随手笔记——2D−2D: 对极几何及代码示例

说明

2D−2D: 对极几何及代码示例

几个重要矩阵

基础矩阵（Fundamental Matrix）F 、本质矩阵（Essential Matrix）E、单应矩阵（Homography）H

几个重要函数

fundamental_matrix = findFundamentalMat(points1, points2, CV_FM_8POINT);
essential_matrix = findEssentialMat(points1, points2, focal_length, principal_point);
homography_matrix = findHomography(points1, points2, RANSAC, 3);

// 此函数仅在Opencv3中提供
recoverPose(essential_matrix, points1, points2, R, t, focal_length, principal_point);

对极约束求解相机运动

整个程序使用opencv提供的算法进行求解
源代码：

#include 
#include 
#include 
#include 
#include 
// #include "extra.h" // use this if in OpenCV2

using namespace std;
using namespace cv;

/****************************************************
 * 本程序演示了如何使用2D-2D的特征匹配估计相机运动
 * **************************************************/

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(
  std::vector keypoints_1,
  std::vector keypoints_2,
  std::vector matches,
  Mat &R, Mat &t);

// 像素坐标转相机归一化坐标
Point2d pixel2cam(const Point2d &p, const Mat &K);

int main(int argc, char **argv) {
  if (argc != 3) {
    cout << "usage: pose_estimation_2d2d 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);
  assert(img_1.data && img_2.data && "Can not load images!");

  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);

  //-- 验证E=t^R*scale
  Mat t_x =
    (Mat_(3, 3) << 0, -t.at(2, 0), t.at(1, 0),
      t.at(2, 0), 0, -t.at(0, 0),
      -t.at(1, 0), t.at(0, 0), 0);

  cout << "t^R=" << endl << t_x * R << endl;

  //-- 验证对极约束
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  for (DMatch m: matches) {
    Point2d pt1 = pixel2cam(keypoints_1[m.queryIdx].pt, K);
    Mat y1 = (Mat_(3, 1) << pt1.x, pt1.y, 1);
    Point2d pt2 = pixel2cam(keypoints_2[m.trainIdx].pt, K);
    Mat y2 = (Mat_(3, 1) << pt2.x, pt2.y, 1);
    Mat d = y2.t() * t_x * R * y1;
    cout << "epipolar constraint = " << d << endl;
  }
  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]);
    }
  }
}

//将像素坐标转换到归一化平面上的坐标
Point2d pixel2cam(const Point2d &p, const Mat &K) {
  return Point2d
    (
      (p.x - K.at(0, 2)) / K.at(0, 0),
      (p.y - K.at(1, 2)) / K.at(1, 1)
    );
}

void pose_estimation_2d2d(std::vector keypoints_1,
                          std::vector keypoints_2,
                          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);
  }

  //-- 计算基础矩阵
  Mat fundamental_matrix;
  fundamental_matrix = findFundamentalMat(points1, points2, CV_FM_8POINT);
  cout << "fundamental_matrix is " << endl << fundamental_matrix << endl;

  //-- 计算本质矩阵
  Point2d principal_point(325.1, 249.7);  //相机光心, TUM dataset标定值
  double focal_length = 521;      //相机焦距, TUM dataset标定值
  Mat essential_matrix;
  essential_matrix = findEssentialMat(points1, points2, focal_length, principal_point);
  cout << "essential_matrix is " << endl << essential_matrix << endl;

  //-- 计算单应矩阵
  //-- 但是本例中场景不是平面，单应矩阵意义不大
  Mat homography_matrix;
  homography_matrix = findHomography(points1, points2, RANSAC, 3);
  cout << "homography_matrix is " << endl << homography_matrix << endl;

  //-- 从本质矩阵中恢复旋转和平移信息.
  // 此函数仅在Opencv3中提供
  recoverPose(essential_matrix, points1, points2, R, t, focal_length, principal_point);
  cout << "R is " << endl << R << endl;
  cout << "t is " << endl << t << endl;

}

注：分解得到的 R, t 一共有 4 种可能性。OpenCV 会替我们使用三角化检测角点的深度是否为正，从而选出正确的解。