Bundle Adjustment

Bundle Adjustment

Bundle Adjustment 问题，是一个最小化重投影误差（Reprojection error）。考虑 n 个三维空间点 P 和它们的投影 p，我们希望计算相机的位姿 \(R,t\)，它的李代 数表示为 \(ξ\)。假设某空间点坐标为\(\boldsymbol{P}_{i}=\left[X_{i}, Y_{i}, Z_{i}\right]^{T}\)，其投影的像素坐标为 \(\boldsymbol{u}_{i}=\left[u_{i}, v_{i}\right]^{T}\)。 根据第五章的内容，像素位置与空间点位置的关系如下：

\(s_{i}\left[\begin{array}{c}u_{i} \\ v_{i} \\ 1\end{array}\right]=\boldsymbol{K} \exp \left(\boldsymbol{\xi}^{\wedge}\right)\left[\begin{array}{c}X_{i} \\ Y_{i} \\ Z_{i} \\ 1\end{array}\right]\)

\(s_{i} \boldsymbol{u}_{i}=\boldsymbol{K} \exp \left(\boldsymbol{\xi}^{\wedge}\right) \boldsymbol{P}_{i}\)

构建最小二乘问题，然后寻找最好的相机位姿，使它最小化:

\(\boldsymbol{\xi}^{*}=\arg \min _{\boldsymbol{\xi}} \frac{1}{2} \sum_{i=1}^{n}\left\|\boldsymbol{u}_{i}-\frac{1}{s_{i}} \boldsymbol{K} \exp \left(\boldsymbol{\xi}^{\wedge}\right) \boldsymbol{P}_{i}\right\|_{2}^{2}\)

首先，记变换 到相机坐标系下的空间点坐标为 \(\boldsymbol{P}^{\prime}\) ，并且把它前三维取出来：

\(\boldsymbol{P}^{\prime}=\left(\exp \left(\boldsymbol{\xi}^{\wedge}\right) \boldsymbol{P}\right)_{1: 3}=\left[X^{\prime}, Y^{\prime}, Z^{\prime}\right]^{T}\)

那么，相机投影模型相对于 $\boldsymbol{P}^{\prime} $ 则为：

\(s \boldsymbol{u}=\boldsymbol{K} \boldsymbol{P}^{\prime}\)

\(\left[\begin{array}{c}s u \\ s v \\ s\end{array}\right]=\left[\begin{array}{ccc}f_{x} & 0 & c_{x} \\ 0 & f_{y} & c_{y} \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{c}X^{\prime} \\ Y^{\prime} \\ Z^{\prime}\end{array}\right]\)

\(u=f_{x} \frac{X^{\prime}}{Z^{\prime}}+c_{x}, \quad v=f_{y} \frac{Y^{\prime}}{Z^{\prime}}+c_{y}\)

\(\frac{\partial \boldsymbol{e}}{\partial \delta \boldsymbol{\xi}}=\lim _{\delta \boldsymbol{\xi} \rightarrow 0} \frac{e(\delta \boldsymbol{\xi} \oplus \boldsymbol{\xi})}{\delta \boldsymbol{\xi}}=\frac{\partial \boldsymbol{e}}{\partial \boldsymbol{P}^{\prime}} \frac{\partial \boldsymbol{P}^{\prime}}{\partial \delta \boldsymbol{\xi}}\)

\(\frac{\partial \boldsymbol{e}}{\partial \boldsymbol{P}^{\prime}}=-\left[\begin{array}{ccc}\frac{\partial u}{\partial X^{\prime}} & \frac{\partial u}{\partial Y^{\prime}} & \frac{\partial u}{\partial Z^{\prime}} \\ \frac{\partial v}{\partial X^{\prime}} & \frac{\partial v}{\partial Y^{\prime}} & \frac{\partial v}{\partial Z^{\prime}}\end{array}\right]=-\left[\begin{array}{ccc}\frac{f_{x}}{Z^{\prime}} & 0 & -\frac{f_{x} X^{\prime}}{Z^{\prime 2}} \\ 0 & \frac{f_{y}}{Z^{\prime}} & -\frac{f_{y} Y^{\prime}}{Z^{\prime 2}}\end{array}\right]\)

