视觉十四讲:第十讲_位姿图

1.简介

带有相机位姿和空间点的图优化称为BA，能够有效的求解大范围的定位与建图问题，但是随着时间，规模越来越大，计算效率会大幅下降。我们发现，特征点在优化问题中占了很大部分，经过若干次迭代之后，特征点就会收敛，此时再进行优化的意义并不大，因此，在优化几次后，可以把特征点固定住，把他们看做位姿估计的约束，不再优化他们的位姿。
位姿图即只考虑位姿，构建一个只有轨迹的图优化，而位姿节点之间的边，由两个关键帧之间通过特征匹配后得到的运动估计来给定初始值，一旦初始值完成，就不再优化路标点的位置，只关心相机位姿之间的联系。

2.位姿优化：

图优化中的节点表示相机位姿，以(xi_{1}, ..., xi_{n})来表示。边，则是两个位姿节点之间的相对运动估计。
(xi_{i})和(xi_{j})之间的一个运动(Delta xi_{ij})。

按李群写法：
构建误差(e_{ij})：

注意优化变量有两个：(xi_{i})和(xi_{j})，因此求(e_{ij})关于这两个变量的导数，按照李代数的求导方式，给两个优化变量各一个左扰动，于是误差变为：
把扰动项移至式子的左侧或右侧，有：

因此，按照李代数上的求导法则，我们求出来误差关于两个位姿的雅可比矩阵。

关于Ti的：

关于Tj的：

由于雅可比矩阵Jr形式过于复杂，我们通常取它的近似。如果误差接近于零，我们设它近似于I或：

得到雅可比矩阵后，剩下的部分就和普通的图优化一样了，所有的位姿顶点和位姿边构成了一个图优化，本质上是一个最小二乘问题，优化变量为各个顶点的位姿，边来自于位姿观测约束。则总体的目标函数为：

3.实践

++ g2o自带的数据类型，默认使用四元数和平移向量表达位姿。
顶点：(ID, t_{x}, t_{y}, t_{z}, q_{x}, q_{y}, q_{z}, q_{w})。前面为平移向量，后面为旋转的单位四元数。
边：(两个节点的ID, t_{x}, t_{y}, t_{z}, q_{x}, q_{y}, q_{z}, q_{w})，信息矩阵的右上角
(信息矩阵为对称矩阵，只保留一半即可)。

#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;

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>()));
    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;
}

++ 自定义李代数数据类型

#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.hpp>

using namespace std;
using namespace Eigen;
using Sophus::SE3d;
using Sophus::SO3d;

typedef Matrix<double, 6, 6> Matrix6d;

// 给定误差求J_R^{-1}的近似
Matrix6d JRInv(const SE3d &e) {
    Matrix6d J;
    J.block(0, 0, 3, 3) = SO3d::hat(e.so3().log());
    J.block(0, 3, 3, 3) = SO3d::hat(e.translation());
    J.block(3, 0, 3, 3) = Matrix3d::Zero(3, 3);
    J.block(3, 3, 3, 3) = SO3d::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, SE3d> {
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(SE3d(
            Quaterniond(data[6], data[3], data[4], data[5]),
            Vector3d(data[0], data[1], data[2])
        ));
    }

    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 = SE3d();
    }

    // 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 = SE3d::exp(upd) * _estimate;
    }
};

// 两个李代数节点之边
class EdgeSE3LieAlgebra : public g2o::BaseBinaryEdge<6, SE3d, 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(SE3d(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() << " ";
        SE3d 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 {
        SE3d v1 = (static_cast<VertexSE3LieAlgebra *> (_vertices[0]))->estimate();
        SE3d v2 = (static_cast<VertexSE3LieAlgebra *> (_vertices[1]))->estimate();
        _error = (_measurement.inverse() * v1.inverse() * v2).log();
    }

    // 雅可比计算
    virtual void linearizeOplus() override {
        SE3d v1 = (static_cast<VertexSE3LieAlgebra *> (_vertices[0]))->estimate();
        SE3d v2 = (static_cast<VertexSE3LieAlgebra *> (_vertices[1]))->estimate();
        Matrix6d J = JRInv(SE3d::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>()));
    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;
}

CMakeLists.txt

cmake_minimum_required(VERSION 2.8)
project(pose_graph)

set(CMAKE_BUILD_TYPE "Release")
set(CMAKE_CXX_FLAGS "-std=c++11 -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}
        )

add_executable(pose_graph_g2o_lie pose_graph_g2o_lie_algebra.cpp)
target_link_libraries(pose_graph_g2o_lie
        g2o_core g2o_stuff
        ${CHOLMOD_LIBRARIES}
        ${Sophus_LIBRARIES}
        )