第9讲 后端1

滤波器 EKF

前端视觉里程计： 短时间内的轨迹和地图。
后端优化： 长时间内的最优轨迹和地图

9.1 概述
9.1.1 状态估计的概率解释

渐进的(Incremental)： 当前的状态 只由 过去的时刻决定，甚至只由前一个决定。
批量的(Batch)：不仅使用过去的信息更新自己的状态，也会用未来的信息来更新。

SfM: Structure from Motion.

马尔可夫性： $k$ 时刻状态仅与 $k-1$ 时刻状态 有关。 【扩展卡尔曼滤波(EKF)】
考虑 $k$ 时刻状态 与 之前所有状态 的关系。 【非线性优化】

• 视觉 SLAM 主流

9.1.2 线性系统和 KF

经典卡尔曼滤波：从概率角度出发的最大后验概率估计方式。

在线性高斯系统中，卡尔曼滤波器 构成了 该系统中的最大后验概率估计。
卡尔曼滤波器构成了线性系统的最优无偏估计。

9.1.3 非线性系统 和 EKF

把卡尔曼滤波器 的结果 扩展到 非线性系统中， 扩展卡尔曼滤波器。

9.1.4 扩展卡尔曼滤波器 EKF 不足

同等计算量的情况下， 非线性优化能取得更好的效果。
精度和鲁棒性更好。

9.2 BA 与 图优化

视觉三维重建

Bundle Adjustment (BA)：从视觉图像中提炼出最优的3D模型和相机参数(内参和外差)。
考虑从任意特征点 发射出来的几束光线(bundles of light rays)， 它们会在几个相机的成像平面上变成像素或是检测到的特征点，如果我们调整(adjustment) 各相机姿态和各自特征点的空间位置，使得这些光纤最终收束到 相机的光心，称为 BA

其它译法： 光束法平差、捆集调整

9.2.1 投影模型和 BA 代价函数

9.2.2 BA 的求解

9.2.3 稀疏性 和 边缘化

Schur消元 Marginalization(边缘化)

9.2.4 鲁棒核函数

1、构造 BA 问题
2、设置 Schur 消元
3、调用 稠密或稀疏 矩阵求解器对变量进行优化

9.3 实践： Ceres BA 【Code】

定义 投影误差模型

sudo apt-get install meshlab

本讲 CMakeLists.txt

cmake_minimum_required(VERSION 2.8)set(CMAKE_CXX_STANDARD 17)
project(bundle_adjustment)
set(CMAKE_BUILD_TYPE "Release")
set(CMAKE_CXX_FLAGS "-O3 -std=c++14")LIST(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake)Find_Package(G2O REQUIRED)
Find_Package(Eigen3 REQUIRED)
Find_Package(Ceres REQUIRED)
Find_Package(Sophus REQUIRED)
Find_Package(CSparse REQUIRED)SET(G2O_LIBS g2o_csparse_extension g2o_stuff g2o_core cxsparse)include_directories(${PROJECT_SOURCE_DIR} ${EIGEN3_INCLUDE_DIR} ${CSPARSE_INCLUDE_DIR})add_library(bal_common common.cpp)add_executable(bundle_adjustment_ceres bundle_adjustment_ceres.cpp)
target_link_libraries(bundle_adjustment_ceres ${CERES_LIBRARIES} bal_common)add_executable(bundle_adjustment_g2o bundle_adjustment_g2o.cpp)
target_link_libraries(bundle_adjustment_g2o ${G2O_LIBS} g2o_solver_csparse g2o_csparse_extension${Sophus_LIBRARIES}bal_common)  

mkdir build && cd build
cmake ..
make 
./bundle_adjustment_ceres ../problem-16-22106-pre.txt

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 20 --delay=5 -c  /home/xixi/myGIF/test.gif

