• 视觉里程计：根据相邻图像的信息估计出粗略的相机运动，给后端提供较好的初始值。两帧间的视觉里程计，也称为两视图几何。
  算法主要有特征点法、直接法。
• 特征点：图像中一些特别的地方，例如角点、边缘、区块。由关键点和描述子组成。

特征提取和匹配

以ORB特征为例。ORB特征组成：

• 关键点（Oriented FAST）：指该特征点在图像里的位置，有些还具有朝向、大小等信息。
  – 特征点计算：比较像素间亮度差异。
  – ORB尺度：图像金字塔方法。
  – ORB旋转：灰度质心法。
• 描述子（BRIEF）：通常是要给向量，描述该关键点周围像素的信息。特点：
  – 随机选点比较。
  – 二进制表达。
  – 适用于实时图像匹配。

Fast关键点的检测过程、尺度描述、旋转描述方法，具体见书籍《视觉SLAM十四讲：从理论到实践（第2版）》第155~157页。

OpenCV的ORB特征

使用OpenCV自带的方法进行ORB特征提取和匹配。

• 检测 Oriented FAST 角点位置;
• 根据角点位置计算 BRIEF 描述子;
• 对两幅图像中的BRIEF描述子进行匹配，使用Hamming 距离;
• 匹配点对筛选，筛选依据：汉明距离小于最小距离的2倍;

ch7/orb_cv.cpp

int main(int argc, char **argv) {
  if (argc != 3) {
    cout << "usage: feature_extraction 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 != nullptr && img_2.data != nullptr);

  //-- 初始化
  std::vector<KeyPoint> keypoints_1, keypoints_2;
  Mat descriptors_1, descriptors_2;
  Ptr<FeatureDetector> detector = ORB::create();
  Ptr<DescriptorExtractor> descriptor = ORB::create();
  Ptr<DescriptorMatcher> matcher = DescriptorMatcher::create("BruteForce-Hamming");

  //-- 第一步:检测 Oriented FAST 角点位置
  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  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);
  chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
  chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "extract ORB cost = " << time_used.count() << " seconds. " << endl;

  Mat outimg1;
  drawKeypoints(img_1, keypoints_1, outimg1, Scalar::all(-1), DrawMatchesFlags::DEFAULT);
  imshow("ORB features", outimg1);

  //-- 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> matches;
  t1 = chrono::steady_clock::now();
  matcher->match(descriptors_1, descriptors_2, matches);
  t2 = chrono::steady_clock::now();
  time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "match ORB cost = " << time_used.count() << " seconds. " << endl;

  //-- 第四步:匹配点对筛选
  // 计算最小距离和最大距离
  auto min_max = minmax_element(matches.begin(), matches.end(),
                                [](const DMatch &m1, const DMatch &m2) { return m1.distance < m2.distance; });
  double min_dist = min_max.first->distance;
  double max_dist = min_max.second->distance;

  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.
  std::vector<DMatch> good_matches;
  for (int i = 0; i < descriptors_1.rows; i++) {
    if (matches[i].distance <= max(2 * min_dist, 30.0)) {
      good_matches.push_back(matches[i]);
    }
  }

  //-- 第五步:绘制匹配结果
  Mat img_match;
  Mat img_goodmatch;
  drawMatches(img_1, keypoints_1, img_2, keypoints_2, matches, img_match);
  drawMatches(img_1, keypoints_1, img_2, keypoints_2, good_matches, img_goodmatch);
  imshow("all matches", img_match);
  imshow("good matches", img_goodmatch);
  waitKey(0);

  return 0;
}

手写ORB特征

ch7/orb_self.cpp

ORB提取

使用三角函数计算特征点角度、计算描述子；

// compute the descriptor
void ComputeORB(const cv::Mat &img, vector<cv::KeyPoint> &keypoints, vector<DescType> &descriptors) {
  const int half_patch_size = 8;
  const int half_boundary = 16;
  int bad_points = 0;
  for (auto &kp: keypoints) {
    if (kp.pt.x < half_boundary || kp.pt.y < half_boundary ||
        kp.pt.x >= img.cols - half_boundary || kp.pt.y >= img.rows - half_boundary) {
      // outside
      bad_points++;
      descriptors.push_back({});
      continue;
    }

    float m01 = 0, m10 = 0;
    for (int dx = -half_patch_size; dx < half_patch_size; ++dx) {
      for (int dy = -half_patch_size; dy < half_patch_size; ++dy) {
        uchar pixel = img.at<uchar>(kp.pt.y + dy, kp.pt.x + dx);
        m01 += dx * pixel;
        m10 += dy * pixel;
      }
    }

