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opencv2d-3d-conversion

Converting 2D point to 3D location

I have a fixed camera with known cameraMatrix and distCoeffs. I also have a chessboard which is fixed too, and transform and rotation vector are also calculated using solvePnP.

I'm wondering how is possible to get 3D location of a 2D point on the same plane that the chessboard is located, like the picture below:

enter image description here

One thing for sure is that the Z of that point is 0, but how to get X and Y of that point.

Solution

  • You can solve this with 3 simple steps:

    Step 1:

    Compute the 3d direction vector, expressed in the coordinate frame of the camera, of the ray corresponding to the given 2d image point by inverting the camera projection model:

    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)

    Step 2:

    Compute the 3d direction of the vector of this ray in the coordinate frame attached to the chessboard, using the relative pose between the camera and the chessboard:

    // 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;

    Step 3:

    Find the desired 3d point by computing the intersection between the 3d ray and the chessboard plane with Z=0:

    // 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;