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 30 --delay=5 -c  /home/xixi/myGIF/test.gif

bundle_adjustment_ceres.cpp

#include <iostream>
#include <ceres/ceres.h>
#include "common.h"
#include "SnavelyReprojectionError.h"using namespace std;void SolveBA(BALProblem &bal_problem);int main(int argc, char **argv) {if (argc != 2) {cout << "usage: bundle_adjustment_ceres bal_data.txt" << endl;return 1;}BALProblem bal_problem(argv[1]);bal_problem.Normalize();bal_problem.Perturb(0.1, 0.5, 0.5);bal_problem.WriteToPLYFile("initial.ply");SolveBA(bal_problem);bal_problem.WriteToPLYFile("final.ply");return 0;
}void SolveBA(BALProblem &bal_problem) {const int point_block_size = bal_problem.point_block_size();const int camera_block_size = bal_problem.camera_block_size();double *points = bal_problem.mutable_points();double *cameras = bal_problem.mutable_cameras();// Observations is 2 * num_observations long array observations// [u_1, u_2, ... u_n], where each u_i is two dimensional, the x// and y position of the observation.const double *observations = bal_problem.observations();ceres::Problem problem;for (int i = 0; i < bal_problem.num_observations(); ++i) {ceres::CostFunction *cost_function;// Each Residual block takes a point and a camera as input// and outputs a 2 dimensional Residualcost_function = SnavelyReprojectionError::Create(observations[2 * i + 0], observations[2 * i + 1]);// If enabled use Huber's loss function.ceres::LossFunction *loss_function = new ceres::HuberLoss(1.0);// Each observation corresponds to a pair of a camera and a point// which are identified by camera_index()[i] and point_index()[i]// respectively.double *camera = cameras + camera_block_size * bal_problem.camera_index()[i];double *point = points + point_block_size * bal_problem.point_index()[i];problem.AddResidualBlock(cost_function, loss_function, camera, point);}// show some information here ...std::cout << "bal problem file loaded..." << std::endl;std::cout << "bal problem have " << bal_problem.num_cameras() << " cameras and "<< bal_problem.num_points() << " points. " << std::endl;std::cout << "Forming " << bal_problem.num_observations() << " observations. " << std::endl;std::cout << "Solving ceres BA ... " << endl;ceres::Solver::Options options;options.linear_solver_type = ceres::LinearSolverType::SPARSE_SCHUR;options.minimizer_progress_to_stdout = true;ceres::Solver::Summary summary;ceres::Solve(options, &problem, &summary);std::cout << summary.FullReport() << "\n";
}

9.4 实践：g2o 求解 BA 【Code】

报错：

/home/xixi/Downloads/slambook2-master/ch9/bundle_adjustment_g2o.cpp:10:10: fatal error: sophus/se3.hpp: No such file or directory10 | #include "sophus/se3.hpp"|          ^~~~~~~~~~~~~~~~

SO3d 去掉 d 。 共3个
报错2：

/usr/local/include/g2o/stuff/tuple_tools.h:41:46: error: ‘tuple_size_v’ is not a member of ‘std’; did you mean ‘tuple_size’?41 |       f, t, i, std::make_index_sequence<std::tuple_size_v<std::decay_t<T>>>());

改动3：

第143-147行 更换成下述代码

    std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverCSparse<g2o::BlockSolverX::PoseMatrixType>());std::unique_ptr<g2o::BlockSolverX> solver_ptr (new g2o::BlockSolverX(std::move(linearSolver)));g2o::OptimizationAlgorithmLevenberg* solver = new g2o::OptimizationAlgorithmLevenberg(std::move(solver_ptr));

其它问题： 链接库的问题， 看 CMakeLists.txt

cd build
cmake ..
make 
./bundle_adjustment_g2o ../problem-16-22106-pre.txt

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 30 --delay=5 -c  /home/xixi/myGIF/test.gif

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 20 --delay=5 -c  /home/xixi/myGIF/test.gif

