多项式曲线拟合

给定数据点pi(xi,yi)，其中i=1,2,…,m。求近似曲线y= φ(x)。并且使得近似曲线与y=f(x)的偏差最小。近似曲线在点pi处的偏差δi= φ(xi)-y，i=1,2,...,m。

损失计算：

1.使偏差绝对值之和最小

2.使偏差绝对值最大的最小

3.使偏差平方和最小

推导：

拟合多项式： 

计算误差：  

求参数，求偏导：

。。。。

得到：

#include <cv.h>

#include <highgui.h>

using namespace std;

bool polynomialCurveFit(vector<cv::Point> & training, int n, cv::Mat& A)
{
    int N = training.size();
    cv::Mat x = cv::Mat::zeros(n + 1, n + 1, CV_64FC1);
    for (int i = 0; i < n + 1; i++)
    {
        for (int j = 0; j < n + 1; j++)
        {
            for (int k = 0; k < N; k++)
            {
                x.at<double>(i, j) = x.at<double>(i, j) + pow(training[k].x, i + j);
            }
        }
    }
    cv::Mat Y = cv::Mat::zeros(n + 1, 1, CV_64FC1);
    for (int i = 0; i < n + 1; i++)
    {
        for (int k = 0; k < N; k++)
        {
            Y.at<double>(i, 0) = Y.at<double>(i, 0) + pow(training[k].x, i)*training[k].y;
        }
    }

    A = cv::Mat::zeros(n + 1, 1, CV_64FC1);
    cv::solve(x, Y, A, cv::DECOMP_LU);//solve 是OpenCV中专用于求解线性方程的函数。。 x左矩阵，y右矩阵，A系数矩阵， method 估算方法。。 参数少于点数，LU分解
    return true;

}

int main()

{

    cv::Mat image = cv::Mat::zeros(480, 640, CV_8UC3);
    image.setTo(cv::Scalar(0, 0, 0));
    vector<cv::Point> points;
    points.push_back(cv::Point(100., 58.));
    points.push_back(cv::Point(150., 70.));
    points.push_back(cv::Point(200., 90.));
    points.push_back(cv::Point(252., 140.));
    points.push_back(cv::Point(300., 220.));
    points.push_back(cv::Point(350., 400.));
    for (int i = 0; i < points.size(); i++)
        cv::circle(image, points[i], 5, cv::Scalar(0, 0, 255), 2, 8, 0);
    cv::polylines(image, points, false, cv::Scalar(0, 255, 0), 1, 8, 0);
    cv::Mat A;
    polynomialCurveFit(points, 3, A);
    cout << A << endl;
    vector<cv::Point>points_fitted;
    for (int x = 0; x < 400; x++)
    {
        double y = A.at<double>(0, 0) + A.at<double>(1, 0)*x + A.at<double>(2, 0)*pow(x, 2) + A.at<double>(3, 0)*pow(x, 3);
        points_fitted.push_back(cv::Point(x, y));
    }
    cv::polylines(image, points_fitted, false, cv::Scalar(0, 255, 255), 1, 8, 0);
    cv::imshow("image", image);
    cv::waitKey(0);
    return 0;

}

拟合结果