第10讲 后端2

滑动窗口优化
位姿图优化【简化的BA】
带IMU 紧耦合 的优化
g2o 的位姿图

第9讲 以BA为主的图优化。
BA能精确地优化每个相机位姿与特征点位置。
在大场景中，大量特征点 会严重降低计算效率，计算量越来越大 ——> 无法实时化。
改进： 简化BA 【位姿图】

10.1 滑动窗口滤波 和 优化

BA：带有相机位姿和空间点的图优化。

控制 BA 规模：仅保留 离当前时刻最近的 N 个关键帧。【滑动窗口法】

ORB-SLAM2 ：
共视图(Covisibility graph) : 与现在的相机 存在共同观测的关键帧构成的图。

10.1.2 滑动窗口法

滑动窗口法 比较适合VO系统，不适合大规模建图系统。

10.2 位姿图

10.3 实践： 位姿图优化

本讲 CMakeLists.txt

cmake_minimum_required(VERSION 2.8)
project(pose_graph)

set(CMAKE_BUILD_TYPE "Release")
set(CMAKE_CXX_FLAGS "-std=c++17 -O2")

list(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake_modules)

# Eigen
include_directories("/usr/include/eigen3")


# sophus 
find_package(Sophus REQUIRED)
include_directories(${Sophus_INCLUDE_DIRS})


# g2o 
find_package(G2O REQUIRED)
include_directories(${G2O_INCLUDE_DIRS})

add_executable(pose_graph_g2o_SE3 pose_graph_g2o_SE3.cpp)
target_link_libraries(pose_graph_g2o_SE3
        g2o_core g2o_stuff g2o_types_slam3d ${CHOLMOD_LIBRARIES}
        )

SET(G2O_LIBS g2o_csparse_extension g2o_stuff g2o_core cxsparse)

add_executable(pose_graph_g2o_lie pose_graph_g2o_lie_algebra.cpp)
target_link_libraries(pose_graph_g2o_lie
        ${G2O_LIBS}
        ${CHOLMOD_LIBRARIES}
        ${Sophus_LIBRARIES})

——————————
改动1：
SE3d 和 SO3d 去掉d

改动2：
CMakeLists.txt 加这一句 使满足 C++17标准， 最新版 g2o 需要

改动3： CMakeLists.txt 添加 csparse 相关的链接库

    std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverEigen<g2o::BlockSolverX::PoseMatrixType>()); // 这里直接用 前面提到的Eigen 线性方程求解库 也可以

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

10.3.1 g2o 原生位姿图 【Code】

查看待优化的位姿图。注意在文件sphere.g2o所在文件夹打开命令行窗口。

g2o_viewer sphere.g2o   

mkdir build && cd build
cmake ..
make 
./pose_graph_g2o_SE3 ../sphere.g2o
g2o_viewer result.g2o

pose_graph_g2o_SE3.cpp

#include <iostream>
#include <fstream>
#include <string>

#include <g2o/types/slam3d/types_slam3d.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/optimization_algorithm_levenberg.h>
#include <g2o/solvers/eigen/linear_solver_eigen.h>

using namespace std;

/************************************************
 * 本程序演示如何用g2o solver进行位姿图优化
 * sphere.g2o是人工生成的一个Pose graph，我们来优化它。
 * 尽管可以直接通过load函数读取整个图，但我们还是自己来实现读取代码，以期获得更深刻的理解
 * 这里使用g2o/types/slam3d/中的SE3表示位姿，它实质上是四元数而非李代数.
 * **********************************************/

int main(int argc, char **argv) {
    if (argc != 2) {
        cout << "Usage: pose_graph_g2o_SE3 sphere.g2o" << endl;
        return 1;
    }
    ifstream fin(argv[1]);
    if (!fin) {
        cout << "file " << argv[1] << " does not exist." << endl;
        return 1;
    }

    // 设定g2o
   /*typedef g2o::BlockSolver<g2o::BlockSolverTraits<6, 6>> BlockSolverType;
    typedef g2o::LinearSolverEigen<BlockSolverType::PoseMatrixType> LinearSolverType;
    auto solver = new g2o::OptimizationAlgorithmLevenberg(
        g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));
*/ 
    std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverEigen<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;     // 图模型
    optimizer.setAlgorithm(solver);   // 设置求解器
    optimizer.setVerbose(true);       // 打开调试输出

