视觉十四讲：第七讲_2D-2D:对极几何估计姿态

1.对极几何

从2张图片中,得到若干个配对好的2d特征点,就可以运用对极几何来恢复出两帧之间的运动.

设P的空间坐标为: $P=[X,Y,Z]^{T}$

两个像素点$p_{1},p_{2}$的像素坐标为: $s_{1}p_{1}=KP, s_{2}p_{2}=K(RP+t)$

K为相机内参,R,t为图像1到图像2的旋转矩阵和平移矩阵.

取$x_{1}=k^{-1}p_{1}, x_{2}=k^{-1}p_{2}$ (x1,x2是两个像素坐标在归一化平面上的坐标)
$x_{2}=Rx_{1}+t$,两侧同时左乘$x^{T}_{2}$t^{^}
$x^{T}_{2}$t^{^{</sup>$x_{2}$=$x^{T}_{2}$t<sup>}}$Rx_{1}$,等式左边为0
$x^{T}_{2}$t^{^}$Rx_{1}=0$
带入$p_{1},p_{2}$得$p_{2}^{T}K^{-T}$t^{^}$RK^{-1}p_{1} = 0$
取基础矩阵$F=K^{-T}EK^{-1}$,取本质矩阵$E=$t^{^}$R$
$x_{2}^{T}Ex_{1} = p_{2}^{T}Fp_{1} = 0$

相机姿态估计问题变成以下两步:

根据配对点的像素位置求出R或者F
根据E或F求出R,t

2.本质矩阵

根据本质矩阵$E=$t^{^}$R$定义,这是一个3*3的矩阵,经典是使用8点法来求解.求解出E后,可通过奇异值分解得到相机的运动R和t.

注意:求出的E和t具有尺度一致性,通常把t进行归一化.

3.尺度不确定性

对t的长度归一化,直接导致单目视觉的尺度不确定性.解决办法可以通过SLAM的初始化来解决,初始时,使机器人平移一段距离,然后以此距离作为平移的单位.初始化之后,就可以使用3D-2D来计算相机运动了

工程中,通常匹配的点比较多,这时可以通过构造最小二乘法来进行求解E,但是由于存在误匹配的情况,所以更多的是使用随机采样一致性(RANSAC)来求解

4.三角测距来测量深度

根据对极几何的定义,$x_{1},x_{2}$为两个特征点归一化的坐标,则满足:

$s_{1}x_{1}=s_{2}Rx_{2}+t$,两边同时左乘$x_{1}$^{^}得
$s_{2}$$x_{1}$^{^{</sup>$Rx_{2}+$$x_{1}$<sup>}}t = 0
其中R和t在上面已经求出,故该式为$s_{2}的$方程.
由于噪声存在,通常可以使用最小二乘法来求解$s_{2}$,从而$s_{1}$也能求出

#include <iostream>

#include <opencv2/opencv.hpp>

// #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<KeyPoint> &keypoints_1,

  std::vector<KeyPoint> &keypoints_2,

  std::vector<DMatch> &matches);

void pose_estimation_2d2d(

  const std::vector<KeyPoint> &keypoints_1,

  const std::vector<KeyPoint> &keypoints_2,

  const std::vector<DMatch> &matches,

  Mat &R, Mat &t);

void triangulation(

  const vector<KeyPoint> &keypoint_1,

  const vector<KeyPoint> &keypoint_2,

  const std::vector<DMatch> &matches,

  const Mat &R, const Mat &t,

  vector<Point3d> &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<KeyPoint> keypoints_1, keypoints_2;

  vector<DMatch> 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<Point3d> points;

  triangulation(keypoints_1, keypoints_2, matches, R, t, points);

  //-- 验证三角化点与特征点的重投影关系

  Mat K = (Mat_<double>(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_<double>(3, 1) << points[i].x, points[i].y, points[i].z) + t;

    float depth2 = pt2_trans.at<double>(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<KeyPoint> &keypoints_1,

                          std::vector<KeyPoint> &keypoints_2,

                          std::vector<DMatch> &matches) {

  //-- 初始化

  Mat descriptors_1, descriptors_2;

  // used in OpenCV3

  Ptr<FeatureDetector> detector = ORB::create();

  Ptr<DescriptorExtractor> descriptor = ORB::create();

  // use this if you are in OpenCV2

  // Ptr<FeatureDetector> detector = FeatureDetector::create ( "ORB" );

  // Ptr<DescriptorExtractor> descriptor = DescriptorExtractor::create ( "ORB" );

  Ptr<DescriptorMatcher> 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<DMatch> 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<KeyPoint> &keypoints_1,

  const std::vector<KeyPoint> &keypoints_2,

  const std::vector<DMatch> &matches,

  Mat &R, Mat &t) {

  // 相机内参,TUM Freiburg2

  Mat K = (Mat_<double>(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);

  //-- 把匹配点转换为vector<Point2f>的形式

  vector<Point2f> points1;

  vector<Point2f> 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);

}

void triangulation(

  const vector<KeyPoint> &keypoint_1,

  const vector<KeyPoint> &keypoint_2,

  const std::vector<DMatch> &matches,

  const Mat &R, const Mat &t,

  vector<Point3d> &points) {

  Mat T1 = (Mat_<float>(3, 4) <<

    1, 0, 0, 0,

    0, 1, 0, 0,

    0, 0, 1, 0);

  Mat T2 = (Mat_<float>(3, 4) <<

    R.at<double>(0, 0), R.at<double>(0, 1), R.at<double>(0, 2), t.at<double>(0, 0),

    R.at<double>(1, 0), R.at<double>(1, 1), R.at<double>(1, 2), t.at<double>(1, 0),

    R.at<double>(2, 0), R.at<double>(2, 1), R.at<double>(2, 2), t.at<double>(2, 0)

  );

  Mat K = (Mat_<double>(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);

  vector<Point2f> 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;

//得到深度点,为4维齐次方程

//输入是两个图片的位姿,以及特征点在两个相机中的坐标,归一化坐标

//输出是第一个图片的特征点在相机中的坐标

  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);  //取列信息

    x /= x.at<float>(3, 0); // 归一化

    Point3d p(

      x.at<float>(0, 0),  //得到非齐次的3D点

      x.at<float>(1, 0),

      x.at<float>(2, 0)

    );

    points.push_back(p);

  }

}

Point2f pixel2cam(const Point2d &p, const Mat &K) {

  return Point2f

    (

      (p.x - K.at<double>(0, 2)) / K.at<double>(0, 0),

      (p.y - K.at<double>(1, 2)) / K.at<double>(1, 1)

    );

}

CMakeLists.txt:

cmake_minimum_required(VERSION 2.8)

project(orb)

set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=c++11")

list(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake)

include_directories(inc)

aux_source_directory(src DIR_SRCS)

SET(SOUR_FILE ${DIR_SRCS})

find_package(OpenCV 3 REQUIRED)

find_package(G2O REQUIRED)

find_package(Sophus REQUIRED)

include_directories(

        ${OpenCV_INCLUDE_DIRS}

        ${G2O_INCLUDE_DIRS}

        ${Sophus_INCLUDE_DIRS}

        "/usr/include/eigen3/"

)

add_executable(orb ${SOUR_FILE})

target_link_libraries(orb ${OpenCV_LIBS})