对极几何与三角化求3D空间坐标

一，使用对极几何约束求R,T

第一步：特征匹配。提取出有效的匹配点

void find_feature_matches(const Mat &img_1, const Mat &img_2,
                          std::vector &keypoints_1,
                          std::vector &keypoints_2,
                          std::vector &matches) {
  //-- 初始化
  Mat descriptors_1, descriptors_2;
  // used in OpenCV3
  Ptr detector = ORB::create();
  Ptr descriptor = ORB::create();
  // use this if you are in OpenCV2
  // Ptr detector = FeatureDetector::create ( "ORB" );
  // Ptr descriptor = DescriptorExtractor::create ( "ORB" );
  Ptr matcher = DescriptorMatcher::create("BruteForce-Hamming");
  //-- 第一步:检测 Oriented FAST 角点位置
  detector->detect(img_1, keypoints_1);
  detector->detect(img_2, keypoints_2);
 
  //-- 第二步:根据角点位置计算 BRIEF 描述子
  descriptor->compute(img_1, keypoints_1, descriptors_1);
  descriptor->compute(img_2, keypoints_2, descriptors_2);
 
  //-- 第三步:对两幅图像中的BRIEF描述子进行匹配，使用 Hamming 距离
  vector match;
  // BFMatcher matcher ( NORM_HAMMING );
  matcher->match(descriptors_1, descriptors_2, match);
 
  //-- 第四步:匹配点对筛选
  double min_dist = 10000, max_dist = 0;
 
  //找出所有匹配之间的最小距离和最大距离, 即是最相似的和最不相似的两组点之间的距离
  for (int i = 0; i < descriptors_1.rows; i++) {
    double dist = match[i].distance;
    if (dist < min_dist) min_dist = dist;
    if (dist > max_dist) max_dist = dist;
  }
 
  printf("-- Max dist : %f \n", max_dist);
  printf("-- Min dist : %f \n", min_dist);
 
  //当描述子之间的距离大于两倍的最小距离时,即认为匹配有误.但有时候最小距离会非常小,设置一个经验值30作为下限.
  for (int i = 0; i < descriptors_1.rows; i++) {
    if (match[i].distance <= max(2 * min_dist, 30.0)) {
      matches.push_back(match[i]);
    }
  }
}

二、使用本质矩阵求解R，T

第二步：根据匹配点对，依据对极几何约束原理，求相机运动的R,t

void pose_estimation_2d2d(
  const std::vector &keypoints_1,
  const std::vector &keypoints_2,
  const std::vector &matches,
  Mat &R, Mat &t) {
  // 相机内参,TUM Freiburg2
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
 
  //-- 把匹配点转换为vector的形式
  vector points1;
  vector points2;
 
  for (int i = 0; i < (int) matches.size(); i++) {
    points1.push_back(keypoints_1[matches[i].queryIdx].pt);
    points2.push_back(keypoints_2[matches[i].trainIdx].pt);
  }
 
  //-- 计算本质矩阵
  Point2d principal_point(325.1, 249.7);        //相机主点, TUM dataset标定值
  int focal_length = 521;            //相机焦距, TUM dataset标定值
  Mat essential_matrix;
  essential_matrix = findEssentialMat(points1, points2, focal_length, principal_point);
 
  //-- 从本质矩阵中恢复旋转和平移信息.
  recoverPose(essential_matrix, points1, points2, R, t, focal_length, principal_point);
}

三、由R，T三角化空间坐标

第三步：根据针孔相机模型的公式，由 R,t估计特征点的空间坐标

//三角化,根据匹配点和求解到的三维点。存储在points中
void triangulation(
  const vector &keypoint_1,
  const vector &keypoint_2,
  const std::vector &matches,
  const Mat &R, const Mat &t,
  vector &points) {
  Mat T1 = (Mat_(3, 4) <<
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0);
    //根据求解到的RT构造T2矩阵
  Mat T2 = (Mat_(3, 4) <<
    R.at(0, 0), R.at(0, 1), R.at(0, 2), t.at(0, 0),
    R.at(1, 0), R.at(1, 1), R.at(1, 2), t.at(1, 0),
    R.at(2, 0), R.at(2, 1), R.at(2, 2), t.at(2, 0)
  );
  //相机内参
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  vector pts_1, pts_2;
  for (DMatch m:matches) {
    // 将像素坐标转换至相机坐标
    pts_1.push_back(pixel2cam(keypoint_1[m.queryIdx].pt, K));
    pts_2.push_back(pixel2cam(keypoint_2[m.trainIdx].pt, K));
  }
 
  Mat pts_4d;
  cv::triangulatePoints(T1, T2, pts_1, pts_2, pts_4d);
 
