将2D点转换为3D位置

Question

我有一台固定相机，已知cameraMatrix和distCoeffs。我也有一个固定的棋盘，transform并且rotationvector也使用来计算solvePnP。

我想知道如何在棋盘所在的平面上获得2D点的3D位置，如下图所示：

可以肯定的是，该点的Z为0，但如何获得该点的X和Y。

Answer 1

您可以通过3个简单的步骤解决此问题：

步骤1：

通过反转相机投影模型，计算与给定2d图像点相对应的射线的3d方向矢量（以相机的坐标系表示）：

std::vector<cv::Point2f> imgPt = {{u,v}}; // Input image point
std::vector<cv::Point2f> normPt;
cv::undistortPoints     (imgPt, normPt, cameraMatrix, distCoeffs);
cv::Matx31f ray_dir_cam(normPt[0].x, normPt[0].y, 1);
// 'ray_dir_cam' is the 3d direction of the ray in camera coordinate frame
// In camera coordinate frame, this ray originates from the camera center at (0,0,0)

第2步：

使用相机和棋盘之间的相对姿势，在附在棋盘上的坐标系中计算此射线矢量的3d方向：

// solvePnP typically gives you 'rvec_cam_chessboard' and 'tvec_cam_chessboard'
// Inverse this pose to get the pose mapping camera coordinates to chessboard coordinates
cv::Matx33f R_cam_chessboard;
cv::Rodrigues(rvec_cam_chessboard, R_cam_chessboard);
cv::Matx33f R_chessboard_cam = R_cam_chessboard.t();
cv::Matx31f t_cam_chessboard = tvec_cam_chessboard;
cv::Matx31f pos_cam_wrt_chessboard = -R_chessboard_cam*t_cam_chessboard;
// Map the ray direction vector from camera coordinates to chessboard coordinates
cv::Matx31f ray_dir_chessboard = R_chessboard_cam * ray_dir_cam;

第三步：

通过计算Z = 0的3d射线与棋盘平面之间的交点找到所需的3d点：

// Expressed in the coordinate frame of the chessboard, the ray originates from the
// 3d position of the camera center, i.e. 'pos_cam_wrt_chessboard', and its 3d
// direction vector is 'ray_dir_chessboard'
// Any point on this ray can be expressed parametrically using its depth 'd':
// P(d) = pos_cam_wrt_chessboard + d * ray_dir_chessboard
// To find the intersection between the ray and the plane of the chessboard, we
// compute the depth 'd' for which the Z coordinate of P(d) is equal to zero
float d_intersection = -pos_cam_wrt_chessboard.val[2]/ray_dir_chessboard.val[2];
cv::Matx31f intersection_point = pos_cam_wrt_chessboard + d_intersection * ray_dir_chessboard;