7.BA优化中的单节点高斯牛顿方程推导以及g2o写法

作者: 光能蜗牛 | 来源:发表于2022-09-26 11:05 被阅读0次

image.png

从世界坐标系下的坐标点 $p$ 到像素点 $(u_s,v_s)$ 的流程如上图，其中 $r_c^2=u_c^2+v_c^2$
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

z=h(x,y)

,其中

x

指代

R,t

,而

y

指代三维点

p

，而

z

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

\xi ^{\wedge}=[R|t]

于是

z_{ij}=h(\xi^{\wedge}_{i},p_j)

,表示从姿态

\xi ^{\wedge}_i

去观测世界坐标点

p_j

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
$error_{ij}=z_{ij观测}-z_{ij}=z_{ij观测}-h(\xi^{\wedge}_{i},p_j)$

所以表示上图的整体误差函数应该是
$\frac{1}{2}\sum_{i=1}^m\sum_{j=1}^n||error_{ij}||^2=\frac{1}{2}\sum_{i=1}^m\sum_{j=1}^n||z_{ij观测}-h(\xi^{\wedge}_i,p_j)||^2$

我们知道对于 $n$ 元函数泰勒一阶展开有
$F(X^{k+1})\approx F(X^{k})+J(X^k)^T\Delta X$
也可以写成
$F(X^{k}+\Delta X)\approx F(X^{k})+J(X^k)^T\Delta X$
对于此处实际的函数
可写成 $error_{ij}(X_{ij}^k+\Delta X_{ij})\approx error_{ij}(X_{ij}^k)+\frac{\partial error_{ij}(X_{ij}^k)}{\partial \Delta X_{ij}}\Delta X_{ij}$

待优化变量 $X_{ij}=[\xi_i,p_j]^T=[]$

对原函数进行展开
$error_{ij}=z_{ij观测}-h(\xi_{i},p_j)$
$=z_{ij观测}-\frac{1}{s}Kp'_j$
$=z_{ij观测}-\frac{1}{s}Ke^{\xi_i^{\wedge}}p_j$

其中 $p'_j=e^{\xi_i^{\wedge}}p_j$ ，， $p'_j$ 为相机坐标 $[X',Y',Z']^T$

在不考虑畸变的情况下对 $p_j$ 投影到像素坐标即:

$\begin{bmatrix}u\\v\\1\end{bmatrix}=\frac{1}{s}\begin{bmatrix}f_x&0&c_x\\0&f_y&c_y\\0&0&1\end{bmatrix}\begin{bmatrix}X'\\Y'\\Z'\end{bmatrix}$

得到 $u=f_x\frac{X'}{Z'}+c_x$ ， $v=f_y\frac{Y'}{Z'}+c_y$

$\frac{\partial error_{ij}}{\partial \Delta X}$ 分为两个部分，一部分对李代数 $\xi_{i}$ 求导，另一部分是对 $p_j$ 求导

对于第一部分利用链式法则
$\frac{\partial error_{ij}}{\partial \delta \xi_i}=\frac{\partial error_{ij}}{\partial p'_j}.\frac{\partial p'_j}{\partial \delta \xi_i}$

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

$=-\begin{bmatrix}\frac{f_x}{Z'}&0&-\frac{f_xX'}{Z'^2}\\ 0&\frac{f_y}{Z'}&-\frac{f_yY'}{Z'^2} \end{bmatrix}$

而第二项
$\frac{\partial p'_j}{\partial \delta \xi_i}=\frac{\partial(e^{\xi_i^{\wedge}}p_j)}{\partial \delta \xi_i}=\frac{\partial (Tp_j)}{\partial \delta \xi_i}$ 注： $\xi=\begin{bmatrix}\phi\\\rho\end{bmatrix}$

$=\begin{bmatrix}I&-(Rp_j+t)^{\wedge}\\0^T&0^T\end{bmatrix}_{4\times 6} \triangleq (Tp_j)^\bigodot$ （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）

