7.BA优化中的单节点高斯牛顿方程推导以及g2o写法

image.png

从世界坐标系下的坐标点到像素点的流程如上图,其中
根据上面的流程可以抽象出如下信息

image.png

如上所示,我们可以定义观测流程 ,其中 指代 ,而 指代三维点 ,而 表示实际测得的特征像素点坐标

image.png

如上图为了让问题便于后面的求解,这里采用\kexi 表示姿态的李代数表示
于是 ,表示从姿态 去观测世界坐标点

可定义我们得到某个姿态观测某个世界坐标点的 误差函数

所以表示上图的整体误差函数应该是

我们知道对于元函数泰勒一阶展开有

也可以写成

对于此处实际的函数
可写成

待优化变量

对原函数进行展开


其中 ,,为相机坐标

在不考虑畸变的情况下对 投影到像素坐标即:

得到,

分为两个部分,一部分对李代数求导,另一部分是对求导

对于第一部分利用链式法则

我们知道前一项
\frac{\partial error_{ij}}{\partial p'_j} =\frac{\partial(z_{ij观测}-\frac{1}{s}Kp'_j)}{\partial p'_j} =\frac{\partial\begin{bmatrix}u_{ij观测}-f_x\frac{X'}{Z'}-c_x\\v_{ij观测}-f_y\frac{Y'}{Z'}-c_y\end{bmatrix}}{\partial\begin{bmatrix}X'&Y'&Z'\end{bmatrix}}

而第二项
注:

(此处比较复杂,结论由slam14讲,4.3.5给出,这个坑太大一下子填不过来,先摘抄了

因为此处是三维,因此,结果也只取三行

于是最终第一部分的雅可比矩阵如下

=-\begin{bmatrix} \frac{f_x}{Z'}&0&-\frac{f_xX'}{Z'^2}&-\frac{f_xX'Y'}{Z'^2}&f_x+\frac{f_xX'^2}{Z'^2}&-\frac{f_xY'}{Z'}\\ 0&\frac{f_y}{Z'}&-\frac{f_yY'}{Z'^2}&-f_y-\frac{f_yY'^2}{Z'^2}&\frac{f_yX'Y'}{Z'^2}&\frac{f_yX'}{Z'} \end{bmatrix}

接下来是对第二部分,即特征点空间位置的雅可比偏导数矩阵
依旧根据链式求导法则


上式的第一项已经推导过,而第二项
我们知道

所以

这样,我们对于位姿和世界坐标点的雅可比矩阵就都有了

根据第5小节的内容,我们知道

高斯牛顿对于单节点多条边的情况是这么处理的

具体来讲应该是这样

即,对单节点的所有边的误差迭代,且这里只优化了

对应到这边具体的slam问题,通常是多个节点,即多个观察pose的问题,优化的也不仅仅是pose了,还得包括观察的路标,这里暂时不讲等下一节
下面的例子同样的也只针对单节点考虑优化问题

在g2o优化中,以防误解,这里再把节点和边的概念补充一下,如下图,每个节点Pose,比如节点由 表示,节点能看到多个不同的 ,也就是和每一个将构成单个的边,所以单个Pose的观测会有多条边

image.png

实际的代码实现方式在slam14讲ch7/pose_estimation_3d2d.cpp的有所实现,确实和我们理解的一样
注意这个代码实现包含了好几种方式
包含手写高斯牛顿方式,opencv的求解方式,以及g2o方式,
这里我们重点参考手写高斯牛顿以及g2o的写法的类似
以下是代码的参考

#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#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);

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

// BA by g2o
typedef vector> VecVector2d;
typedef vector> VecVector3d;

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

// BA by gauss-newton
void bundleAdjustmentGaussNewton(
  const VecVector3d &points_3d,
  const VecVector2d &points_2d,
  const Mat &K,
  Sophus::SE3d &pose
);

int main(int argc, char **argv) {
  if (argc != 5) {
    cout << "usage: pose_estimation_3d2d img1 img2 depth1 depth2" << 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;

  // 建立3D点
  Mat d1 = imread(argv[3], CV_LOAD_IMAGE_UNCHANGED);       // 深度图为16位无符号数,单通道图像
  Mat K = (Mat_(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  vector pts_3d;
  vector pts_2d;
  for (DMatch m:matches) {
    ushort d = d1.ptr(int(keypoints_1[m.queryIdx].pt.y))[int(keypoints_1[m.queryIdx].pt.x)];
    if (d == 0)   // bad depth
      continue;
    float dd = d / 5000.0;
    Point2d p1 = pixel2cam(keypoints_1[m.queryIdx].pt, K);
    pts_3d.push_back(Point3f(p1.x * dd, p1.y * dd, dd));
    pts_2d.push_back(keypoints_2[m.trainIdx].pt);
  }

  cout << "3d-2d pairs: " << pts_3d.size() << endl;

  chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
  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
  chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
  chrono::duration time_used = chrono::duration_cast>(t2 - t1);
  cout << "solve pnp in opencv cost time: " << time_used.count() << " seconds." << endl;

  cout << "R=" << endl << R << endl;
  cout << "t=" << endl << t << endl;

  VecVector3d pts_3d_eigen;
  VecVector2d pts_2d_eigen;
  for (size_t i = 0; i < pts_3d.size(); ++i) {
    pts_3d_eigen.push_back(Eigen::Vector3d(pts_3d[i].x, pts_3d[i].y, pts_3d[i].z));
    pts_2d_eigen.push_back(Eigen::Vector2d(pts_2d[i].x, pts_2d[i].y));
  }

  cout << "calling bundle adjustment by gauss newton" << endl;
  Sophus::SE3d pose_gn;
  t1 = chrono::steady_clock::now();
  bundleAdjustmentGaussNewton(pts_3d_eigen, pts_2d_eigen, K, pose_gn);
  t2 = chrono::steady_clock::now();
  time_used = chrono::duration_cast>(t2 - t1);
  cout << "solve pnp by gauss newton cost time: " << time_used.count() << " seconds." << endl;

  cout << "calling bundle adjustment by g2o" << endl;
  Sophus::SE3d pose_g2o;
  t1 = chrono::steady_clock::now();
  bundleAdjustmentG2O(pts_3d_eigen, pts_2d_eigen, K, pose_g2o);
  t2 = chrono::steady_clock::now();
  time_used = chrono::duration_cast>(t2 - t1);
  cout << "solve pnp by g2o cost time: " << time_used.count() << " seconds." << 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]);
    }
  }
}

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

  for (int iter = 0; iter < iterations; iter++) {
    Eigen::Matrix H = Eigen::Matrix::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 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;
}

/// 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 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) {}

  virtual void computeError() override {
    const VertexPose *v = static_cast (_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>();
  }

  virtual void linearizeOplus() override {
    const VertexPose *v = static_cast (_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;
};

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> BlockSolverType;  // pose is 6, landmark is 3
  typedef g2o::LinearSolverDense 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(g2o::make_unique()));
  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(0, 0), K.at(0, 1), K.at(0, 2),
    K.at(1, 0), K.at(1, 1), K.at(1, 2),
    K.at(2, 0), K.at(2, 1), K.at(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 time_used = chrono::duration_cast>(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();
}

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