我使用OpenCV来测量物体的长度，使用摄像头。（物体依赖于平面）
对于摄像机校准，我使用了棋盘。

// position array of checker corners in real world coordinate
3d_checker_position 

// possition array of checker corners in image coordinate
2d_checker_position

// find camera matrix, distortion coefficients, 
// camera rotation vector, camera translation vector
camera_mat, dist_coeffs, rvec, tvec = cv.calibrateCamera(3d_checker_position, 2d_checker_position)

// 3D -> 2D
test_2d_checker_position 
  = cv.projectPoints(3d_checker_position, rvec, tvec, camera_mat, dist_coeffs)
// test_2d_checker_position == 2d_checker_position

// 2D -> 3D
test_3d_checker_position 
  = what_function_should_I_use(2d_checker_position, rvec, tvec, camera_mat, dist_coeffs)

如何实现what_function_should_I_use，或者OpenCV中有任何函数吗？

编辑23.04.07

为了充分理解世界到图像坐标的转换，我首先手工实现了3D -〉2D，这是基于下面的针孔模型方程。

sP = I * E * W

其中P是2通道（x，y）图像点，I是内部参数矩阵（相机矩阵），E是外部参数矩阵（变换），W是3通道（x，y，z）世界点。
下面是从3D到2D的实现代码。s是用于将齐次坐标转换为图像坐标的加权因子。

// Firstly, assumed that intrinsic parameters(camera matrix), 
// extrinsic parameters(rotation and translation vector), 
// and distortion parameters are pre-calculated from 
// cv::calibrateCamera function.
cv::Mat intrinsic; // (3x3)
cv::Mat rvec; // (3x1)
cv::Mat tvec; // (3x1) 
std::vector<float> distortion;
// ...

// Secondly, make a extrinsic matrix which is [R | t].
cv::Mat extrinsic= cv::Mat(cv::Size(4, 3), CV_64F);
cv::Mat rot_mat;
cv::Rodrigues(rvec, rot_mat);
rot_mat.copyTo(extrinsic(cv::Rect(0, 0, 3, 3)));
tvec.copyTo(extrinsic(cv::Rect(3, 0, 1, 3)));

// Thirdly, make a pinhole camera model matrix 
// by merging intrinsic and extrinsic parameters.
cv::Mat pinhole_model = intrinsic * extrinsic;

// Finally, convert 3D point to 2D point.

cv::Mat world_point_homogeneous = cv::Mat(cv::Size(1, 4), CV_64F);
world_point_homogeneous.at<double>(0, 0) = world_point.x;
world_point_homogeneous.at<double>(1, 0) = world_point.y;
world_point_homogeneous.at<double>(2, 0) = world_point.z;
world_point_homogeneous.at<double>(3, 0) = 1;

cv::Mat projected_homogeneous 
  = pinhole_model * world_point_homogeneous;

// Convert homogeneous coordinate to image coordinate.
// This equals dividing 's' from 'P' in the above equation.
const double w = projected_homogeneous.at<double>(2, 0);
cv::Point2d image_point;
image_point.x = projected_homogeneous.at<double>(0, 0) / w;
image_point.y = projected_homogeneous.at<double>(1, 0) / w;

结果

上面的代码确实将3D点转换为2D点，这与cv::projectPoints()的结果非常接近。

问题

1.何时以及如何将distortion参数应用于流程？
1.回到原点问题，在将2D点转换为3D点之前，我需要w因子（在等式中，s）。我如何获得它？