$=\begin{bmatrix}I&-(p'_j)^{\wedge}\\0^T&0^T\end{bmatrix}$

因为此处 $p'_j$ 是三维，因此，结果也只取三行
$\frac{\partial p'_j}{\partial \delta \xi_i}=\begin{bmatrix}I&-(p'_j)^{\wedge}\end{bmatrix}_{3\times 6}=\begin{bmatrix} 1&0&0&0&Z'&-Y'\\ 0&1&0&-Z'&0&X'\\ 0&0&1&Y'&-X'&0\\ \end{bmatrix}$

于是最终第一部分的雅可比矩阵如下

$\frac{\partial error_{ij}}{\partial \delta \xi_i}=\frac{\partial error_{ij}}{\partial p'_j}.\frac{\partial p'_j}{\partial \delta \xi_i}$

$=-\begin{bmatrix}\frac{f_x}{Z'}&0&-\frac{f_xX'}{Z'^2}\\ 0&\frac{f_y}{Z'}&-\frac{f_yY'}{Z'^2} \end{bmatrix} \begin{bmatrix} 1&0&0&0&Z'&-Y'\\ 0&1&0&-Z'&0&X'\\ 0&0&1&Y'&-X'&0\\ \end{bmatrix}$

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

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

$\frac{\partial error_{ij}}{\partial p_j}=\frac{\partial error_{ij}}{\partial p'_j}.\frac{\partial p'_j}{\partial p_j}$
上式的第一项已经推导过，而第二项
我们知道

$p_j'=Rp_j+t$

所以 $\frac{\partial p'_j}{\partial p_j}=R$

$\frac{\partial error_{ij}}{\partial p_j}=-\begin{bmatrix}\frac{f_x}{Z'}&0&-\frac{f_xX'}{Z'^2}\\ 0&\frac{f_y}{Z'}&-\frac{f_yY'}{Z'^2} \end{bmatrix}R$

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
$\Big(\sum_{j=1}^N\big(J(X^{(k)})J(X^{(k)})^T\big) \Big).\Delta X=-\sum_{j=1}^NJ(X^{(k)})f(X^{(k)})$
具体来讲应该是这样

$\Big(\sum_{j=1}^N\big(J(\xi_i,p_j)J(\xi_i,p_j)^T\big) \Big).\Delta \xi_i=-\sum_{j=1}^NJ(\xi_i,p_j)e_{ij}(\xi_i,p_j)$

即，对单节点的所有边的误差迭代，且这里只优化了 $\xi_i$

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 $i$ 由 $\xi_i$ 表示，节点 $\xi_i$ 能看到多个不同的 $p_j$ ，也就是 $\xi_i$ 和每一个 $p_j$ 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 <iostream>
#include <opencv2/core/core.hpp>
#include <opencv2/features2d/features2d.hpp>
#include <opencv2/highgui/highgui.hpp>
#include <opencv2/calib3d/calib3d.hpp>
#include <Eigen/Core>
#include <g2o/core/base_vertex.h>
#include <g2o/core/base_unary_edge.h>
#include <g2o/core/sparse_optimizer.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/solver.h>
#include <g2o/core/optimization_algorithm_gauss_newton.h>
#include <g2o/solvers/dense/linear_solver_dense.h>
#include <sophus/se3.hpp>
#include <chrono>

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);

// 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<Eigen::Vector2d, Eigen::aligned_allocator<Eigen::Vector2d>> VecVector2d;
typedef vector<Eigen::Vector3d, Eigen::aligned_allocator<Eigen::Vector3d>> 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<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;

  // 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_<double>(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);
  vector<Point3f> pts_3d;
  vector<Point2f> pts_2d;
  for (DMatch m:matches) {
    ushort d = d1.ptr<unsigned short>(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<double> time_used = chrono::duration_cast<chrono::duration<double>>(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<chrono::duration<double>>(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<chrono::duration<double>>(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<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]);
    }
  }
}

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)
    );
}

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;
}

/// 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) {}

  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>();
  }

  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;
};

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();
}

7.BA优化中的单节点高斯牛顿方程推导以及g2o写法

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