    int vertexCnt = 0, edgeCnt = 0; // 顶点和边的数量
    while (!fin.eof()) {
        string name;
        fin >> name;
        if (name == "VERTEX_SE3:QUAT") {
            // SE3 顶点
            g2o::VertexSE3 *v = new g2o::VertexSE3();
            int index = 0;
            fin >> index;
            v->setId(index);
            v->read(fin);
            optimizer.addVertex(v);
            vertexCnt++;
            if (index == 0)
                v->setFixed(true);
        } else if (name == "EDGE_SE3:QUAT") {
            // SE3-SE3 边
            g2o::EdgeSE3 *e = new g2o::EdgeSE3();
            int idx1, idx2;     // 关联的两个顶点
            fin >> idx1 >> idx2;
            e->setId(edgeCnt++);
            e->setVertex(0, optimizer.vertices()[idx1]);
            e->setVertex(1, optimizer.vertices()[idx2]);
            e->read(fin);
            optimizer.addEdge(e);
        }
        if (!fin.good()) break;
    }

    cout << "read total " << vertexCnt << " vertices, " << edgeCnt << " edges." << endl;

    cout << "optimizing ..." << endl;
    optimizer.initializeOptimization();
    optimizer.optimize(30);

    cout << "saving optimization results ..." << endl;
    optimizer.save("result.g2o");

    return 0;
}

10.3.2 李代数上的位姿优化 【Code】

前端和后端 分开： 跟踪和建图

报错：

xixi@ubuntu:~/Downloads/slambook2-master/ch10/build$ ./pose_graph_g2o_lie ../sphere.g2o
Segmentation fault

./pose_graph_g2o_lie ../sphere.g2o
g2o_viewer result.g2o

pose_graph_g2o_lie_algebra.cpp

#include <iostream>
#include <fstream>
#include <string>
#include <Eigen/Core>

#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/eigen/linear_solver_eigen.h>

#include <sophus/se3.h>

using namespace std;
using namespace Eigen;
using Sophus::SE3;
using Sophus::SO3;

/************************************************
 * 本程序演示如何用g2o solver进行位姿图优化
 * sphere.g2o是人工生成的一个Pose graph，我们来优化它。
 * 尽管可以直接通过load函数读取整个图，但我们还是自己来实现读取代码，以期获得更深刻的理解
 * 本节使用李代数表达位姿图，节点和边的方式为自定义
 * **********************************************/

typedef Matrix<double, 6, 6> Matrix6d;

// 给定误差求J_R^{-1}的近似
Matrix6d JRInv(const SE3 &e) {
    Matrix6d J;
    J.block(0, 0, 3, 3) = SO3::hat(e.so3().log());
    J.block(0, 3, 3, 3) = SO3::hat(e.translation());
    J.block(3, 0, 3, 3) = Matrix3d::Zero(3, 3);
    J.block(3, 3, 3, 3) = SO3::hat(e.so3().log());
    // J = J * 0.5 + Matrix6d::Identity();
    J = Matrix6d::Identity();    // try Identity if you want
    return J;
}

// 李代数顶点
typedef Matrix<double, 6, 1> Vector6d;

class VertexSE3LieAlgebra : public g2o::BaseVertex<6, SE3> {
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    virtual bool read(istream &is) override {
        double data[7];
        for (int i = 0; i < 7; i++)
            is >> data[i];
        setEstimate(SE3(
            Quaterniond(data[6], data[3], data[4], data[5]),
            Vector3d(data[0], data[1], data[2])
        ));
        return true;
    }

    virtual bool write(ostream &os) const override {
        os << id() << " ";
        Quaterniond q = _estimate.unit_quaternion();
        os << _estimate.translation().transpose() << " ";
        os << q.coeffs()[0] << " " << q.coeffs()[1] << " " << q.coeffs()[2] << " " << q.coeffs()[3] << endl;
        return true;
    }

    virtual void setToOriginImpl() override {
        _estimate = SE3();
    }

    // 左乘更新
    virtual void oplusImpl(const double *update) override {
        Vector6d upd;
        upd << update[0], update[1], update[2], update[3], update[4], update[5];
        _estimate = SE3::exp(upd) * _estimate;
    }
};