    // angle should be arc tan(m01/m10);
    float m_sqrt = sqrt(m01 * m01 + m10 * m10) + 1e-18; // avoid divide by zero
    float sin_theta = m01 / m_sqrt;
    float cos_theta = m10 / m_sqrt;

    // compute the angle of this point
    DescType desc(8, 0);
    for (int i = 0; i < 8; i++) {
      uint32_t d = 0;
      for (int k = 0; k < 32; k++) {
        int idx_pq = i * 8 + k;
        cv::Point2f p(ORB_pattern[idx_pq * 4], ORB_pattern[idx_pq * 4 + 1]);
        cv::Point2f q(ORB_pattern[idx_pq * 4 + 2], ORB_pattern[idx_pq * 4 + 3]);

        // rotate with theta
        cv::Point2f pp = cv::Point2f(cos_theta * p.x - sin_theta * p.y, sin_theta * p.x + cos_theta * p.y)
                         + kp.pt;
        cv::Point2f qq = cv::Point2f(cos_theta * q.x - sin_theta * q.y, sin_theta * q.x + cos_theta * q.y)
                         + kp.pt;
        if (img.at<uchar>(pp.y, pp.x) < img.at<uchar>(qq.y, qq.x)) {
          d |= 1 << k;
        }
      }
      desc[i] = d;
    }
    descriptors.push_back(desc);
  }

  cout << "bad/total: " << bad_points << "/" << keypoints.size() << endl;
}

ORB匹配

使用SSE指令集_mm_popcnt_u32计算汉明距离；

// brute-force matching
void BfMatch(const vector<DescType> &desc1, const vector<DescType> &desc2, vector<cv::DMatch> &matches) {
  const int d_max = 40;

  for (size_t i1 = 0; i1 < desc1.size(); ++i1) {
    if (desc1[i1].empty()) continue;
    cv::DMatch m{i1, 0, 256};
    for (size_t i2 = 0; i2 < desc2.size(); ++i2) {
      if (desc2[i2].empty()) continue;
      int distance = 0;
      for (int k = 0; k < 8; k++) {
        distance += _mm_popcnt_u32(desc1[i1][k] ^ desc2[i2][k]);
      }
      if (distance < d_max && distance < m.distance) {
        m.distance = distance;
        m.trainIdx = i2;
      }
    }
    if (m.distance < d_max) {
      matches.push_back(m);
    }
  }
}

主函数调用

int main(int argc, char **argv) {

  // load image
  cv::Mat first_image = cv::imread(first_file, 0);
  cv::Mat second_image = cv::imread(second_file, 0);
  assert(first_image.data != nullptr && second_image.data != nullptr);

  // detect FAST keypoints1 using threshold=40
  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  vector<cv::KeyPoint> keypoints1;
  cv::FAST(first_image, keypoints1, 40);
  vector<DescType> descriptor1;
  ComputeORB(first_image, keypoints1, descriptor1);

  // same for the second
  vector<cv::KeyPoint> keypoints2;
  vector<DescType> descriptor2;
  cv::FAST(second_image, keypoints2, 40);
  ComputeORB(second_image, keypoints2, descriptor2);
  chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
  chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "extract ORB cost = " << time_used.count() << " seconds. " << endl;

  // find matches
  vector<cv::DMatch> matches;
  t1 = chrono::steady_clock::now();
  BfMatch(descriptor1, descriptor2, matches);
  t2 = chrono::steady_clock::now();
  time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "match ORB cost = " << time_used.count() << " seconds. " << endl;
  cout << "matches: " << matches.size() << endl;

  // plot the matches
  cv::Mat image_show;
  cv::drawMatches(first_image, keypoints1, second_image, keypoints2, matches, image_show);
  cv::imshow("matches", image_show);
  cv::imwrite("matches.png", image_show);
  cv::waitKey(0);

  cout << "done." << endl;
  return 0;
}

计算相机运动

针对三种不同场景分别使用不同的方法来求解。

• 单目相机->对极几何；
• 双目或RGB-D相机->ICP；
• 3D+2D->PnP;