  // 转换成非齐次坐标
  for (int i = 0; i < pts_4d.cols; i++) {
    Mat x = pts_4d.col(i);
    x /= x.at(3, 0); // 归一化
    Point3d p(
      x.at(0, 0),
      x.at(1, 0),
      x.at(2, 0)
    );
    points.push_back(p);
  }
}

其中 triangulatePoints()的具体用法为

triangulatePoints(T1, T2, left, right, points_final) ;

Mat T1 = (Mat_(3, 4) <<
		1, 0, 0, 0,
		0, 1, 0, 0,
		0, 0, 1, 0);
Mat T2 = (Mat_(3, 4) <<
		R.at(0, 0), R.at(0, 1), R.at(0, 2), T.at(0, 0),
		R.at(1, 0), R.at(1, 1), R.at(1, 2), T.at(1, 0),
		R.at(2, 0), R.at(2, 1), R.at(2, 2), T.at(2, 0)
		);`
triangulatePoints(T1, T2, left, right, points_final) ;

其中T2为3x4的[R|T]矩阵，left、right为相机坐标系下的归一化坐标，
因此不能直接使用提取到的像素坐标。应首先将像素坐标通过相机内参转化到相机坐标系下。

所以通过函数pixel2cam可将像素坐标转换到归一化相机坐标系下
归一化坐标：X=(u-u0)/fx

//像素坐标到归一化平面相机坐标的转换
Point2f pixel2cam(const Point2f& p, const Mat& K)
{
    return Point2f
    (
        (p.x - K.at(0, 2)) / K.at(0, 0),
        (p.y - K.at(1, 2)) / K.at(1, 1)
    );
}

四、代码demo

总的代码为：

#include 
#include 
// #include "extra.h" // used in opencv2
using namespace std;
using namespace cv;
 
void find_feature_matches(
  const Mat &img_1, const Mat &img_2,
  std::vector &keypoints_1,
  std::vector &keypoints_2,
  std::vector &matches);
 
void pose_estimation_2d2d(
  const std::vector &keypoints_1,
  const std::vector &keypoints_2,
  const std::vector &matches,
  Mat &R, Mat &t);
 
void triangulation(
  const vector &keypoint_1,
  const vector &keypoint_2,
  const std::vector &matches,
  const Mat &R, const Mat &t,
  vector &points
);
 
/// 作图用
inline cv::Scalar get_color(float depth) {
  float up_th = 50, low_th = 10, th_range = up_th - low_th;
  if (depth > up_th) depth = up_th;
  if (depth < low_th) depth = low_th;
  return cv::Scalar(255 * depth / th_range, 0, 255 * (1 - depth / th_range));
}
 
// 像素坐标转相机归一化坐标
Point2f pixel2cam(const Point2d &p, const Mat &K);
 
int main(int argc, char **argv) {
  if (argc != 3) {
    cout << "usage: triangulation img1 img2" << endl;
    return 1;
  }
  //-- 读取图像
  Mat img_1 = imread(argv[1], CV_LOAD_IMAGE_COLOR);
  Mat img_2 = imread(argv[2], CV_LOAD_IMAGE_COLOR);
 
  vector keypoints_1, keypoints_2;
  vector matches;
  find_feature_matches(img_1, img_2, keypoints_1, keypoints_2, matches);
  cout << "一共找到了" << matches.size() << "组匹配点" << endl;
 
  //-- 估计两张图像间运动
  Mat R, t;
  pose_estimation_2d2d(keypoints_1, keypoints_2, matches, R, t);
 
  //-- 三角化
  vector points;
  //tr是三维点
  triangulation(keypoints_1, keypoints_2, matches, R, t, tr);
 
  //-- 验证三角化点与特征点的重投影关系
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  Mat img1_plot = img_1.clone();
  Mat img2_plot = img_2.clone();
  for (int i = 0; i < matches.size(); i++) {
    // 第一个图
    float depth1 = points[i].z;
    cout << "depth: " << depth1 << endl;
    Point2d pt1_cam = pixel2cam(keypoints_1[matches[i].queryIdx].pt, K);
    cv::circle(img1_plot, keypoints_1[matches[i].queryIdx].pt, 2, get_color(depth1), 2);
 