// 两个李代数节点之边
class EdgeSE3LieAlgebra : public g2o::BaseBinaryEdge<6, SE3, VertexSE3LieAlgebra, VertexSE3LieAlgebra> {
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    virtual bool read(istream &is) override {
        double data[7];
        for (int i = 0; i < 7; i++)
            is >> data[i];
        Quaterniond q(data[6], data[3], data[4], data[5]);
        q.normalize();
        setMeasurement(SE3(q, Vector3d(data[0], data[1], data[2])));
        for (int i = 0; i < information().rows() && is.good(); i++)
            for (int j = i; j < information().cols() && is.good(); j++) {
                is >> information()(i, j);
                if (i != j)
                    information()(j, i) = information()(i, j);
            }
        return true;
    }

    virtual bool write(ostream &os) const override {
        VertexSE3LieAlgebra *v1 = static_cast<VertexSE3LieAlgebra *> (_vertices[0]);
        VertexSE3LieAlgebra *v2 = static_cast<VertexSE3LieAlgebra *> (_vertices[1]);
        os << v1->id() << " " << v2->id() << " ";
        SE3 m = _measurement;
        Eigen::Quaterniond q = m.unit_quaternion();
        os << m.translation().transpose() << " ";
        os << q.coeffs()[0] << " " << q.coeffs()[1] << " " << q.coeffs()[2] << " " << q.coeffs()[3] << " ";

        // information matrix 
        for (int i = 0; i < information().rows(); i++)
            for (int j = i; j < information().cols(); j++) {
                os << information()(i, j) << " ";
            }
        os << endl;
        return true;
    }

    // 误差计算与书中推导一致
    virtual void computeError() override {
        SE3 v1 = (static_cast<VertexSE3LieAlgebra *> (_vertices[0]))->estimate();
        SE3 v2 = (static_cast<VertexSE3LieAlgebra *> (_vertices[1]))->estimate();
        _error = (_measurement.inverse() * v1.inverse() * v2).log();
    }

    // 雅可比计算
    virtual void linearizeOplus() override {
        SE3 v1 = (static_cast<VertexSE3LieAlgebra *> (_vertices[0]))->estimate();
        SE3 v2 = (static_cast<VertexSE3LieAlgebra *> (_vertices[1]))->estimate();
        Matrix6d J = JRInv(SE3::exp(_error));
        // 尝试把J近似为I？
        _jacobianOplusXi = -J * v2.inverse().Adj();
        _jacobianOplusXj = J * v2.inverse().Adj();
    }
};

int main(int argc, char **argv) {
    if (argc != 2) {
        cout << "Usage: pose_graph_g2o_SE3_lie sphere.g2o" << endl;
        return 1;
    }
    ifstream fin(argv[1]);
    if (!fin) {
        cout << "file " << argv[1] << " does not exist." << endl;
        return 1;
    }

    // 设定g2o
    /*typedef g2o::BlockSolver<g2o::BlockSolverTraits<6, 6>> BlockSolverType;
    typedef g2o::LinearSolverEigen<BlockSolverType::PoseMatrixType> LinearSolverType;
    auto solver = new g2o::OptimizationAlgorithmLevenberg(
        g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));
    */
    std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverEigen<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;     // 图模型
    optimizer.setAlgorithm(solver);   // 设置求解器
    optimizer.setVerbose(true);       // 打开调试输出

    int vertexCnt = 0, edgeCnt = 0; // 顶点和边的数量

    vector<VertexSE3LieAlgebra *> vectices;
    vector<EdgeSE3LieAlgebra *> edges;
    while (!fin.eof()) {
        string name;
        fin >> name;
        if (name == "VERTEX_SE3:QUAT") {
            // 顶点
            VertexSE3LieAlgebra *v = new VertexSE3LieAlgebra();
            int index = 0;
            fin >> index;
            v->setId(index);
            v->read(fin);
            optimizer.addVertex(v);
            vertexCnt++;
            vectices.push_back(v);
            if (index == 0)
                v->setFixed(true);
        } else if (name == "EDGE_SE3:QUAT") {
            // SE3-SE3 边
            EdgeSE3LieAlgebra *e = new EdgeSE3LieAlgebra();
            int idx1, idx2;     // 关联的两个顶点
            fin >> idx1 >> idx2;
            e->setId(edgeCnt++);
            e->setVertex(0, optimizer.vertices()[idx1]);
            e->setVertex(1, optimizer.vertices()[idx2]);
            e->read(fin);
            optimizer.addEdge(e);
            edges.push_back(e);
        }
        if (!fin.good()) break;
    }

    cout << "read total " << vertexCnt << " vertices, " << edgeCnt << " edges." << endl;

    cout << "optimizing ..." << endl;
    optimizer.initializeOptimization();
    optimizer.optimize(30);

    cout << "saving optimization results ..." << endl;

