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(oldsymbol{P}_{i}=left[X_{i}, Y_{i}, Z_{i} ight]^{T})，其投影的像素坐标为 (oldsymbol{u}_{i}=left[u_{i}, v_{i} ight]^{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[egin{array}{c}u_{i} \ v_{i} \ 1end{array} ight]=oldsymbol{K} exp left(oldsymbol{xi}^{wedge} ight)left[egin{array}{c}X_{i} \ Y_{i} \ Z_{i} \ 1end{array} ight])
(s_{i} oldsymbol{u}_{i}=oldsymbol{K} exp left(oldsymbol{xi}^{wedge} ight) oldsymbol{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:
(oldsymbol{xi}^{*}=arg min _{oldsymbol{xi}} frac{1}{2} sum_{i=1}^{n}left|oldsymbol{u}_{i}-frac{1}{s_{i}} oldsymbol{K} exp left(oldsymbol{xi}^{wedge} ight) oldsymbol{P}_{i} ight|_{2}^{2})
首先，记变换 到相机坐标系下的空间点坐标为 (oldsymbol{P}^{prime}) ，并且把它前三维取出来：
(oldsymbol{P}^{prime}=left(exp left(oldsymbol{xi}^{wedge} ight) oldsymbol{P} ight)_{1: 3}=left[X^{prime}, Y^{prime}, Z^{prime} ight]^{T})
那么，相机投影模型相对于 $oldsymbol{P}^{prime} $ 则为：
(s oldsymbol{u}=oldsymbol{K} oldsymbol{P}^{prime})
(left[egin{array}{c}s u \ s v \ send{array} ight]=left[egin{array}{ccc}f_{x} & 0 & c_{x} \ 0 & f_{y} & c_{y} \ 0 & 0 & 1end{array} ight]left[egin{array}{c}X^{prime} \ Y^{prime} \ Z^{prime}end{array} ight])
(u=f_{x} frac{X^{prime}}{Z^{prime}}+c_{x}, quad v=f_{y} frac{Y^{prime}}{Z^{prime}}+c_{y})
(frac{partial oldsymbol{e}}{partial delta oldsymbol{xi}}=lim _{delta oldsymbol{xi} ightarrow 0} frac{e(delta oldsymbol{xi} oplus oldsymbol{xi})}{delta oldsymbol{xi}}=frac{partial oldsymbol{e}}{partial oldsymbol{P}^{prime}} frac{partial oldsymbol{P}^{prime}}{partial delta oldsymbol{xi}})
(frac{partial oldsymbol{e}}{partial oldsymbol{P}^{prime}}=-left[egin{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} ight]=-left[egin{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} ight])
(frac{partial(oldsymbol{T} oldsymbol{P})}{partial delta oldsymbol{xi}}=(oldsymbol{T} oldsymbol{P})^{odot}=left[egin{array}{cc}oldsymbol{I} & -oldsymbol{P}^{prime wedge} \ mathbf{0}^{T} & mathbf{0}^{T}end{array} ight])
(frac{partial oldsymbol{P}^{prime}}{partial delta oldsymbol{xi}}=left[oldsymbol{I},-oldsymbol{P}^{prime wedge} ight])
(frac{partial e}{partial delta oldsymbol{xi}}=-left[egin{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} ight])
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 oldsymbol{e}}{partial oldsymbol{P}}=frac{partial oldsymbol{e}}{partial oldsymbol{P}^{prime}} frac{partial oldsymbol{P}^{prime}}{partial oldsymbol{P}})
(oldsymbol{P}^{prime}=exp left(oldsymbol{xi}^{wedge} ight) oldsymbol{P}=oldsymbol{R} oldsymbol{P}+oldsymbol{t})
我们发现 P ′ 对 P 求导后只剩下 R。于是
(frac{partial oldsymbol{e}}{partial oldsymbol{P}}=-left[egin{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} ight] oldsymbol{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
", max_dist );
printf ( "-- Min dist : %f
", 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;
}