最小二乘椭圆拟合算法实现opencv

最小二乘椭圆拟合算法实现opencv

椭圆的目标函数：

F(A,B,C,D,E) = XiGeMa(xi^2 + A*xiyi+B*yi^2+C*xi+D*yi+E)^2

分别对A,B,C,D,E 求一阶偏导并令其等于0

得到线性方程组：

|A1 B1 C1 D1 E1 | |A| = |resul1|

|A2 B2 C2 D2 E2 | |B| = |resul2|

|A3 B3 C3 D3 E3 | |C| = |resul3|

|A4 B4 C4 D4 E4 | |D| = |resul4|

|A5 B5 C5 D5 E5 | |E| = |resul5|

求得A,B,C,D,E.

椭圆的五个参数：

center.x = (2*B*C - A*D) /(A*A-4*B);

center.y = (2*D-A*D)/(A*A-4*B);

fenzi = 2*(A*C*D-B*C*C-D*D+4*E*B-A*A*E);

fenmu =(A*A-4*B) * (B- sqrt(A*A+ (1-B) * (1-B) ) +1);

femmu2 =(A*A-4*B) * (B+ sqrt(A*A+ (1-B) * (1-B) ) +1);

long =sqrt(fabs(fenzi/fenmu));

short =sqrt(fabs(fenzi/femmu2));

theta = atan(sqrt((center.x *center.x -center.y*center.y*B)/(center.x *center.x *B-center.y*center.y))+0.0001)*180/cv_pi;;

vector<double> getEllipsepar(vector<CvPoint> vec_point)

{

vector<double> vec_result;

double x3y1 = 0,x1y3= 0,x2y2= 0,yyy4= 0, xxx3= 0,xxx2= 0,x2y1= 0,yyy3= 0,x1y2= 0 ,yyy2= 0,x1y1= 0,xxx1= 0,yyy1= 0;

int N = vec_point.size();

for (int m_i = 0;m_i < N ;++m_i )

{

double xi = vec_point[m_i].x ;

double yi = vec_point[m_i].y;

x3y1 += xi*xi*xi*yi ;

x1y3 += xi*yi*yi*yi;

x2y2 += xi*xi*yi*yi; ;

yyy4 +=yi*yi*yi*yi;

xxx3 += xi*xi*xi ;

xxx2 += xi*xi ;

x2y1 += xi*xi*yi;

x1y2 += xi*yi*yi;

yyy2 += yi*yi;

x1y1 += xi*yi;

xxx1 += xi;

yyy1 += yi;

yyy3 += yi*yi*yi;

}

long double resul1 = -(x3y1);

long double resul2 = -(x2y2);

long double resul3 = -(xxx3);

long double resul4 = -(x2y1);

long double resul5 = -(xxx2);

long double B1 = x1y3, C1 = x2y1, D1 = x1y2, E1 = x1y1, A1 = x2y2; long double B2 = yyy4, C2 = x1y2, D2 = yyy3, E2 = yyy2, A2 = x1y3; long double B3 = x1y2, C3 = xxx2, D3 = x1y1, E3 = xxx1, A3 = x2y1; long double B4 = yyy3, C4 = x1y1, D4 = yyy2, E4 = yyy1, A4 = x1y2; long double B5 = yyy2, C5 = xxx1, D5 = yyy1, E5 = N, A5 = x1y1;

//

CvMat* Ma = cvCreateMat(5,5,CV_64FC1);

CvMat* Md = cvCreateMat(5,1,CV_64FC1);

CvMat* Mb = cvCreateMat(5,1,CV_64FC1);

//

cvmSet(Mb,0,0,resul1);

cvmSet(Mb,1,0,resul2);

cvmSet(Mb,2,0,resul3);

cvmSet(Mb,3,0,resul4);

cvmSet(Mb,4,0,resul5);

cvmSet(Ma,0,0,A1);

cvmSet(Ma,0,1,B1);

cvmSet(Ma,0,2,C1);

cvmSet(Ma,0,3,D1);

cvmSet(Ma,0,4,E1);

cvmSet(Ma,1,0,A2);

cvmSet(Ma,1,1,B2);

cvmSet(Ma,1,2,C2);

cvmSet(Ma,1,3,D2);

cvmSet(Ma,1,4,E2);

cvmSet(Ma,2,0,A3);

cvmSet(Ma,2,1,B3);

cvmSet(Ma,2,2,C3);

cvmSet(Ma,2,3,D3);

cvmSet(Ma,2,4,E3);

cvmSet(Ma,3,0,A4);

cvmSet(Ma,3,1,B4);

cvmSet(Ma,3,2,C4);

cvmSet(Ma,3,3,D4);

cvmSet(Ma,3,4,E4);

cvmSet(Ma,4,0,A5);

cvmSet(Ma,4,1,B5);

cvmSet(Ma,4,2,C5);

cvmSet(Ma,4,3,D5); cvmSet(Ma,4,4,E5);

cvSolve(Ma, Mb, Md);

long double A = cvmGet(Md,0,0);

long double B = cvmGet(Md,1,0);

long double C = cvmGet(Md,2,0);

long double D = cvmGet(Md,3,0);

long double E = cvmGet(Md,4,0);