N)为椭圆轮廓上的N(N≥5)个测量点，依据最小二乘原理，所拟合的目标函数为：F(A,B,C,D,E)=∑i=1N(xi2+Axiyi+Byi2+Cxi+Dyi+E)2欲使F为最小，需使∂F∂A=∂F∂B=∂F∂C=∂F∂D=∂F∂E=0由此可以得方程：[∑❑❑xi2yi2∑❑❑xiyi3∑❑❑xi2yi∑❑❑xiyi2∑❑❑xiyi∑❑❑xiyi3∑❑❑yi4∑❑❑xiyi2∑❑❑yi3∑❑❑yi2∑❑❑xi2yi∑❑❑xiyi2∑❑❑xi3∑❑❑xiyi∑❑❑xi∑❑❑xiyi2∑❑❑yi3∑❑❑xiyi∑❑❑yi2∑❑❑yi2∑❑❑xiyi∑❑❑yi2∑❑❑xi∑❑❑yiN][ABCDE]=-[∑❑❑xi3yi∑❑❑xi2yi2∑❑❑xi3∑❑❑xi2yi∑❑❑xi2]解方程可以得到A，B，C，D，E的值。根据椭圆的几何知识，可以计算出椭圆的五个参数：位置参数(θ,x0,y0)以及形状参数(a,b)。x0=2BC−ADA2−4By0=2D−ADA2−4Ba=❑√2(ACD−BC2−D2+4BE−A2E)(A2−4B)(B−❑√A2+(1−B2)+1)b=❑√2(ACD−BC2−D2+4BE−A2E)(A2−4B)(B+❑√A2+(1−B2)+1)θ=tan−1❑√a2−b2Ba2B−b2附：MATLAB程序function[semimajor_axis,semiminor_axis,x0,y0,phi]=ellipse_fit(x,y)%%Input:%x——avectorofxmeasurements%y——avectorofymeasurements%%Output:%semimajor_axis——Magnitudeofellipselongeraxis%semiminor_axis——Magnitudeofellipseshorteraxis%x0——xcoordinateofellipsecenter%y0——ycoordinateofellipsecenter%phi——Angleofrotationinradianswithrespecttox-axis%%explain%2*b'*x*y+c'*y^2+2*d'*x+2*f'*y+g'=-x^2%M*p=bM=[2*x*yy^22*x2*yones(size(x))],%p=[bcdefg]b=-x^2.%p=pseudoinverse(M)*b.x=x(:);y=y(:);%ConstructMM=[2*x.*yy.^22*x2*yones(size(x))];%Multiply(-X.^2)bypseudoinverse(M)e=M\(-x.^2);%Extractparametersfromvectorea=1;b=e(1);c=e(2);d=e(3);f=e(4);g=e(5);%UseFormulasfromMathworldtofindsemimajor_axis,semiminor_axis,x0,y0,andphidelta=b^2-a*c;x0=(c*d-b*f)/delta;y0=(a*f-b*d)/delta;phi=0.5*acot((c-a)/(2*b));nom=2*(a*f^2+c*d^2+g*b^2-2*b*d*f-a*c*g);s=sqrt(1+(4*b^2)/(a-c)^2);a_prime=sqrt(nom/(delta*((c-a)*s-(c+a))));b_prime=sqrt(nom/(delta*((a-c)*s-(c+a))));semimajor_axis=max(a_prime,b_prime);semiminor_axis=min(a_prime,b_prime);if(a_prime