2D-2D：对极几何

对极约束求解相机运动

求解相机运动

• 求解基础矩阵 findFundamentalMat
• 求解本质矩阵 findEssentialMat
• 求解单应矩阵 findHomography

三个矩阵的计算公式，都是OpenCV自带的。

求解相机运动的函数
ch7/pose_estimation_2d2d.cpp

void pose_estimation_2d2d(std::vector<KeyPoint> keypoints_1,
                          std::vector<KeyPoint> keypoints_2,
                          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;
}

主函数入口

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<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);

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

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

  //-- 验证对极约束
  Mat K = (Mat_<double>(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_<double>(3, 1) << pt1.x, pt1.y, 1);
    Point2d pt2 = pixel2cam(keypoints_2[m.trainIdx].pt, K);
    Mat y2 = (Mat_<double>(3, 1) << pt2.x, pt2.y, 1);
    Mat d = y2.t() * t_x * R * y1;
    cout << "epipolar constraint = " << d << endl;
  }
  return 0;
}

查找匹配点
find_feature_matches,调用opencv实现，与前面有重复，不详细写。

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");
...
}

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<double>(0, 2)) / K.at<double>(0, 0),
      (p.y - K.at<double>(1, 2)) / K.at<double>(1, 1)
    );
}

三角测量

求解特征点的空间位置。
使用OpenCV自带的三角化函数计算出特征点的深度。

ch7/triangulation.cpp

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;
  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<float>(3, 0); // 归一化
    Point3d p(
      x.at<float>(0, 0),
      x.at<float>(1, 0),
      x.at<float>(2, 0)
    );
    points.push_back(p);
  }
}

3D-2D:PnP

求解PnP的方法：

• 线性优化：直接线性变换（DLT）、EPnP(Efficient PnP)、UPnP
• 非线性优化：光束法平差（Bundle Adjustment，BA）

ch7/pose_estimation_3d2d.cpp

OpenCV：EPnP求解位姿

  Mat r, t;
  solvePnP(pts_3d, pts_2d, K, Mat(), r, t, false); // 调用OpenCV 的 PnP 求解，可选择EPNP，DLS等方法
  Mat R;
  cv::Rodrigues(r, R); // r为旋转向量形式，用Rodrigues公式转换为矩阵

手写位姿估计

对应的公式待补充。

void bundleAdjustmentGaussNewton(
  const VecVector3d &points_3d,
  const VecVector2d &points_2d,
  const Mat &K,
  Sophus::SE3d &pose) {
  typedef Eigen::Matrix<double, 6, 1> Vector6d;
  const int iterations = 10;
  double cost = 0, lastCost = 0;
  double fx = K.at<double>(0, 0);
  double fy = K.at<double>(1, 1);
  double cx = K.at<double>(0, 2);
  double cy = K.at<double>(1, 2);

  for (int iter = 0; iter < iterations; iter++) {
    Eigen::Matrix<double, 6, 6> H = Eigen::Matrix<double, 6, 6>::Zero();
    Vector6d b = Vector6d::Zero();

    cost = 0;
    // compute cost
    for (int i = 0; i < points_3d.size(); i++) {
      Eigen::Vector3d pc = pose * points_3d[i];
      double inv_z = 1.0 / pc[2];
      double inv_z2 = inv_z * inv_z;
      Eigen::Vector2d proj(fx * pc[0] / pc[2] + cx, fy * pc[1] / pc[2] + cy);

      Eigen::Vector2d e = points_2d[i] - proj;

      cost += e.squaredNorm();
      Eigen::Matrix<double, 2, 6> J;
      J << -fx * inv_z,
        0,
        fx * pc[0] * inv_z2,
        fx * pc[0] * pc[1] * inv_z2,
        -fx - fx * pc[0] * pc[0] * inv_z2,
        fx * pc[1] * inv_z,
        0,
        -fy * inv_z,
        fy * pc[1] * inv_z2,
        fy + fy * pc[1] * pc[1] * inv_z2,
        -fy * pc[0] * pc[1] * inv_z2,
        -fy * pc[0] * inv_z;

      H += J.transpose() * J;
      b += -J.transpose() * e;
    }

    Vector6d dx;
    dx = H.ldlt().solve(b);

    if (isnan(dx[0])) {
      cout << "result is nan!" << endl;
      break;
    }

    if (iter > 0 && cost >= lastCost) {
      // cost increase, update is not good
      cout << "cost: " << cost << ", last cost: " << lastCost << endl;
      break;
    }