\(\frac{\partial(\boldsymbol{T} \boldsymbol{P})}{\partial \delta \boldsymbol{\xi}}=(\boldsymbol{T} \boldsymbol{P})^{\odot}=\left[\begin{array}{cc}\boldsymbol{I} & -\boldsymbol{P}^{\prime \wedge} \\ \mathbf{0}^{T} & \mathbf{0}^{T}\end{array}\right]\)

\(\frac{\partial \boldsymbol{P}^{\prime}}{\partial \delta \boldsymbol{\xi}}=\left[\boldsymbol{I},-\boldsymbol{P}^{\prime \wedge}\right]\)

\(\frac{\partial e}{\partial \delta \boldsymbol{\xi}}=-\left[\begin{array}{ccccc}\frac{f_{x}}{Z^{\prime}} & 0 & -\frac{f_{x} X^{\prime}}{Z^{\prime 2}} & -\frac{f_{x} X^{\prime} Y^{\prime}}{Z^{\prime 2}} & f_{x}+\frac{f_{x} X^{2}}{Z^{\prime 2}} & -\frac{f_{x} Y^{\prime}}{Z^{\prime}} \\ 0 & \frac{f_{y}}{Z^{\prime}} & -\frac{f_{y} Y^{\prime}}{Z^{\prime 2}} & -f_{y}-\frac{f_{y} Y^{\prime 2}}{Z^{\prime 2}} & \frac{f_{y} X^{\prime} Y^{\prime}}{Z^{\prime 2}} & \frac{f_{y} X^{\prime}}{Z^{\prime}}\end{array}\right]\)

我们还希望优化特征点的空间位置。因此，需要讨论 \(e\) ,关于空间点 \(P\) 的导数。

\(\frac{\partial \boldsymbol{e}}{\partial \boldsymbol{P}}=\frac{\partial \boldsymbol{e}}{\partial \boldsymbol{P}^{\prime}} \frac{\partial \boldsymbol{P}^{\prime}}{\partial \boldsymbol{P}}\)

\(\boldsymbol{P}^{\prime}=\exp \left(\boldsymbol{\xi}^{\wedge}\right) \boldsymbol{P}=\boldsymbol{R} \boldsymbol{P}+\boldsymbol{t}\)

我们发现 P ′ 对 P 求导后只剩下 R。于是

\(\frac{\partial \boldsymbol{e}}{\partial \boldsymbol{P}}=-\left[\begin{array}{ccc}\frac{f_{x}}{Z^{\prime}} & 0 & -\frac{f_{x} X^{\prime}}{Z^{\prime 2}} \\ 0 & \frac{f_{y}}{Z^{\prime}} & -\frac{f_{y} Y^{\prime}}{Z^{\prime 2}}\end{array}\right] \boldsymbol{R}\)

#include <iostream>
#include <opencv2/core/core.hpp>
#include <opencv2/features2d/features2d.hpp>
#include <opencv2/highgui/highgui.hpp>
#include <opencv2/calib3d/calib3d.hpp>
#include <Eigen/Core>
#include <Eigen/Geometry>
#include <Eigen/SVD>
#include <g2o/core/base_vertex.h>
#include <g2o/core/base_unary_edge.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/optimization_algorithm_gauss_newton.h>
#include <g2o/solvers/eigen/linear_solver_eigen.h>
#include <g2o/types/sba/types_six_dof_expmap.h>
#include <chrono>

using namespace std;
using namespace cv;

void find_feature_matches (
    const Mat& img_1, const Mat& img_2,
    std::vector<KeyPoint>& keypoints_1,
    std::vector<KeyPoint>& keypoints_2,
    std::vector< DMatch >& matches );

// 像素坐标转相机归一化坐标
Point2d pixel2cam ( const Point2d& p, const Mat& K );

void pose_estimation_3d3d (
    const vector<Point3f>& pts1,
    const vector<Point3f>& pts2,
    Mat& R, Mat& t
);

void bundleAdjustment(
    const vector<Point3f>& points_3d,
    const vector<Point3f>& points_2d,
    Mat& R, Mat& t
);

