最小二乘法拟合圆公式推导及matlab实现

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;}