圆拟合算法（距离之和最小）

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建立了一个类，叫做 “CircleFitSolver”。把所有的计算都封装到这个类中。

#ifndef CIRCLEFITSOLVER_H
#define CIRCLEFITSOLVER_H

#include <complex>
#include <vector>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_multimin.h>

using namespace std;

typedef complex<int> POINT;

bool circleLeastFit(const std::vector<POINT> &points, double &center_x, double &center_y, double &radius);

class CircleFitSolver
{
public:
    CircleFitSolver();
    ~CircleFitSolver();
    void setMaxIter(int iter) {m_max_iter = iter;}

    /**
     * @brief circleFitL1  拟合圆，拟合判据为数据点到拟合圆的距离绝对值之和最小。
     * @param points 输入参数，存储各个数据点。
     * @param center_x radius > 0 时作为迭代算法的初始值。计算完成后返回拟合圆的圆心 X 坐标
     * @param center_y radius > 0 时作为迭代算法的初始值。计算完成后返回拟合圆的圆心 Y 坐标
     * @param radius   radius < 0 时，用最小二乘拟合的结果作为迭代算法的初始值。计算完成后返回拟合圆的半径。
     * @return true 表示拟合成功，否则拟合失败。
     */
    bool circleFitL1(const vector<POINT> &points, double &center_x, double &center_y, double &radius);
private:
    gsl_multimin_function m_function;
    gsl_multimin_fminimizer * m_fminimizer;

    int m_max_iter; // 迭代算法的最大迭代次数

    gsl_vector *m_start_point; // 迭代算法的初始值
    gsl_vector *m_step_size; // 迭代算法的初始步长

    void setStartPoint(double center_x, double center_y, double radius);

    static double L1_distance(const gsl_vector * v, void * params);

};

#endif // CIRCLEFITSOLVER_H
  
具体的实现代码如下：

#include "circlefitsolver.h"
#include <cmath>

using namespace std;

/**
* 最小二乘法拟合圆
* 拟合出的圆以圆心坐标和半径的形式表示
* 此代码改编自 newsmth.net 的 jingxing 在 Graphics 版贴出的代码。
* 版权归 jingxing， 我只是搬运工外加一些简单的修改工作。
*/
bool circleLeastFit(const std::vector<POINT> &points, double &center_x, double &center_y, double &radius)
{
     center_x = 0.0f;
     center_y = 0.0f;
     radius = 0.0f;
     if (points.size() < 3)
     {
         return false;
     }

     double sum_x = 0.0f, sum_y = 0.0f;
     double sum_x2 = 0.0f, sum_y2 = 0.0f;
     double sum_x3 = 0.0f, sum_y3 = 0.0f;
     double sum_xy = 0.0f, sum_x1y2 = 0.0f, sum_x2y1 = 0.0f;

     int N = points.size();
     for (int i = 0; i < N; i++)
     {
         double x = points[i].real();
         double y = points[i].imag();
         double x2 = x * x;
         double y2 = y * y;
         sum_x += x;
         sum_y += y;
         sum_x2 += x2;
         sum_y2 += y2;
         sum_x3 += x2 * x;
         sum_y3 += y2 * y;
         sum_xy += x * y;
         sum_x1y2 += x * y2;
         sum_x2y1 += x2 * y;
     }

     double C, D, E, G, H;
     double a, b, c;

     C = N * sum_x2 - sum_x * sum_x;
     D = N * sum_xy - sum_x * sum_y;
     E = N * sum_x3 + N * sum_x1y2 - (sum_x2 + sum_y2) * sum_x;
     G = N * sum_y2 - sum_y * sum_y;
     H = N * sum_x2y1 + N * sum_y3 - (sum_x2 + sum_y2) * sum_y;
     a = (H * D - E * G) / (C * G - D * D);
     b = (H * C - E * D) / (D * D - G * C);
     c = -(a * sum_x + b * sum_y + sum_x2 + sum_y2) / N;

     center_x = a / (-2);
     center_y = b / (-2);
     radius = sqrt(a * a + b * b - 4 * c) / 2;
     return true;
}

double CircleFitSolver::L1_distance(const gsl_vector * v, void * params)
{
    vector<POINT> *vect = (vector<POINT> *)params;
    int N  = vect->size();

    double a, b, r;
    a = gsl_vector_get(v, 0);
    b = gsl_vector_get(v, 1);
    r = gsl_vector_get(v, 2);

    double sum = 0;
    for(int i = 0; i < N; i++)
    {
        const POINT p = vect->at(i);
        double xi = p.real() - a;
        double yi = p.imag() - b;
        double dist = sqrt(xi * xi + yi * yi) - r;
        sum += fabs(dist);
    }
    return sum;
}


inline void CircleFitSolver::setStartPoint(double center_x, double center_y, double radius)
{
    gsl_vector_set (m_start_point, 0, center_x);
    gsl_vector_set (m_start_point, 1, center_y);
    gsl_vector_set (m_start_point, 2, radius);
}

bool CircleFitSolver::circleFitL1(const vector<POINT> &points, double &center_x, double &center_y, double &radius)
{
    m_function.params = (void *)&points;

    if( radius < 0 )
    {
        // 用最小二乘拟合的结果作为初始值
        if( !circleLeastFit(points, center_x, center_y, radius) )
        {
            return false;
        }
    }

    setStartPoint(center_x, center_y, radius);
    /* 经验值，初始步长设置为半径的十分之一 */
    gsl_vector_set (m_step_size, 0, radius / 10.0);
    gsl_vector_set (m_step_size, 1, radius / 10.0);
    gsl_vector_set (m_step_size, 2, radius / 10.0);

    gsl_multimin_fminimizer_set(m_fminimizer, &m_function, m_start_point, m_step_size);

    int iter = 0;
    int status;
    do
    {
        iter++;
        status = gsl_multimin_fminimizer_iterate(m_fminimizer);
        if (status == GSL_ENOPROG ) // 表示无法找到更好的解了
        {
            break;
        }
        double size = gsl_multimin_fminimizer_size (m_fminimizer);
        status = gsl_multimin_test_size (size, 1e-2);
    }
    while (status == GSL_CONTINUE && iter < m_max_iter);

    gsl_vector * out = gsl_multimin_fminimizer_x(m_fminimizer);

    center_x = gsl_vector_get(out, 0);
    center_y = gsl_vector_get(out, 1);
    radius = gsl_vector_get(out, 2);

    return true;
}

CircleFitSolver::CircleFitSolver()
{
    m_max_iter = 100; // 默认最大迭代 100 步

    m_function.n = 3;
    m_function.f = L1_distance;

    m_start_point = gsl_vector_alloc (m_function.n);
    m_step_size = gsl_vector_alloc (m_function.n);

    m_fminimizer = gsl_multimin_fminimizer_alloc(gsl_multimin_fminimizer_nmsimplex, 3);
}

CircleFitSolver::~CircleFitSolver()
{
    gsl_vector_free(m_start_point);
    gsl_vector_free(m_step_size);

    gsl_multimin_fminimizer_free(m_fminimizer);
}

[code]原文链接：https://blog.csdn.net/liyuanbhu/article/details/50890587[/code]