相机畸变模型
我们计算畸变都是在归一化平面上进行的，下面的(x,y), (x_distort,y_distort)都是在归一化坐标，相机坐标(X,Y,Z)的归一化坐标(X/Z, Y/Z, 1)

1、径向畸变
由透镜形状引起的，畸变系数k1, k2, k3，畸变模型：

2、切向畸变
由镜片安装和成像平面不平行引起的，畸变系数p1, p2，切向畸变模型：

3、总的畸变模型

求解思路
我们得到的原图是畸变后的图像(x_distort,y_distort)，要计算畸变之前的真实图像(x,y)，不是用逆运算，太难麻烦了，而是计算真是图像畸变后会投影在哪，对应过去。先把原图像设置为一个空的图像，把一个个像素畸变投影过去，找到和畸变后图像像素点的对应关系（可以参考下面代码方法1理解）。

代码示例

 注意，OpenCV畸变系数矩阵的默认顺序 [k1, k2, p1, p2, k3]

#include <opencv2/opencv.hpp>
#include <chrono>
using namespace std;

int main(int argc, char **argv)
{
  // 内参
  double fx = 458.654, fy = 457.296, cx = 367.215, cy = 248.375;
  /**内参矩阵K
   * fx  0  cx
   * 0  fy  cy
   * 0   0   1
   */
  // 畸变参数
  double k1 = -0.28340811, k2 = 0.07395907, p1 = 0.00019359, p2 = 1.76187114e-05;

  cv::Mat image = cv::imread(argv[1], 0); // 图像是灰度图，CV_8UC1
  int rows = image.rows, cols = image.cols;
  cv::Mat image_undistort = cv::Mat(rows, cols, CV_8UC1); // 方法1去畸变以后的图
  cv::Mat image_undistort2 = cv::Mat(rows, cols, CV_8UC1);  // 方法2 OpenCV去畸变以后的图

  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  //! 方法1. 自己写计算去畸变后图像的内容
  for (int v = 0; v < rows; v++)
  {
    for (int u = 0; u < cols; u++)
    {
      double x = (u - cx) / fx, y = (v - cy) / fy; //要求解的真实图，归一化平面上的坐标
      double r = sqrt(x * x + y * y);
      double x_distorted = x * (1 + k1 * r * r + k2 * r * r * r * r) + 2 * p1 * x * y + p2 * (r * r + 2 * x * x); //畸变后归一化坐标
      double y_distorted = y * (1 + k1 * r * r + k2 * r * r * r * r) + p1 * (r * r + 2 * y * y) + 2 * p2 * x * y;
      double u_distorted = fx * x_distorted + cx; //畸变后像素坐标，即原图
      double v_distorted = fy * y_distorted + cy;
      // 投影赋值
      if (u_distorted >= 0 && v_distorted >= 0 && u_distorted < cols && v_distorted < rows) //真实图畸变后仍然在图上的
      {
        image_undistort.at<uchar>(v, u) = image.at<uchar>((int)v_distorted, (int)u_distorted);
      }
      else
      {
        image_undistort.at<uchar>(v, u) = 0; //这里最好用插值法
      }
    }
  }
  chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
  chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "time = " << time_used.count() << endl;

  //! 方法2. OpenCV自带的undistort函数，更快速
  cv::Mat K = cv::Mat::eye(3, 3, CV_32FC1); //内参矩阵
  K.at<float>(0, 0) = fx;
  K.at<float>(1, 1) = fy;
  K.at<float>(0, 2) = cx;
  K.at<float>(1, 2) = cy;
  cv::Mat distort_coeffs = cv::Mat::zeros(1, 5, CV_32FC1); //畸变系数矩阵 顺序是[k1, k2, p1, p2, k3]
  distort_coeffs.at<float>(0, 0) = k1;
  distort_coeffs.at<float>(0, 1) = k2;
  distort_coeffs.at<float>(0, 2) = p1;
  distort_coeffs.at<float>(0, 3) = p2;
  cout << "K = " << endl
       << K << endl;
  cout << "distort_coeffs = " << endl
       << distort_coeffs << endl;

  t1 = chrono::steady_clock::now();
  cv::undistort(image, image_undistort2, K, distort_coeffs); //去畸变
  t2 = chrono::steady_clock::now();
  time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "time = " << time_used.count() << endl;

  // 展示去畸变后图像
  cv::imshow("distorted", image);
  cv::imshow("undistorted", image_undistort);
  cv::imshow("image_undistort2", image_undistort2);
  cv::waitKey(0);
  return 0;
}