1.ICP

假設有一組配對好的3D點， \(P={P_{1}, ..., P_{N}}\) , \(P^{'}={P_{1}^{'}, ..., P_{N}^{'}}\)。
有一個歐式變換R,t，使得： $ p_{i} = Rp^{'}_{i} + t $
該問題可以用迭代最近點(ICP)來求解。注意考慮兩組3D點的變換時，和相機沒有關係。
ICP求解線性代數的求解(SVD)和非線性優化方式求解(類似於BA)

2.SVD求解：

定義誤差項： \(e_{i} = p_{i} - ( Rp^{'}_{i} + t )\)
構建最小二乘問題，使誤差平方和達到極小的R,t
定義兩組點的質心：
\(p = \frac{1}{n} \sum^{n}_{i=1} (p_{i}), p^{'} = \frac{1}{n} \sum^{n}_{i=1} (p_{i}^{'}),\)

步驟：

• 計算兩組點的質心位置 \(p,p^{'}\),然後再計算每個點的去質心座標：
  \(q_{i} = p_{i} - p, q_{i}^{'} = p_{i}^{'} - p^{'}\)

• 計算 \(R^{*} = argmin \frac{1}{2} \sum^{n}_{i=1} || q_{i} - Rq_{i}^{'} ||^{2}\)

• 將上式展開，優化函式變為求解 \(-tr( R \sum^{n}_{i=1} q_{i}^{'} q_{i}^{T} )\)
  定義 \(W=\sum^{n}_{i=1} q_{i}^{'} q_{i}^{T}\),對W進行SVD分解，得到\(W=U \Sigma V^{T}\)

• \Sigma 為奇異值組成的對角矩陣，對角線元素從大到小排列，而U和V為正交矩陣。當W滿秩時，\(R=UV^{T}\)

• 根據求出的R，計算t： \(t^{*} = p - Rp^{'}\)

3.程式碼：

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

  // 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;

  // 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;

  Eigen::Matrix3d R_ = U * (V.transpose());
  if (R_.determinant() < 0) {
    R_ = -R_;
  }
  Eigen::Vector3d t_ = Eigen::Vector3d(p1.x, p1.y, p1.z) - R_ * Eigen::Vector3d(p2.x, p2.y, p2.z);

  // convert to cv::Mat
  //推導是按第二張圖到第一張圖的變化，
  //此處進行逆變換，即為第一張圖到第二張圖的變化
  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));
}
 

4.非線性優化方法：

/// 節點，優化變數維度和資料型別
class VertexPose : public g2o::BaseVertex<6, Sophus::SE3d> {
public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;

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

  //更新估計值
  /// 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 {}
};

/// 邊，誤差模型      觀測維度，觀測資料型別，  連結節點型別
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 {
	//獲取姿態估計值  
    const VertexPose *pose = static_cast<const VertexPose *> ( _vertices[0] );
	//計算誤差，測量值-轉換值
    _error = _measurement - pose->estimate() * _point;
  }

  //計算雅可比矩陣
  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;
};

//將頂點和邊加入g2o
oid bundleAdjustment(
  const vector<Point3f> &pts1,
  const vector<Point3f> &pts2,
  Mat &R, Mat &t) {
  // 構建圖優化，先設定g2o
  typedef g2o::BlockSolverX BlockSolverType;
  typedef g2o::LinearSolverDense<BlockSolverType::PoseMatrixType> LinearSolverType; // 線性求解器型別
  // 梯度下降方法，可以從GN, LM, DogLeg 中選
  auto solver = new g2o::OptimizationAlgorithmLevenberg(
    g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));
  g2o::SparseOptimizer optimizer;     // 圖模型
  optimizer.setAlgorithm(solver);   // 設定求解器
  optimizer.setVerbose(true);       // 開啟除錯輸出

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

  // edges
  for (size_t i = 0; i < pts1.size(); i++) {
    EdgeProjectXYZRGBDPoseOnly *edge = new EdgeProjectXYZRGBDPoseOnly(
      Eigen::Vector3d(pts2[i].x, pts2[i].y, pts2[i].z));
    edge->setVertex(0, pose);
    edge->setMeasurement(Eigen::Vector3d(
      pts1[i].x, pts1[i].y, pts1[i].z));
    edge->setInformation(Eigen::Matrix3d::Identity());
    optimizer.addEdge(edge);
  }

  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  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 << endl << "after optimization:" << endl;
  cout << "T=\n" << pose->estimate().matrix() << endl;

  // convert to cv::Mat
  Eigen::Matrix3d R_ = pose->estimate().rotationMatrix();
  Eigen::Vector3d t_ = pose->estimate().translation();
  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));
}