// g2o edge
class EdgeProjectXYZRGBDPoseOnly : public g2o::BaseUnaryEdge<3, Eigen::Vector3d, g2o::VertexSE3Expmap>
{
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
    EdgeProjectXYZRGBDPoseOnly( const Eigen::Vector3d& point ) : _point(point) {}

    virtual void computeError()
    {
        const g2o::VertexSE3Expmap* pose = static_cast<const g2o::VertexSE3Expmap*> ( _vertices[0] );
        // measurement is p, point is p'
        _error = _measurement - pose->estimate().map( _point );
    }

    virtual void linearizeOplus()
    {
        g2o::VertexSE3Expmap* pose = static_cast<g2o::VertexSE3Expmap *>(_vertices[0]);
        g2o::SE3Quat T(pose->estimate());
        Eigen::Vector3d xyz_trans = T.map(_point);
        double x = xyz_trans[0];
        double y = xyz_trans[1];
        double z = xyz_trans[2];

        _jacobianOplusXi(0,0) = 0;
        _jacobianOplusXi(0,1) = -z;
        _jacobianOplusXi(0,2) = y;
        _jacobianOplusXi(0,3) = -1;
        _jacobianOplusXi(0,4) = 0;
        _jacobianOplusXi(0,5) = 0;

        _jacobianOplusXi(1,0) = z;
        _jacobianOplusXi(1,1) = 0;
        _jacobianOplusXi(1,2) = -x;
        _jacobianOplusXi(1,3) = 0;
        _jacobianOplusXi(1,4) = -1;
        _jacobianOplusXi(1,5) = 0;

        _jacobianOplusXi(2,0) = -y;
        _jacobianOplusXi(2,1) = x;
        _jacobianOplusXi(2,2) = 0;
        _jacobianOplusXi(2,3) = 0;
        _jacobianOplusXi(2,4) = 0;
        _jacobianOplusXi(2,5) = -1;
    }

    bool read ( istream& in ) {}
    bool write ( ostream& out ) const {}
protected:
    Eigen::Vector3d _point;
};

int main ( int argc, char** argv )
{
    if ( argc != 5 )
    {
        cout<<"usage: pose_estimation_3d3d img1 img2 depth1 depth2"<<endl;
        return 1;
    }
    //-- 读取图像
    Mat img_1 = imread ( argv[1], CV_LOAD_IMAGE_COLOR );
    Mat img_2 = imread ( argv[2], CV_LOAD_IMAGE_COLOR );

    vector<KeyPoint> keypoints_1, keypoints_2;
    vector<DMatch> matches;
    find_feature_matches ( img_1, img_2, keypoints_1, keypoints_2, matches );
    cout<<"一共找到了"<<matches.size() <<"组匹配点"<<endl;

    // 建立3D点
    Mat depth1 = imread ( argv[3], CV_LOAD_IMAGE_UNCHANGED );       // 深度图为16位无符号数，单通道图像
    Mat depth2 = imread ( argv[4], CV_LOAD_IMAGE_UNCHANGED );       // 深度图为16位无符号数，单通道图像
    Mat K = ( Mat_<double> ( 3,3 ) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1 );
    vector<Point3f> pts1, pts2;

    for ( DMatch m:matches )
    {
        ushort d1 = depth1.ptr<unsigned short> ( int ( keypoints_1[m.queryIdx].pt.y ) ) [ int ( keypoints_1[m.queryIdx].pt.x ) ];
        ushort d2 = depth2.ptr<unsigned short> ( int ( keypoints_2[m.trainIdx].pt.y ) ) [ int ( keypoints_2[m.trainIdx].pt.x ) ];
        if ( d1==0 || d2==0 )   // bad depth
            continue;
        Point2d p1 = pixel2cam ( keypoints_1[m.queryIdx].pt, K );
        Point2d p2 = pixel2cam ( keypoints_2[m.trainIdx].pt, K );
        float dd1 = float ( d1 ) /5000.0;
        float dd2 = float ( d2 ) /5000.0;
        pts1.push_back ( Point3f ( p1.x*dd1, p1.y*dd1, dd1 ) );
        pts2.push_back ( Point3f ( p2.x*dd2, p2.y*dd2, dd2 ) );
    }

    cout<<"3d-3d pairs: "<<pts1.size() <<endl;
    Mat R, t;
    pose_estimation_3d3d ( pts1, pts2, R, t );
    cout<<"ICP via SVD results: "<<endl;
    cout<<"R = "<<R<<endl;
    cout<<"t = "<<t<<endl;
    cout<<"R_inv = "<<R.t() <<endl;
    cout<<"t_inv = "<<-R.t() *t<<endl;

    cout<<"calling bundle adjustment"<<endl;

    bundleAdjustment( pts1, pts2, R, t );