    // 因为用了自定义顶点且没有向g2o注册，这里保存自己来实现
    // 伪装成 SE3 顶点和边，让 g2o_viewer 可以认出
    ofstream fout("result_lie.g2o");
    for (VertexSE3LieAlgebra *v:vectices) {
        fout << "VERTEX_SE3:QUAT ";
        v->write(fout);
    }
    for (EdgeSE3LieAlgebra *e:edges) {
        fout << "EDGE_SE3:QUAT ";
        e->write(fout);
    }
    fout.close();
    return 0;
}

习题10

题1 【没推完】

如果将位姿图中的误差定义为 $\Delta {\mathbf{\xi }}_{ij}={\mathbf{\xi }}_i\circ {\mathbf{\xi }}_j^{-1}$ ，推导按照此定义的左乘扰动雅克比矩阵。

本题中 位姿图的误差定义为 $\Delta {\mathbf{\xi }}_{ij}={\mathbf{\xi }}_i\circ {\mathbf{\xi }}_j^{-1}=\ln ({\mathbf{T}}_i{\mathbf{T}}_j^{-1}{)}^{\vee }$

• 和 P271 的定义区别在于 是第二个 求逆。这里是相对第 $j$ 个位姿的坐标系进行位姿变化讨论
• $\mathbf{T}$ 对应的李代数为 $\mathbf{\xi }$
  

对应的李群写法为： ${\mathbf{T}}_{ji}={\mathbf{T}}_i{\mathbf{T}}_j^{-1}$

构建误差 ${\mathbf{e}}_{ij}$

• 这样构建的误差理想下是0，因为 ln($\mathbf{I})=0$。怎么感觉应该是下面这样的🤔

$${\mathbf{e}}_{ij}=\ln ({\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_i{\mathbf{T}}_j^{-1}{)}^{\vee }$$
其中

${\mathbf{T}}_{ji}={\mathbf{T}}_i{\mathbf{T}}_j^{-1}$
${\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_{ji}={\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_i{\mathbf{T}}_j^{-1}$
$\mathbf{I}={\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_i{\mathbf{T}}_j^{-1}$

给 ${\mathbf{\xi }}_i$ 和 ${\mathbf{\xi }}_j$ 各一个左扰动

$$\begin{array}{rl}{\hat{\mathbf{e}}}_{ij}& =\ln ({\mathbf{T}}_{ji}^{-1}\exp (\Delta {\mathbf{\xi }}_i^{\wedge }){\mathbf{T}}_i(\exp (\Delta {\mathbf{\xi }}_j^{\wedge }){\mathbf{T}}_j{)}^{-1}{)}^{\vee }\\ & =\ln ({\mathbf{T}}_{ji}^{-1}\exp (\Delta {\mathbf{\xi }}_i^{\wedge }){\mathbf{T}}_i{\mathbf{T}}_j^{-1}(\exp (\Delta {\mathbf{\xi }}_j^{\wedge }{)}^{-1}{)}^{\vee }\\ & 由式(10.7)\\ & =\ln ({\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_i\exp ((\mathrm{Ad}({\mathbf{T}}_i^{-1})\Delta {\mathbf{\xi }}_i{)}^{\wedge })\exp ((-\mathrm{Ad}({\mathbf{T}}_j^{-1})\Delta {\mathbf{\xi }}_j{)}^{\wedge }){\mathbf{T}}_j^{-1}{)}^{\vee }\\ & =\ln ({\mathbf{T}}_{ji}^{-1}{\mathbf{T}}_i{\mathbf{T}}_j^{-1}\exp ((\mathrm{Ad}({\mathbf{T}}_i^{-1})\Delta {\mathbf{\xi }}_i{)}^{\wedge })\exp ((-\mathrm{Ad}({\mathbf{T}}_j^{-1})\Delta {\mathbf{\xi }}_j{)}^{\wedge }){)}^{\vee }\\ & =\ln (\exp ({\mathbf{e}}_{ij})\exp ((\mathrm{Ad}({\mathbf{T}}_i^{-1})\Delta {\mathbf{\xi }}_i{)}^{\wedge })\exp ((-\mathrm{Ad}({\mathbf{T}}_j^{-1})\Delta {\mathbf{\xi }}_j{)}^{\wedge }){)}^{\vee }\end{array}$$

由公式：

对于 ${\mathbf{T}}_i$
$\ln (\exp ({\mathbf{e}}_{ij})\exp ((\mathrm{Ad}({\mathbf{T}}_i^{-1})\Delta {\mathbf{\xi }}_i{)}^{\wedge })\exp ((-\mathrm{Ad}({\mathbf{T}}_j^{-1})\Delta {\mathbf{\xi }}_j{)}^{\wedge }){)}^{\vee }\approx$

LaTex

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