使用OpenCV给图像去畸变

1. 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
   

2. 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）。

3. 代码示例

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

   #include 
   #include 
   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;
   }