问题陈述

假设我知道每个点的距离，我试图将2D点重新投影到其原始3D坐标。遵循OpenCV文档，我设法使其与零失真一起使用。但是，如果存在变形，则结果不正确。

目前的方法

因此，我们的想法是扭转以下情况：

分为以下内容：

通过：

1. 摆脱任何扭曲使用 cv::undistortPoints
2. 通过反转上面的第二个方程，使用内在函数返回归一化的相机坐标
3. 乘以z反向标准化。

问题

1. 为什么我需要减去f_x并f_y返回归一化的相机坐标（在测试时凭经验找到）？在下面的代码中，在第2步中，如果我不减去-即使未失真的结果也关闭了，这是我的错误-我弄乱了索引。
2. 如果包含变形，结果将是错误的-我在做什么错？

样例代码（C ++）

#include <iostream>
#include <opencv2/calib3d/calib3d.hpp>
#include <opencv2/core/core.hpp>
#include <opencv2/imgproc/imgproc.hpp>
#include <vector>

std::vector<cv::Point2d> Project(const std::vector<cv::Point3d>& points,
                                 const cv::Mat& intrinsic,
                                 const cv::Mat& distortion) {
  std::vector<cv::Point2d> result;
  if (!points.empty()) {
    cv::projectPoints(points, cv::Mat(3, 1, CV_64F, cvScalar(0.)),
                      cv::Mat(3, 1, CV_64F, cvScalar(0.)), intrinsic,
                      distortion, result);
  }
  return result;
}

std::vector<cv::Point3d> Unproject(const std::vector<cv::Point2d>& points,
                                   const std::vector<double>& Z,
                                   const cv::Mat& intrinsic,
                                   const cv::Mat& distortion) {
  double f_x = intrinsic.at<double>(0, 0);
  double f_y = intrinsic.at<double>(1, 1);
  double c_x = intrinsic.at<double>(0, 2);
  double c_y = intrinsic.at<double>(1, 2);
  // This was an error before:
  // double c_x = intrinsic.at<double>(0, 3);
  // double c_y = intrinsic.at<double>(1, 3);

  // Step 1. Undistort
  std::vector<cv::Point2d> points_undistorted;
  assert(Z.size() == 1 || Z.size() == points.size());
  if (!points.empty()) {
    cv::undistortPoints(points, points_undistorted, intrinsic,
                        distortion, cv::noArray(), intrinsic);
  }

  // Step 2. Reproject
  std::vector<cv::Point3d> result;
  result.reserve(points.size());
  for (size_t idx = 0; idx < points_undistorted.size(); ++idx) {
    const double z = Z.size() == 1 ? Z[0] : Z[idx];
    result.push_back(
        cv::Point3d((points_undistorted[idx].x - c_x) / f_x * z,
                    (points_undistorted[idx].y - c_y) / f_y * z, z));
  }
  return result;
}

int main() {
  const double f_x = 1000.0;
  const double f_y = 1000.0;
  const double c_x = 1000.0;
  const double c_y = 1000.0;
  const cv::Mat intrinsic =
      (cv::Mat_<double>(3, 3) << f_x, 0.0, c_x, 0.0, f_y, c_y, 0.0, 0.0, 1.0);
  const cv::Mat distortion =
      // (cv::Mat_<double>(5, 1) << 0.0, 0.0, 0.0, 0.0);  // This works!
      (cv::Mat_<double>(5, 1) << -0.32, 1.24, 0.0013, 0.0013);  // This doesn't!

  // Single point test.
  const cv::Point3d point_single(-10.0, 2.0, 12.0);
  const cv::Point2d point_single_projected = Project({point_single}, intrinsic,
                                                     distortion)[0];
  const cv::Point3d point_single_unprojected = Unproject({point_single_projected},
                                    {point_single.z}, intrinsic, distortion)[0];

  std::cout << "Expected Point: " << point_single.x;
  std::cout << " " << point_single.y;
  std::cout << " " << point_single.z << std::endl;
  std::cout << "Computed Point: " << point_single_unprojected.x;
  std::cout << " " << point_single_unprojected.y;
  std::cout << " " << point_single_unprojected.z << std::endl;
}

相同代码（Python）

import cv2
import numpy as np

def Project(points, intrinsic, distortion):
  result = []
  rvec = tvec = np.array([0.0, 0.0, 0.0])
  if len(points) > 0:
    result, _ = cv2.projectPoints(points, rvec, tvec,
                                  intrinsic, distortion)
  return np.squeeze(result, axis=1)

def Unproject(points, Z, intrinsic, distortion):
  f_x = intrinsic[0, 0]
  f_y = intrinsic[1, 1]
  c_x = intrinsic[0, 2]
  c_y = intrinsic[1, 2]
  # This was an error before
  # c_x = intrinsic[0, 3]
  # c_y = intrinsic[1, 3]

  # Step 1. Undistort.
  points_undistorted = np.array([])
  if len(points) > 0:
    points_undistorted = cv2.undistortPoints(np.expand_dims(points, axis=1), intrinsic, distortion, P=intrinsic)
  points_undistorted = np.squeeze(points_undistorted, axis=1)

  # Step 2. Reproject.
  result = []
  for idx in range(points_undistorted.shape[0]):
    z = Z[0] if len(Z) == 1 else Z[idx]
    x = (points_undistorted[idx, 0] - c_x) / f_x * z
    y = (points_undistorted[idx, 1] - c_y) / f_y * z
    result.append([x, y, z])
  return result

f_x = 1000.
f_y = 1000.
c_x = 1000.
c_y = 1000.

intrinsic = np.array([
  [f_x, 0.0, c_x],
  [0.0, f_y, c_y],
  [0.0, 0.0, 1.0]
])

distortion = np.array([0.0, 0.0, 0.0, 0.0])  # This works!
distortion = np.array([-0.32, 1.24, 0.0013, 0.0013])  # This doesn't!

point_single = np.array([[-10.0, 2.0, 12.0],])
point_single_projected = Project(point_single, intrinsic, distortion)
Z = np.array([point[2] for point in point_single])
point_single_unprojected = Unproject(point_single_projected,
                                     Z,
                                     intrinsic, distortion)
print "Expected point:", point_single[0]
print "Computed point:", point_single_unprojected[0]

零失真的结果（如上所述）是正确的：

Expected Point: -10 2 12
Computed Point: -10 2 12

但是，当包含变形时，结果将变为：

Expected Point: -10 2 12
Computed Point: -4.26634 0.848872 12

更新1.澄清

这是一个用于图像投影的相机-我假设3D点在相机框架坐标中。

更新2。弄清楚第一个问题

好的，我弄清楚了f_xand的减法f_y-我很愚蠢，无法弄乱索引。更新了代码以进行更正。另一个问题仍然成立。

更新3.添加了Python等效代码

为了增加可见性，请添加Python代码，因为它具有相同的错误。

...
const double f_x = 2746.;
const double f_y = 2748.;
const double c_x = 991.;
const double c_y = 619.;
...
const cv::Point3d point_single(10.0, -2.0, 30.0);
...

...
double f_x = intrinsic.at<double>(0, 0);
double f_y = intrinsic.at<double>(1, 1);
double c_x = intrinsic.at<double>(0, 3);
double c_y = intrinsic.at<double>(1, 3);
...