    // update your estimation
    pose = Sophus::SE3d::exp(dx) * pose;
    lastCost = cost;

    cout << "iteration " << iter << " cost=" << std::setprecision(12) << cost << endl;
    if (dx.norm() < 1e-6) {
      // converge
      break;
    }
  }

  cout << "pose by g-n: \n" << pose.matrix() << endl;
}

g2o进行BA优化

节点和边的定义(这里是自定义的，g2o也有自带的)

• 节点：第二个相机的位姿节点T
• 边：每个3D点在第二个相机中的投影

/// vertex and edges used in g2o ba
class VertexPose : public g2o::BaseVertex<6, Sophus::SE3d> {
public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;

  virtual void setToOriginImpl() override {
    _estimate = Sophus::SE3d();
  }

  /// left multiplication on SE3
  virtual void oplusImpl(const double *update) override {
    Eigen::Matrix<double, 6, 1> update_eigen;
    update_eigen << update[0], update[1], update[2], update[3], update[4], update[5];
    _estimate = Sophus::SE3d::exp(update_eigen) * _estimate;
  }

  virtual bool read(istream &in) override {}

  virtual bool write(ostream &out) const override {}
};

class EdgeProjection : public g2o::BaseUnaryEdge<2, Eigen::Vector2d, VertexPose> {
public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;

  EdgeProjection(const Eigen::Vector3d &pos, const Eigen::Matrix3d &K) : _pos3d(pos), _K(K) {}

计算重投影误差

  virtual void computeError() override {
    const VertexPose *v = static_cast<VertexPose *> (_vertices[0]);
    Sophus::SE3d T = v->estimate();
    Eigen::Vector3d pos_pixel = _K * (T * _pos3d);
    pos_pixel /= pos_pixel[2];
    _error = _measurement - pos_pixel.head<2>();
  }

计算雅可比矩阵

  virtual void linearizeOplus() override {
    const VertexPose *v = static_cast<VertexPose *> (_vertices[0]);
    Sophus::SE3d T = v->estimate();
    Eigen::Vector3d pos_cam = T * _pos3d;
    double fx = _K(0, 0);
    double fy = _K(1, 1);
    double cx = _K(0, 2);
    double cy = _K(1, 2);
    double X = pos_cam[0];
    double Y = pos_cam[1];
    double Z = pos_cam[2];
    double Z2 = Z * Z;
    _jacobianOplusXi
      << -fx / Z, 0, fx * X / Z2, fx * X * Y / Z2, -fx - fx * X * X / Z2, fx * Y / Z,
      0, -fy / Z, fy * Y / (Z * Z), fy + fy * Y * Y / Z2, -fy * X * Y / Z2, -fy * X / Z;
  }

  virtual bool read(istream &in) override {}

  virtual bool write(ostream &out) const override {}

private:
  Eigen::Vector3d _pos3d;
  Eigen::Matrix3d _K;
};

使用g2o定义并求解图优化问题

void bundleAdjustmentG2O(
  const VecVector3d &points_3d,
  const VecVector2d &points_2d,
  const Mat &K,
  Sophus::SE3d &pose) {

  // 构建图优化，先设定g2o
  typedef g2o::BlockSolver<g2o::BlockSolverTraits<6, 3>> BlockSolverType;  // pose is 6, landmark is 3
  typedef g2o::LinearSolverDense<BlockSolverType::PoseMatrixType> LinearSolverType; // 线性求解器类型
  // 梯度下降方法，可以从GN, LM, DogLeg 中选
  auto solver = new g2o::OptimizationAlgorithmGaussNewton(
    g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));
  g2o::SparseOptimizer optimizer;     // 图模型
  optimizer.setAlgorithm(solver);   // 设置求解器
  optimizer.setVerbose(true);       // 打开调试输出

  // vertex
  VertexPose *vertex_pose = new VertexPose(); // camera vertex_pose
  vertex_pose->setId(0);
  vertex_pose->setEstimate(Sophus::SE3d());
  optimizer.addVertex(vertex_pose);

  // K
  Eigen::Matrix3d K_eigen;
  K_eigen <<
          K.at<double>(0, 0), K.at<double>(0, 1), K.at<double>(0, 2),
    K.at<double>(1, 0), K.at<double>(1, 1), K.at<double>(1, 2),
    K.at<double>(2, 0), K.at<double>(2, 1), K.at<double>(2, 2);