    // 第二个图
    Mat pt2_trans = R * (Mat_(3, 1) << points[i].x, points[i].y, points[i].z) + t;
    float depth2 = pt2_trans.at(2, 0);
    cv::circle(img2_plot, keypoints_2[matches[i].trainIdx].pt, 2, get_color(depth2), 2);
  }
  cv::imshow("img 1", img1_plot);
  cv::imshow("img 2", img2_plot);
  cv::waitKey();
 
  return 0;
}
 
void find_feature_matches(const Mat &img_1, const Mat &img_2,
                          std::vector &keypoints_1,
                          std::vector &keypoints_2,
                          std::vector &matches) {
  //-- 初始化
  Mat descriptors_1, descriptors_2;
  // used in OpenCV3
  Ptr detector = ORB::create();
  Ptr descriptor = ORB::create();
  // use this if you are in OpenCV2
  // Ptr detector = FeatureDetector::create ( "ORB" );
  // Ptr descriptor = DescriptorExtractor::create ( "ORB" );
  Ptr matcher = DescriptorMatcher::create("BruteForce-Hamming");
  //-- 第一步:检测 Oriented FAST 角点位置
  detector->detect(img_1, keypoints_1);
  detector->detect(img_2, keypoints_2);
 
  //-- 第二步:根据角点位置计算 BRIEF 描述子
  descriptor->compute(img_1, keypoints_1, descriptors_1);
  descriptor->compute(img_2, keypoints_2, descriptors_2);
 
  //-- 第三步:对两幅图像中的BRIEF描述子进行匹配，使用 Hamming 距离
  vector match;
  // BFMatcher matcher ( NORM_HAMMING );
  matcher->match(descriptors_1, descriptors_2, match);
 
  //-- 第四步:匹配点对筛选
  double min_dist = 10000, max_dist = 0;
 
  //找出所有匹配之间的最小距离和最大距离, 即是最相似的和最不相似的两组点之间的距离
  for (int i = 0; i < descriptors_1.rows; i++) {
    double dist = match[i].distance;
    if (dist < min_dist) min_dist = dist;
    if (dist > max_dist) max_dist = dist;
  }
 
  printf("-- Max dist : %f \n", max_dist);
  printf("-- Min dist : %f \n", min_dist);
 
  //当描述子之间的距离大于两倍的最小距离时,即认为匹配有误.但有时候最小距离会非常小,设置一个经验值30作为下限.
  for (int i = 0; i < descriptors_1.rows; i++) {
    if (match[i].distance <= max(2 * min_dist, 30.0)) {
      matches.push_back(match[i]);
    }
  }
}
 
void pose_estimation_2d2d(
  const std::vector &keypoints_1,
  const std::vector &keypoints_2,
  const std::vector &matches,
  Mat &R, Mat &t) {
  // 相机内参,TUM Freiburg2
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
 
  //-- 把匹配点转换为vector的形式
  vector points1;
  vector points2;
 
  for (int i = 0; i < (int) matches.size(); i++) {
    points1.push_back(keypoints_1[matches[i].queryIdx].pt);
    points2.push_back(keypoints_2[matches[i].trainIdx].pt);
  }
 
  //-- 计算本质矩阵
  Point2d principal_point(325.1, 249.7);        //相机主点, TUM dataset标定值
  int focal_length = 521;            //相机焦距, TUM dataset标定值
  Mat essential_matrix;
  essential_matrix = findEssentialMat(points1, points2, focal_length, principal_point);
 
  //-- 从本质矩阵中恢复旋转和平移信息.
  recoverPose(essential_matrix, points1, points2, R, t, focal_length, principal_point);
}
 
//三角化,根据匹配点和求解到的三维点。存储在points中
void triangulation(
  const vector &keypoint_1,
  const vector &keypoint_2,
  const std::vector &matches,
  const Mat &R, const Mat &t,
  vector &points) {
  Mat T1 = (Mat_(3, 4) <<
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0);
    //根据求解到的RT构造T2矩阵
  Mat T2 = (Mat_(3, 4) <<
    R.at(0, 0), R.at(0, 1), R.at(0, 2), t.at(0, 0),
    R.at(1, 0), R.at(1, 1), R.at(1, 2), t.at(1, 0),
    R.at(2, 0), R.at(2, 1), R.at(2, 2), t.at(2, 0)
  );
  //相机内参
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  vector pts_1, pts_2;
  for (DMatch m:matches) {
    // 将像素坐标转换至相机坐标
    pts_1.push_back(pixel2cam(keypoint_1[m.queryIdx].pt, K));
    pts_2.push_back(pixel2cam(keypoint_2[m.trainIdx].pt, K));
  }
 
  Mat pts_4d;
  cv::triangulatePoints(T1, T2, pts_1, pts_2, pts_4d);
 
  // 转换成非齐次坐标
  for (int i = 0; i < pts_4d.cols; i++) {
    Mat x = pts_4d.col(i);
    x /= x.at(3, 0); // 归一化
    Point3d p(
      x.at(0, 0),
      x.at(1, 0),
      x.at(2, 0)
    );
    points.push_back(p);
  }
}
 
Point2f pixel2cam(const Point2d &p, const Mat &K) {
  return Point2f
    (
      (p.x - K.at(0, 2)) / K.at(0, 0),
      (p.y - K.at(1, 2)) / K.at(1, 1)
    );
}