最小二乘法拟合椭圆（椭圆拟合线）
参考文章：
最小二乘法拟合椭圆——MATLAB和Qt-C++实现
https://blog.csdn.net/sinat_21107433/article/details/80877758
以上文章中，C++代码有问题。因此参考如下文章，得到正确的结果。
矩阵求逆-高斯消元法介绍及其实现
https://blog.csdn.net/qithon/article/details/80100029
代码如下：
1. 建立类：EclipseFitting
import android.util.Log;
public class EclipseFitting {
/**
*
* 功能说明： 对平面上的一些列点给出最小二乘的椭圆拟合，利用消元法
* 解得最小二乘解作为椭圆参数。
*
*调用形式： EclipseFitting2（arrayx,arrayy,Result）；
*参数说明： arrayx: arrayx[n] 为第n个点的x坐标
arrayy: arrayy[n]为第n点的y坐标
n : 点的个数
Result : Result[0][],椭圆的五个参数，分别为center.x,center.y,2a,2b,xtheta
Result[1][],椭圆方程的的五个系数（A,B,C,D,E），X^2+A*XY+B*Y^2+C*X+D*Y+E=0
esp: 解精度，通常取1e-6,这个是解方程用的说
*
*
* @param arrayx
* @param arrayy
* @param n
* @param Result
*/
void EclipseFitting2(int [] arrayx, int [] arrayy,int N,double [][] Result)
{
double A = 0.00,B = 0.00,C = 0.00,D = 0.00,E = 0.00;
double x2y2=0.0,x1y3=0.0,x2y1=0.0,x1y2=0.0,x1y1=0.0,yyy4=0.0,yyy3=0.0,yyy2=0.0,xxx2=0.0,xxx1=0.0,yyy1=0.0,x3y1=0.0,xxx3=0.0;
//平面上任意位置的一个椭圆，其中心坐标为（x0，y0），半长轴a，半短轴b，长轴偏角为θ，方程通式为
//方程为：X^2+A*XY+B*Y^2+C*X+D*Y+E=0
for(int i = 0; i < N; i++)
{
double xi = arrayx[i], yi = arrayy[i];
x2y2 += xi*xi*yi*yi;
x1y3 += xi*yi*yi*yi;
x2y1 += xi*xi*yi;
x1y2 += xi*yi*yi;
x1y1 += xi*yi;
yyy4 += yi*yi*yi*yi;
yyy3 += yi*yi*yi;
yyy2 += yi*yi;
xxx2 += xi*xi;
xxx1 += xi;
yyy1 += yi;
x3y1 += xi*xi*xi*yi;
xxx3 += xi*xi*xi;
}
//5*5 matrix
double[][] matrix={{x2y2,x1y3,x2y1,x1y2,x1y1},
{x1y3,yyy4,x1y2,yyy3,yyy2},
{x2y1,x1y2,xxx2,x1y1,xxx1},
{x1y2,yyy3,x1y1,yyy2,yyy1},
{x1y1,yyy2,xxx1,yyy1,N}
};
double[] matrix2={x3y1,x2y2,xxx3,x2y1,xxx2};
double[] matrix3 = {A,B,C,D,E};
Log.i("EclipseFitting2","系数矩阵");
LogMatrix(matrix,5);
求矩阵matrix的逆，结果为InverseMatrix
///
double[][] mInverseMatrix=InverseMatrix(matrix,5);
///求参数A,B,C,D,E
for(int i=0;i<5;i++)
{
for(int j=0;j<5;j++)
{
matrix3[i] += mInverseMatrix[i][j]*(-matrix2[j]);
}
}
A = matrix3[0];
B = matrix3[1];
C = matrix3[2];
D = matrix3[3];
E = matrix3[4];
///求拟合结果重要参数
double Xc = (2*B*C-A*D)/(A*A-4*B);
double Yc = (2*D-A*D)/(A*A-4*B);
double a = Math.sqrt(Math.abs(2*(A*C*D-B*C*C-D*D+4*B*E-A*A*E)/((A*A-4*B)*(B-Math.sqrt(A*A+(1-B)*(1-B))+1))));
double b = Math.sqrt(Math.abs(2*(A*C*D-B*C*C-D*D+4*B*E-A*A*E)/((A*A-4*B)*(B+Math.sqrt(A*A+(1-B)*(1-B))+1))));
double theta = Math.atan2(a*a-b*b*B,a*a*B-b*b);
Result[0][0]=Xc; Result[0][1]=Yc;
Result[0][2]=a; Result[0][3]=b;
Result[0][4]=theta*180/3.1415926;
Result[1][0]=A; Result[1][1]=B; Result[1][2]=C;
Result[1][3]=D; Result[1][4]=E;
}
/**
matrix为所求矩阵
dim为矩阵的维度
*/
public static double[][] InverseMatrix(double[][] matrix, int dim)
{
double eps = 1e-6;
double[][] mat = new double [dim][ dim * 2];
//构造增广矩阵W
for(int i = 0;i < dim; i++)
