将 2D 点转换为 3D 位置

Question

我有一个固定摄像头，已知 cameraMatrix 和 distCoeffs。我也有一个固定的棋盘，transform 和rotation 向量也是使用solvePnP 计算的。

我想知道如何在棋盘所在的同一平面上获得 2D 点的 3D 位置，如下图所示：

有一点是肯定的，那个点的Z是0，但是如何得到那个点的X和Y。

Answer 1

您可以通过 3 个简单的步骤解决此问题：

第 1 步：

通过反转相机投影模型，计算对应于给定 2d 图像点的射线的 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位姿，计算此射线在附加到棋盘的坐标系中的向量的 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 3 步：

通过计算 3d 射线与 Z=0 的棋盘平面之间的交点来找到所需的 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;

Answer 2

由于您的情况仅限于平原，简单的方法是使用 Homography。

首先undistort您的图像。然后使用findHomography 计算将像素坐标（图像）转换为真实坐标（欧几里得空间，例如以厘米为单位）的单应矩阵。类似于这个的东西：

#include <opencv2/calib3d.hpp>
//...

//points on undistorted image (in pixel). more is better
vector<Point2f>  src_points = { Point2f(123,321), Point2f(456,654), Point2f(789,987), Point2f(123,321) };
//points on chessboard (e.g. in cm)
vector<Point2f>  dst_points = { Point2f(0, 0), Point2f(12.5, 0), Point2f(0, 16.5), Point2f(12.5, 16.5) }; 
Mat H = findHomography(src_points, dst_points, RANSAC);

//print euclidean coordinate of new point on undistorted image (in pixel)
cout << H * Mat(Point3d(125, 521, 0)) << endl;