    // verify p1 = R*p2 + t
    for ( int i=0; i<5; i++ )
    {
        cout<<"p1 = "<<pts1[i]<<endl;
        cout<<"p2 = "<<pts2[i]<<endl;
        cout<<"(R*p2+t) = "<<
            R * (Mat_<double>(3,1)<<pts2[i].x, pts2[i].y, pts2[i].z) + t
            <<endl;
        cout<<endl;
    }
}

void find_feature_matches ( const Mat& img_1, const Mat& img_2,
                            std::vector<KeyPoint>& keypoints_1,
                            std::vector<KeyPoint>& keypoints_2,
                            std::vector< DMatch >& matches )
{
    //-- 初始化
    Mat descriptors_1, descriptors_2;
    // used in OpenCV3
    Ptr<FeatureDetector> detector = ORB::create();
    Ptr<DescriptorExtractor> descriptor = ORB::create();
    // use this if you are in OpenCV2
    // Ptr<FeatureDetector> detector = FeatureDetector::create ( "ORB" );
    // Ptr<DescriptorExtractor> descriptor = DescriptorExtractor::create ( "ORB" );
    Ptr<DescriptorMatcher> matcher  = DescriptorMatcher::create("BruteForce-Hamming");
    //-- 第一步:检测 Oriented FAST 角点位置
    detector->detect ( img_1,keypoints_1 );
    detector->detect ( img_2,keypoints_2 );

    //-- 第二步:根据角点位置计算 BRIEF 描述子
    descriptor->compute ( img_1, keypoints_1, descriptors_1 );
    descriptor->compute ( img_2, keypoints_2, descriptors_2 );

    //-- 第三步:对两幅图像中的BRIEF描述子进行匹配，使用 Hamming 距离
    vector<DMatch> match;
   // BFMatcher matcher ( NORM_HAMMING );
    matcher->match ( descriptors_1, descriptors_2, match );

    //-- 第四步:匹配点对筛选
    double min_dist=10000, max_dist=0;

    //找出所有匹配之间的最小距离和最大距离, 即是最相似的和最不相似的两组点之间的距离
    for ( int i = 0; i < descriptors_1.rows; i++ )
    {
        double dist = match[i].distance;
        if ( dist < min_dist ) min_dist = dist;
        if ( dist > max_dist ) max_dist = dist;
    }

    printf ( "-- Max dist : %f \n", max_dist );
    printf ( "-- Min dist : %f \n", min_dist );

    //当描述子之间的距离大于两倍的最小距离时,即认为匹配有误.但有时候最小距离会非常小,设置一个经验值30作为下限.
    for ( int i = 0; i < descriptors_1.rows; i++ )
    {
        if ( match[i].distance <= max ( 2*min_dist, 30.0 ) )
        {
            matches.push_back ( match[i] );
        }
    }
}

Point2d pixel2cam ( const Point2d& p, const Mat& K )
{
    return Point2d
           (
               ( p.x - K.at<double> ( 0,2 ) ) / K.at<double> ( 0,0 ),
               ( p.y - K.at<double> ( 1,2 ) ) / K.at<double> ( 1,1 )
           );
}

void pose_estimation_3d3d (
    const vector<Point3f>& pts1,
    const vector<Point3f>& pts2,
    Mat& R, Mat& t
)
{
    Point3f p1, p2;     // center of mass
    int N = pts1.size();
    for ( int i=0; i<N; i++ )
    {
        p1 += pts1[i];
        p2 += pts2[i];
    }
    p1 = Point3f( Vec3f(p1) /  N);
    p2 = Point3f( Vec3f(p2) / N);
    vector<Point3f>     q1 ( N ), q2 ( N ); // remove the center
    for ( int i=0; i<N; i++ )
    {
        q1[i] = pts1[i] - p1;
        q2[i] = pts2[i] - p2;
    }