bundle_adjustment_g2o.cpp

#include <g2o/core/base_vertex.h>
#include <g2o/core/base_binary_edge.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/optimization_algorithm_levenberg.h>
#include <g2o/solvers/csparse/linear_solver_csparse.h>
#include <g2o/core/robust_kernel_impl.h>
#include <iostream>#include "common.h"
#include "sophus/se3.h"using namespace Sophus;
using namespace Eigen;
using namespace std;/// 姿态和内参的结构
struct PoseAndIntrinsics {PoseAndIntrinsics() {}/// set from given data addressexplicit PoseAndIntrinsics(double *data_addr) {rotation = SO3::exp(Vector3d(data_addr[0], data_addr[1], data_addr[2]));translation = Vector3d(data_addr[3], data_addr[4], data_addr[5]);focal = data_addr[6];k1 = data_addr[7];k2 = data_addr[8];}/// 将估计值放入内存void set_to(double *data_addr) {auto r = rotation.log();for (int i = 0; i < 3; ++i) data_addr[i] = r[i];for (int i = 0; i < 3; ++i) data_addr[i + 3] = translation[i];data_addr[6] = focal;data_addr[7] = k1;data_addr[8] = k2;}SO3 rotation;Vector3d translation = Vector3d::Zero();double focal = 0;double k1 = 0, k2 = 0;
};/// 位姿加相机内参的顶点，9维，前三维为so3，接下去为t, f, k1, k2
class VertexPoseAndIntrinsics : public g2o::BaseVertex<9, PoseAndIntrinsics> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;VertexPoseAndIntrinsics() {}virtual void setToOriginImpl() override {_estimate = PoseAndIntrinsics();}virtual void oplusImpl(const double *update) override {_estimate.rotation = SO3::exp(Vector3d(update[0], update[1], update[2])) * _estimate.rotation;_estimate.translation += Vector3d(update[3], update[4], update[5]);_estimate.focal += update[6];_estimate.k1 += update[7];_estimate.k2 += update[8];}/// 根据估计值投影一个点Vector2d project(const Vector3d &point) {Vector3d pc = _estimate.rotation * point + _estimate.translation;pc = -pc / pc[2];double r2 = pc.squaredNorm();double distortion = 1.0 + r2 * (_estimate.k1 + _estimate.k2 * r2);return Vector2d(_estimate.focal * distortion * pc[0],_estimate.focal * distortion * pc[1]);}virtual bool read(istream &in) {}virtual bool write(ostream &out) const {}
};class VertexPoint : public g2o::BaseVertex<3, Vector3d> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;VertexPoint() {}virtual void setToOriginImpl() override {_estimate = Vector3d(0, 0, 0);}virtual void oplusImpl(const double *update) override {_estimate += Vector3d(update[0], update[1], update[2]);}virtual bool read(istream &in) {}virtual bool write(ostream &out) const {}
};class EdgeProjection :public g2o::BaseBinaryEdge<2, Vector2d, VertexPoseAndIntrinsics, VertexPoint> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;virtual void computeError() override {auto v0 = (VertexPoseAndIntrinsics *) _vertices[0];auto v1 = (VertexPoint *) _vertices[1];auto proj = v0->project(v1->estimate());_error = proj - _measurement;}// use numeric derivativesvirtual bool read(istream &in) {}virtual bool write(ostream &out) const {}};void SolveBA(BALProblem &bal_problem);int main(int argc, char **argv) {if (argc != 2) {cout << "usage: bundle_adjustment_g2o bal_data.txt" << endl;return 1;}BALProblem bal_problem(argv[1]);bal_problem.Normalize();bal_problem.Perturb(0.1, 0.5, 0.5);bal_problem.WriteToPLYFile("initial.ply");SolveBA(bal_problem);bal_problem.WriteToPLYFile("final.ply");return 0;
}void SolveBA(BALProblem &bal_problem) {const int point_block_size = bal_problem.point_block_size();const int camera_block_size = bal_problem.camera_block_size();double *points = bal_problem.mutable_points();double *cameras = bal_problem.mutable_cameras();// pose dimension 9, landmark is 3/*typedef g2o::BlockSolver<g2o::BlockSolverTraits<9, 3>> BlockSolverType;typedef g2o::LinearSolverCSparse<BlockSolverType::PoseMatrixType> LinearSolverType;// use LMauto solver = new g2o::OptimizationAlgorithmLevenberg(g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));*/std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverCSparse<g2o::BlockSolverX::PoseMatrixType>());std::unique_ptr<g2o::BlockSolverX> solver_ptr (new g2o::BlockSolverX(std::move(linearSolver)));g2o::OptimizationAlgorithmLevenberg* solver = new g2o::OptimizationAlgorithmLevenberg(std::move(solver_ptr));g2o::SparseOptimizer optimizer;optimizer.setAlgorithm(solver);optimizer.setVerbose(true);/// build g2o problemconst double *observations = bal_problem.observations();// vertexvector<VertexPoseAndIntrinsics *> vertex_pose_intrinsics;vector<VertexPoint *> vertex_points;for (int i = 0; i < bal_problem.num_cameras(); ++i) {VertexPoseAndIntrinsics *v = new VertexPoseAndIntrinsics();double *camera = cameras + camera_block_size * i;v->setId(i);v->setEstimate(PoseAndIntrinsics(camera));optimizer.addVertex(v);vertex_pose_intrinsics.push_back(v);}for (int i = 0; i < bal_problem.num_points(); ++i) {VertexPoint *v = new VertexPoint();double *point = points + point_block_size * i;v->setId(i + bal_problem.num_cameras());v->setEstimate(Vector3d(point[0], point[1], point[2]));// g2o在BA中需要手动设置待Marg的顶点v->setMarginalized(true);optimizer.addVertex(v);vertex_points.push_back(v);}// edgefor (int i = 0; i < bal_problem.num_observations(); ++i) {EdgeProjection *edge = new EdgeProjection;edge->setVertex(0, vertex_pose_intrinsics[bal_problem.camera_index()[i]]);edge->setVertex(1, vertex_points[bal_problem.point_index()[i]]);edge->setMeasurement(Vector2d(observations[2 * i + 0], observations[2 * i + 1]));edge->setInformation(Matrix2d::Identity());edge->setRobustKernel(new g2o::RobustKernelHuber());optimizer.addEdge(edge);}optimizer.initializeOptimization();optimizer.optimize(40);// set to bal problemfor (int i = 0; i < bal_problem.num_cameras(); ++i) {double *camera = cameras + camera_block_size * i;auto vertex = vertex_pose_intrinsics[i];auto estimate = vertex->estimate();estimate.set_to(camera);}for (int i = 0; i < bal_problem.num_points(); ++i) {double *point = points + point_block_size * i;auto vertex = vertex_points[i];for (int k = 0; k < 3; ++k) point[k] = vertex->estimate()[k];}
}