{
for(int j = 0;j < 2 * dim; j++)
{
if(j < dim)
{
mat[i][j] = matrix[i][j];
}
else
{
mat[i][j] = (j - dim == i) ? 1 : 0;
}
}
}
//从上往下消元
for(int i = 0;i < dim; i++)
{
if(Math.abs(mat[i][i]) < eps)
//若mat[i,i]为0，则往下找一个不为0的行，加到第i行
{
int j;
for ( j = i + 1; j < dim; j++)
{
if (Math.abs(mat[j][ i]) > eps) break;
}
if (j == dim) return null;
for(int r = i; r < 2 * dim; r++)
{
mat[i][ r] += mat[j][ r];
}
}
double ep = mat[i][ i];
//将mat[i][i]变为1
for (int r = i; r < 2 * dim; r++)
{
mat[i][ r] /= ep;
}
for(int j = i + 1; j < dim; j++)
{
double e = -1 * (mat[j][ i] / mat[i][ i]);
for(int r = i; r < 2 * dim; r++)
{
mat[j][ r] += e * mat[i][ r];
}
}
}
LogMatrix(mat,10);
//从下往上消元
for(int i = dim - 1; i >= 0; i--)
{
for (int j = i - 1; j >= 0; j--)
{
double e = -1 * (mat[j][ i] / mat[i][ i]);
for (int r = i; r < 2 * dim; r++)
{
mat[j][ r] += e * mat[i][ r];
}
}
}
LogMatrix(mat,10);
double[][] result = new double[dim][ dim];
for(int i = 0;i < dim; i++)
{
for(int r = dim; r < 2 * dim; r++)
{
result[i][ r - dim] = mat[i][ r];
}
}
return result;
}
// 原文：https://blog.csdn.net/qithon/article/details/80100029
private static void LogMatrix(double[][] Matrix, int width){
for(int i=0;i<5;i++){
String information=" ";
for(int j=0;j2.应用代码：
private void PlotEclipse(){
int [] x={10,20,30,50,50,70,90,140,170,170,200,260,260,290,300,310,320};
int [] y={70,50,40,30,110,20,15,130,10,130,130,20,120,30,100,45,70 };
// int [] x={1,2,3,5,5,7,9,14,17,17,20,26,26,29,30,31,32};
// int [] y={7,5,4,3,11,2,1,13,1,13,13,2,12,3,10,4,7 };
int i;
// x=new int [6]; y=new int [6];
double[][] result=new double[2][5];
EclipseFitting mEclipseFitting=new EclipseFitting();
mEclipseFitting.EclipseFitting2(x, y, x.length, result);
Log.i("PlotEclipse","X0, Y0: "+result[0][0]+","+result[0][1]);
Log.i("PlotEclipse","a, b: "+(result[0][2])+","+(result[0][3]));
Log.i("PlotEclipse","倾斜角"+result[0][4]+"degree");
Log.i("PlotEclipse","A, B: "+result[1][0]+","+result[1][1]);
Log.i("PlotEclipse","C, D: "+(result[1][2])+","+(result[1][3]));
Log.i("PlotEclipse","E"+result[1][4]);
float[] mPoints=new float [600*4];
double temp,xx,yy,NewX,NewY;
int X_Offset=300,Y_Offset=200;
double theta=(result[0][4]*3.1415926/180.0f);
for( i=0;iMath.abs(Error2)){
Error=Error2;
}else{
Error=Error1;
}
ErrorSum+=Error*Error;
Log.i("PlotEclipse",i+" Error:"+Error);
}
ErrorMean=Math.sqrt(ErrorSum)/x.length;
Log.i("PlotEclipse","Error:"+ErrorMean);
/
Canvas canvas = faceHolder.lockCanvas();
faceHolder=mSurfaceView.getHolder();
Paint p = new Paint();
p.setAntiAlias(true);
p.setStyle(Style.STROKE);
p.setStrokeWidth(20);
p.setColor(Color.BLUE);
for (i=0;i 3. 效果图：
蓝色为离散点，绿色为一半拟合线。