    // compute q1*q2^T
    Eigen::Matrix3d W = Eigen::Matrix3d::Zero();
    for ( int i=0; i<N; i++ )
    {
        W += Eigen::Vector3d ( q1[i].x, q1[i].y, q1[i].z ) * Eigen::Vector3d ( q2[i].x, q2[i].y, q2[i].z ).transpose();
    }
    cout<<"W="<<W<<endl;

    // SVD on W
    Eigen::JacobiSVD<Eigen::Matrix3d> svd ( W, Eigen::ComputeFullU|Eigen::ComputeFullV );
    Eigen::Matrix3d U = svd.matrixU();
    Eigen::Matrix3d V = svd.matrixV();
    
    if (U.determinant() * V.determinant() < 0)
	{
        for (int x = 0; x < 3; ++x)
        {
            U(x, 2) *= -1;
        }
	}
    
    cout<<"U="<<U<<endl;
    cout<<"V="<<V<<endl;

    Eigen::Matrix3d R_ = U* ( V.transpose() );
    Eigen::Vector3d t_ = Eigen::Vector3d ( p1.x, p1.y, p1.z ) - R_ * Eigen::Vector3d ( p2.x, p2.y, p2.z );

    // convert to cv::Mat
    R = ( Mat_<double> ( 3,3 ) <<
          R_ ( 0,0 ), R_ ( 0,1 ), R_ ( 0,2 ),
          R_ ( 1,0 ), R_ ( 1,1 ), R_ ( 1,2 ),
          R_ ( 2,0 ), R_ ( 2,1 ), R_ ( 2,2 )
        );
    t = ( Mat_<double> ( 3,1 ) << t_ ( 0,0 ), t_ ( 1,0 ), t_ ( 2,0 ) );
}

void bundleAdjustment (
    const vector< Point3f >& pts1,
    const vector< Point3f >& pts2,
    Mat& R, Mat& t )
{
    // 初始化g2o
    typedef g2o::BlockSolver< g2o::BlockSolverTraits<6,3> > Block;  // pose维度为 6, landmark 维度为 3
    Block::LinearSolverType* linearSolver = new g2o::LinearSolverEigen<Block::PoseMatrixType>(); // 线性方程求解器
    Block* solver_ptr = new Block( linearSolver );      // 矩阵块求解器
    g2o::OptimizationAlgorithmGaussNewton* solver = new g2o::OptimizationAlgorithmGaussNewton( solver_ptr );
    g2o::SparseOptimizer optimizer;
    optimizer.setAlgorithm( solver );

    // vertex
    g2o::VertexSE3Expmap* pose = new g2o::VertexSE3Expmap(); // camera pose
    pose->setId(0);
    pose->setEstimate( g2o::SE3Quat(
        Eigen::Matrix3d::Identity(),
        Eigen::Vector3d( 0,0,0 )
    ) );
    optimizer.addVertex( pose );

    // edges
    int index = 1;
    vector<EdgeProjectXYZRGBDPoseOnly*> edges;
    for ( size_t i=0; i<pts1.size(); i++ )
    {
        EdgeProjectXYZRGBDPoseOnly* edge = new EdgeProjectXYZRGBDPoseOnly(
            Eigen::Vector3d(pts2[i].x, pts2[i].y, pts2[i].z) );
        edge->setId( index );
        edge->setVertex( 0, dynamic_cast<g2o::VertexSE3Expmap*> (pose) );
        edge->setMeasurement( Eigen::Vector3d(
            pts1[i].x, pts1[i].y, pts1[i].z) );
        edge->setInformation( Eigen::Matrix3d::Identity()*1e4 );
        optimizer.addEdge(edge);
        index++;
        edges.push_back(edge);
    }

    chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
    optimizer.setVerbose( true );
    optimizer.initializeOptimization();
    optimizer.optimize(10);
    chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
    chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2-t1);
    cout<<"optimization costs time: "<<time_used.count()<<" seconds."<<endl;

    cout<<endl<<"after optimization:"<<endl;
    cout<<"T="<<endl<<Eigen::Isometry3d( pose->estimate() ).matrix()<<endl;

}