商求导公式最小二乘法拟合圆公式推导及matlab实现

2020年10月11日发(作者：常重胤)
2009-01-17 | 最小二乘法拟合圆公式推导及matlab实现

最小二乘法(least squares analysis)是一种数学优化技术，它通过
最小化误差的平方和找到一组数据的最佳函数匹配。最小二乘法是用最
简的方法求得一些绝对不可知的真值，而令误差平方之和为最小。 最
小二乘法通常用于曲线拟合 (least squares fitting) 。
这里有拟合圆曲线 的公式推导过程和vc实现。






matlab 实现：
function[R,A,B]=irc(x,y,N)
%x,y是平面点的坐标，N是点个数
%R是拟合半径，A，B是圆心的平面坐标
x1=0;
x2=0;
x3=0;
y1=0;
y2=0;
y3=0;
x1y1=0;
x1y2=0;
x2y1=0;
for i=1:N
x1=x1+x(i);
x2=x2+x(i)*x(i);
x3=x3+x(i)*x(i)*x(i);
y1=y1+y(i);
y2=y2+y(i)*y(i);
y3=y3+y(i)*y(i)*y(i);
x1y1=x1y1+x(i)*y(i);
x1y2=x1y2+x(i)*y(i)*y(i);
x2y1=x2y1+x(i)*x(i)*y(i);
end
C=N*x2-x1*x1;
D=N*x1y1-x1*y1;
E=N*x3+N*x1y2-(x2+y2)*x1;
G=N*y2-y1*y1;
H=N*x2y1+N*y3-(x2+y2)*y1;
a=(H*D-E*G)(C*G-D*D);
b=(H*C-E*D)(D*D-G*C);
c=-(a*x1+b*y1+x2+y2)N;
A=a(-2);
B=b(-2);
R=sqrt(a*a+b*b-4*c)2;







VC
void CViewActionImageTool::LeastSquaresFitting()
{
if (m_nNum<3)
{ return; }
int i=0;
double X1=0;
double Y1=0;
double X2=0;
double Y2=0;
double X3=0;
double Y3=0;
double X1Y1=0;
double X1Y2=0;
double X2Y1=0;
for (i=0;i {
X1 = X1 + m_points[i].x;
Y1 = Y1 + m_points[i].y;
X2 = X2 + m_points[i].x*m_points[i].x;
Y2 = Y2 + m_points[i].y*m_points[i].y;
X3 = X3 + m_points[i].x*m_points[i].x*m_points[i].x;
Y3 = Y3 + m_points[i].y*m_points[i].y*m_points[i].y;
X1Y1 = X1Y1 + m_points[i].x*m_points[i].y;
X1Y2 = X1Y2 + m_points[i].x*m_points[i].y*m_points[i].y;
X2Y1 = X2Y1 + m_points[i].x*m_points[i].x*m_points[i].y;
}
double C,D,E,G,H,N;
double a,b,c;
N = m_nNum;
C = N*X2 - X1*X1;
D = N*X1Y1 - X1*Y1;
E = N*X3 + N*X1Y2 - (X2+Y2)*X1;
G = N*Y2 - Y1*Y1;
H = N*X2Y1 + N*Y3 - (X2+Y2)*Y1;
a = (H*D-E*G)(C*G-D*D);
b = (H*C-E*D)(D*D-G*C);
c = -(a*X1 + b*Y1 + X2 + Y2)N;
double A,B,R;
A = a(-2);
B = b(-2);
R = sqrt(a*a+b*b-4*c)2;
m_fCenterX = A;
m_fCenterY = B;
m_fRadius = R; return;}