《视觉 SLAM 十四讲》V2 第 9 讲 后端优化1 【扩展卡尔曼滤波器 EKF && BA+非线性优化(Ceres、g2o)】

文章目录

• [第9讲 后端1](#第9讲 后端1)
• • • [9.1.2 线性系统和 KF](#9.1.2 线性系统和 KF)
    • [9.1.4 扩展卡尔曼滤波器 EKF 不足](#9.1.4 扩展卡尔曼滤波器 EKF 不足)
  • [9.2 BA 与 图优化](#9.2 BA 与 图优化)
  • • [9.2.1 投影模型和 BA 代价函数](#9.2.1 投影模型和 BA 代价函数)
    • [9.2.2 BA 的求解](#9.2.2 BA 的求解)
    • [9.2.3 稀疏性 和 边缘化](#9.2.3 稀疏性 和 边缘化)
    • [9.2.4 鲁棒核函数](#9.2.4 鲁棒核函数)
  • [9.3 实践： Ceres BA 【Code】](#9.3 实践： Ceres BA 【Code】)
  • • • • [本讲 CMakeLists.txt](#本讲 CMakeLists.txt)
  • [9.4 实践：g2o 求解 BA 【Code】](#9.4 实践：g2o 求解 BA 【Code】)
  • 习题
  • • [√ 题 1](#√ 题 1)

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 k k 时刻状态仅与 k − 1 k-1 k−1 时刻状态 有关。 【扩展卡尔曼滤波(EKF)】

考虑 k k 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 Residual
        cost_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 directory
   10 | #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 address
    explicit 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 derivatives
    virtual 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 LM
    auto 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 problem
    const double *observations = bal_problem.observations();
    // vertex
    vector<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);
    }

    // edge
    for (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 problem
    for (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 K \bm{K} K 的求解 可以不依靠 P ^ k \bm{\hat{P}}_k P^k。

即 用 别的已知量替换 K \bm{K} K 定义中的 P ^ k \bm{\hat{P}}_k P^k

State estimation for robotics: A matrix lie group approach.

来自上述书籍的公式：

---

证明:
K = P ^ k C k T Q − 1 由式 ( 9.16 ) = ( C k T Q − 1 C k + P ˇ k − 1 ) − 1 C k T Q − 1 由式 ( 2.75 c ) 由右到左 其中： A ⇒ P ˇ k , B ⇒ C k T , C ⇒ C k , D ⇒ Q = P ˇ k C k T ( Q + C k P ˇ k C k T ) − 1 \begin{align*} \bm{K} & = \bm{\hat{P}}_k\bm{C}_k^T\bm{Q}^{-1} \\ & 由 式 (9.16) \\ &= (\bm{C}_k^T\bm{Q}^{-1}\bm{C}_k+\bm{\check{P}}_k^{-1})^{-1}\bm{C}_k^T\bm{Q}^{-1} \\ & 由 式 (2.75c) 由右到左 \\ & 其中： A \Rightarrow \bm{\check{P}}_k, B \Rightarrow \bm{C}_k^T, C \Rightarrow \bm{C}_k, D \Rightarrow \bm{Q}\\ & = \bm{\check{P}}_k\bm{C}_k^T(\bm{Q} + \bm{C}_k\bm{\check{P}}_k\bm{C}_k^T)^{-1} \end{align*} K=P^kCkTQ−1由式(9.16)=(CkTQ−1Ck+Pˇk−1)−1CkTQ−1由式(2.75c)由右到左其中：A⇒Pˇk,B⇒CkT,C⇒Ck,D⇒Q=PˇkCkT(Q+CkPˇkCkT)−1

证毕。

---

$\check{A}$

A ˇ \check{A} Aˇ

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 S \bm{S} S 矩阵为 半正定矩阵。

• 待 证
  
  S = B − E C − 1 E T \bm{S} = \bm{B}-\bm{EC}^{-1}\bm{E}^T S=B−EC−1ET