  // edges
  int index = 1;
  for (size_t i = 0; i < points_2d.size(); ++i) {
    auto p2d = points_2d[i];
    auto p3d = points_3d[i];
    EdgeProjection *edge = new EdgeProjection(p3d, K_eigen);
    edge->setId(index);
    edge->setVertex(0, vertex_pose);
    edge->setMeasurement(p2d);
    edge->setInformation(Eigen::Matrix2d::Identity());
    optimizer.addEdge(edge);
    index++;
  }

  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  optimizer.setVerbose(true);
  optimizer.initializeOptimization();
  optimizer.optimize(10);
  chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
  chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
  cout << "optimization costs time: " << time_used.count() << " seconds." << endl;
  cout << "pose estimated by g2o =\n" << vertex_pose->estimate().matrix() << endl;
  pose = vertex_pose->estimate();
}

3D-3D:ICP

ch7/pose_estimation_3d3d.cpp

SVD法（线性解法）

1、计算两组点的质心位置，然后计算每个点的去质心坐标。

void pose_estimation_3d3d(const vector<Point3f> &pts1,
                          const vector<Point3f> &pts2,
                          Mat &R, Mat &t) {
  Point3f p1, p2;     // center of mass
  int N = pts1.size();
  for (int i = 0; i < N; i++) {
    p1 += pts1[i];
    p2 += pts2[i];
  }
  p1 = Point3f(Vec3f(p1) / N);
  p2 = Point3f(Vec3f(p2) / N);
  vector<Point3f> q1(N), q2(N); // remove the center
  for (int i = 0; i < N; i++) {
    q1[i] = pts1[i] - p1;
    q2[i] = pts2[i] - p2;
  }

2、计算旋转矩阵
定义矩阵W

  // compute q1*q2^T
  Eigen::Matrix3d W = Eigen::Matrix3d::Zero();
  for (int i = 0; i < N; i++) {
    W += Eigen::Vector3d(q1[i].x, q1[i].y, q1[i].z) * Eigen::Vector3d(q2[i].x, q2[i].y, q2[i].z).transpose();
  }
  cout << "W=" << W << endl;

对W进行SVD分解

  // SVD on W
  Eigen::JacobiSVD<Eigen::Matrix3d> svd(W, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Eigen::Matrix3d U = svd.matrixU();
  Eigen::Matrix3d V = svd.matrixV();

  cout << "U=" << U << endl;
  cout << "V=" << V << endl;

求解R

  Eigen::Matrix3d R_ = U * (V.transpose());
  if (R_.determinant() < 0) {
    R_ = -R_;
  }

求解t

  Eigen::Vector3d t_ = Eigen::Vector3d(p1.x, p1.y, p1.z) - R_ * Eigen::Vector3d(p2.x, p2.y, p2.z);

  // convert to cv::Mat
  R = (Mat_<double>(3, 3) <<
    R_(0, 0), R_(0, 1), R_(0, 2),
    R_(1, 0), R_(1, 1), R_(1, 2),
    R_(2, 0), R_(2, 1), R_(2, 2)
  );
  t = (Mat_<double>(3, 1) << t_(0, 0), t_(1, 0), t_(2, 0));
}

非线性优化法（BA解法）

结点定义、BA调用的代码，与PnP的BA优化方法类似。这里仅列出有差异地方。
边定义

/// g2o edge
class EdgeProjectXYZRGBDPoseOnly : public g2o::BaseUnaryEdge<3, Eigen::Vector3d, VertexPose> {
public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;

  EdgeProjectXYZRGBDPoseOnly(const Eigen::Vector3d &point) : _point(point) {}

  virtual void computeError() override {
    const VertexPose *pose = static_cast<const VertexPose *> ( _vertices[0] );
    _error = _measurement - pose->estimate() * _point;
  }

  virtual void linearizeOplus() override {
    VertexPose *pose = static_cast<VertexPose *>(_vertices[0]);
    Sophus::SE3d T = pose->estimate();
    Eigen::Vector3d xyz_trans = T * _point;
    _jacobianOplusXi.block<3, 3>(0, 0) = -Eigen::Matrix3d::Identity();
    _jacobianOplusXi.block<3, 3>(0, 3) = Sophus::SO3d::hat(xyz_trans);
  }

  bool read(istream &in) {}

  bool write(ostream &out) const {}

protected:
  Eigen::Vector3d _point;
};

参考资料：
1、书籍：《视觉SLAM十四讲：从理论到实践（第2版）》
2、代码：https://github.com/gaoxiang12/slambook2