视觉SLAM十四讲作业练习（5）对极几何

两帧图像，从蓝色到绿色的运动为 R 与 t ，设两个坐标系 O 与 O’ ，在 O 坐标系下点 X 在归一化平面的坐标为 x ，在 O’ 坐标系下 X 在归一化平面的坐标为 x’ 。目的是将第一个坐标系下的坐标转换到第二个坐标系下的在坐标，即用 O‘ 坐标系来描述 O 坐标系下的点。所以计算的 R 是 R O ′ O R_{O'O} RO′O​ 和 t O ′ O t_{O'O} tO′O​，所以图上的 t 方向是由 O‘ 指向 O 。

由三角形上的向量关系可得：
x ′ ⋅ ( t × x ) = 0 x' cdot(t times x)=0 x′⋅(t×x)=0
矩阵形式：
x ′    T ( t × R    x ) = x ′    T t Λ R    x = 0 x';^T (ttimes R;x)=x';^T t^Lambda R;x=0 x′T(t×Rx)=x′TtΛRx=0
关键点便在于 x = R    x x = R;x x=Rx 这里，原因：x是 O坐标系下的坐标，需要转换到 O‘ 坐标系下，所以左乘 R。

此时得到本质矩阵E：
E = t Λ R E =t^Lambda R E=tΛR
算了基础矩阵F就先不推导了，总之先记住：
F = K − T E K − 1 F=K^{-T}EK^{-1} F=K−TEK−1
且
x 2 T E x 1 = p 2 T F p 1 = 0 x_2^T Ex_1 = p_2^TFp_1 = 0 x2T​Ex1​=p2T​Fp1​=0
其中， x 1 x_1 x1​， x 2 x_2 x2​为归一化平面上的点， p 1 p_1 p1​， p 2 p_2 p2​为像素平面上的点。

本质矩阵E可以通过奇异值分解得到相机运动的R与t。

主程序:

验证对极约束

#include 
#include 
#include 
#include 
#include 

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(
    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) 
{

    //-- 1.读取图像
    Mat img_1 = imread("../1.png");
    Mat img_2 = imread("../2.png");

    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;

    //-- 2.估计两张图像间运动
    Mat R, t;
    pose_estimation_2d2d(keypoints_1, keypoints_2, matches, R, t);

    //-- 3.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;

    //-- 4.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;
}

像素坐标转相机归一化坐标

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)
    );
}

靠，怎么归一化的啊。。。。

求本质矩阵E，基本矩阵F，R以及t：

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, -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;

}

#include 
#include 
#include 

using namespace Eigen;

#include 

#include 

using namespace std;

int main(int argc, char **argv) {
    // Essential matrix
    Matrix3d E;
    E << -0.0203618550523477, -0.4007110038118445, -0.03324074249824097,
            0.3939270778216369, -0.03506401846698079, 0.5857110303721015,
            -0.006788487241438284, -0.5815434272915686, -0.01438258684486258;

    // R,t to be recovered
    Matrix3d R;
    Vector3d t;

    // SVD and fix sigular values
    // 1.定义一个SVD分解
    JacobiSVD svd (E, ComputeFullV | ComputeFullU);
    Matrix3d U = svd.matrixU();
    Matrix3d V = svd.matrixV();
    Vector3d singular_values = svd.singularValues();
    
    // 2.处理奇异值
    double singular_value = (singular_values[0] + singular_values[1]) / 2;
    singular_values = Vector3d(singular_value, singular_value, 0);
    DiagonalMatrix sigma (singular_values);	//得到diag()矩阵

    // set t1, t2, R1, R2 
    // 3.四个可能的解
    Matrix3d Rz1 = AngleAxisd(M_PI/2, Vector3d(0,0,1)).toRotationMatrix();
    Matrix3d Rz2 = AngleAxisd(-M_PI/2, Eigen::Vector3d(0,0,1)).toRotationMatrix();

    Matrix3d t_wedge1 = U * Rz1 * sigma * U.transpose(); 
    Matrix3d t_wedge2 = U * Rz2 * sigma * U.transpose();
    
    Matrix3d R1 = U * Rz1.transpose() * V.transpose();
    Matrix3d R2 = U * Rz2.transpose() * V.transpose();

    cout << "R1 = " << R1 << endl;
    cout << "R2 = " << R2 << endl;
    cout << "t1 = " << Sophus::SO3d::vee(t_wedge1) << endl;
    cout << "t2 = " << Sophus::SO3d::vee(t_wedge2) << endl;

    // check t^R=E up to scale
    Matrix3d tR = t_wedge1 * R1;
    cout << "t^R = " << tR << endl;

    return 0;
}

具体求解算法参见《SLAM–三角测量SVD分解法、最小二乘法及R t矩阵的判断》http://t.csdn.cn/YFbBb

主程序

int main(int argc, char **argv) {
  //-- 1.读取图像
  Mat img_1 = imread("../1.png");
  Mat img_2 = imread("../2.png");

  //-- 2.匹配
  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;

  //-- 3.估计两张图像间运动
  Mat R, t;
  pose_estimation_2d2d(keypoints_1, keypoints_2, matches, R, t);	//得到R，t

  //-- 4.三角化
  vector points;
  triangulation(keypoints_1, keypoints_2, matches, R, t, points);

  //-- 5.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;
}

通过匹配及对极几何得到R，t求得T1，T2；K，两个相机坐标系下归一化的点pts，使用cv内置函数：cv::triangulatePoints(T1, T2, pts_1, pts_2, pts_4d);求得pst_4d然后转换成齐次坐标得到 p 点。

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);
  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)
  );

  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);
  }
}