1 重投影方程

直观数学理解

2 李代数表现形式 - 便于求导和减少约束条件

https://blog.csdn.net/weixin_49804978/article/details/121922128

由于相机位姿未知以及观测点的噪声，该等式存在一个误差。我们将误差求和，构建最小二乘问题，然后寻找做好的相机位姿，使它最小化：

该问题的误差项，是将像素坐标（观测到的投影位置）与 3D 点按照当前估计的位姿进行投影得到的位置相比较得到的误差，所以称之为重投影误差。
（用高斯牛顿法/列文伯格-马夸尔特方法求解）

3 误差方程求解

3-1 构造最小二乘解框架

高斯牛顿法举例

其中x 是位姿Rt 和三维点xyz ，而非原来的 (u,v)观测值

3-2 具体求解一阶雅可比方程

预测值

观测值

投影方程

三维点

最终

1（u,v）对dT 求导

T被转化为扰动E 所以T分为两个部分

1 像素坐标（u,v）对 Rt求导

2 T对离地阿叔扰动dT求导

推导过程

第一项求导

第二项求导

跟新流程

1*6维度 3维位姿 3为位置

具体过程

1 Rt转化为dT

dT_1*6 = g2o::SE3Quat (R_3*3,t_1*3)

2初始值

_estimate = SE3Quat();

3设置值

pose->setEstimate (dT_1*6)

4每次跟新

1*6维度 3维位姿 3为位置

Eigen::Map<const Vector6> update(update_);

5转化

SE3Quat::exp(update)

6跟新

setEstimate(SE3Quat::exp(update)*estimate()); //更新方式

联合

实际代码对应

位姿一阶导数J代码

  _jacobianOplusXj(0, 0) = x * y / z_2 * cam->focal_length;
  _jacobianOplusXj(0, 1) = -(1 + (x * x / z_2)) * cam->focal_length;
  _jacobianOplusXj(0, 2) = y / z * cam->focal_length;
  _jacobianOplusXj(0, 3) = -1. / z * cam->focal_length;
  _jacobianOplusXj(0, 4) = 0;
  _jacobianOplusXj(0, 5) = x / z_2 * cam->focal_length;
 
  _jacobianOplusXj(1, 0) = (1 + y * y / z_2) * cam->focal_length;
  _jacobianOplusXj(1, 1) = -x * y / z_2 * cam->focal_length;
  _jacobianOplusXj(1, 2) = -x / z * cam->focal_length;
  _jacobianOplusXj(1, 3) = 0;
  _jacobianOplusXj(1, 4) = -1. / z * cam->focal_length;
  _jacobianOplusXj(1, 5) = y / z_2 * cam->focal_length;

对空间点求导

三维点一阶导数J代码

  Eigen::Matrix<double, 2, 3, Eigen::ColMajor> tmp;
  tmp(0, 0) = cam->focal_length;
  tmp(0, 1) = 0;
  tmp(0, 2) = -x / z * cam->focal_length;
 
  tmp(1, 0) = 0;
  tmp(1, 1) = cam->focal_length;
  tmp(1, 2) = -y / z * cam->focal_length;
 
  _jacobianOplusXi = -1. / z * tmp * T.rotation().toRotationMatrix();

位姿和三维点联合在一个边中代码

edge_project_xyz2uv.cpp

// g2o - General Graph Optimization
 
 
#include "edge_project_xyz2uv.h"
 
namespace g2o {
 
EdgeProjectXYZ2UV::EdgeProjectXYZ2UV()
    : BaseBinaryEdge<2, Vector2, VertexPointXYZ, VertexSE3Expmap>() {
  _cam = 0;
  resizeParameters(1);
  installParameter(_cam, 0);
}
 
bool EdgeProjectXYZ2UV::read(std::istream& is) {
  readParamIds(is);
  internal::readVector(is, _measurement);
  return readInformationMatrix(is);
}
 
bool EdgeProjectXYZ2UV::write(std::ostream& os) const {
  writeParamIds(os);
  internal::writeVector(os, measurement());
  return writeInformationMatrix(os);
}
 
void EdgeProjectXYZ2UV::computeError() {
  const VertexSE3Expmap* v1 = static_cast<const VertexSE3Expmap*>(_vertices[1]);
  const VertexPointXYZ* v2 = static_cast<const VertexPointXYZ*>(_vertices[0]);
  const CameraParameters* cam =
      static_cast<const CameraParameters*>(parameter(0));
  _error = measurement() - cam->cam_map(v1->estimate().map(v2->estimate()));
}
 
void EdgeProjectXYZ2UV::linearizeOplus() {
  VertexSE3Expmap* vj = static_cast<VertexSE3Expmap*>(_vertices[1]);
  SE3Quat T(vj->estimate());
  VertexPointXYZ* vi = static_cast<VertexPointXYZ*>(_vertices[0]);
  Vector3 xyz = vi->estimate();
  Vector3 xyz_trans = T.map(xyz);
 
  double x = xyz_trans[0];
  double y = xyz_trans[1];
  double z = xyz_trans[2];
  double z_2 = z * z;
 
  const CameraParameters* cam =
      static_cast<const CameraParameters*>(parameter(0));
 
  Eigen::Matrix<double, 2, 3, Eigen::ColMajor> tmp;
  tmp(0, 0) = cam->focal_length;
  tmp(0, 1) = 0;
  tmp(0, 2) = -x / z * cam->focal_length;
 
  tmp(1, 0) = 0;
  tmp(1, 1) = cam->focal_length;
  tmp(1, 2) = -y / z * cam->focal_length;
 
  _jacobianOplusXi = -1. / z * tmp * T.rotation().toRotationMatrix();
 
  _jacobianOplusXj(0, 0) = x * y / z_2 * cam->focal_length;
  _jacobianOplusXj(0, 1) = -(1 + (x * x / z_2)) * cam->focal_length;
  _jacobianOplusXj(0, 2) = y / z * cam->focal_length;
  _jacobianOplusXj(0, 3) = -1. / z * cam->focal_length;
  _jacobianOplusXj(0, 4) = 0;
  _jacobianOplusXj(0, 5) = x / z_2 * cam->focal_length;
 
  _jacobianOplusXj(1, 0) = (1 + y * y / z_2) * cam->focal_length;
  _jacobianOplusXj(1, 1) = -x * y / z_2 * cam->focal_length;
  _jacobianOplusXj(1, 2) = -x / z * cam->focal_length;
  _jacobianOplusXj(1, 3) = 0;
  _jacobianOplusXj(1, 4) = -1. / z * cam->focal_length;
  _jacobianOplusXj(1, 5) = y / z_2 * cam->focal_length;
}
 
}  // namespace g2o