习题

√ 题 1

证明式 (9.25) 成立。提示：你可能会用到 SMW 公式，参考文献 [6, 76].

根据 SMW(Sherman-Morrison-Woodbury) 恒等式 证明 $\mathbf{K}$ 的求解 可以不依靠 ${\hat{\mathbf{P}}}_k$。
即 用 别的已知量替换 $\mathbf{K}$ 定义中的 ${\hat{\mathbf{P}}}_k$

State estimation for robotics: A matrix lie group approach.
来自上述书籍的公式：

——————————

证明:
$$\begin{array}{rl}\mathbf{K}& ={\hat{\mathbf{P}}}_k{\mathbf{C}}_k^T{\mathbf{Q}}^{-1}\\ & 由式(9.16)\\ & =({\mathbf{C}}_k^T{\mathbf{Q}}^{-1}{\mathbf{C}}_k+{\check{\mathbf{P}}}_k^{-1}{)}^{-1}{\mathbf{C}}_k^T{\mathbf{Q}}^{-1}\\ & 由式(2.75c)由右到左\\ & 其中：A\Rightarrow {\check{\mathbf{P}}}_k,B\Rightarrow {\mathbf{C}}_k^T,C\Rightarrow {\mathbf{C}}_k,D\Rightarrow \mathbf{Q}\\ & ={\check{\mathbf{P}}}_k{\mathbf{C}}_k^T(\mathbf{Q}+{\mathbf{C}}_k{\check{\mathbf{P}}}_k{\mathbf{C}}_k^T{)}^{-1}\end{array}$$

证毕。

————————————————————

$\check{A}$

$\check{A}$

关于 Q 的补充：

感觉上面这个 记法 比较清楚， 十四讲里有点混乱。

题 4
证明 $\mathbf{S}$ 矩阵为 半正定矩阵。

• 待 证
  
  $$\mathbf{S}=\mathbf{B}-{\mathbf{EC}}^{-1}{\mathbf{E}}^T$$