六、3D-3D ICP问题线性SVD解法与非线性BA解法
ICP（Iterative Closest Point）求解点云数据对的变换矩阵
误差模型
#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 )
{
//-- 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 )
);
}
主函数
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 );
// 1. 找关键点和匹配关系
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;
// 2. 建立两幅图的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;
//poxel2cam()像素坐标P转到相机归一化坐标系x; .pt转化为像素坐标
Point2d p1 = pixel2cam ( keypoints_1[m.queryIdx].pt, K );
Point2d p2 = pixel2cam ( keypoints_2[m.trainIdx].pt, K );
float dd1 = float ( d1 ) /5000.0; //自定义的深度尺度，方便画图，无实际意义。
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;
// 3. 位姿估计使用SVD
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;
// 4.位姿估计使用 BA非线性优化
bundleAdjustment( pts1, pts2, R, t );
//5 验证 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;
}
}
/
SVD解决ICP问题
void pose_estimation_3d3d (
const vector<Point3f>& pts1,
const vector<Point3f>& pts2,
Mat& R, Mat& t
)
{
//1. 去质心坐标q1 q2
Point3f p1, p2; // center of mass(Point3f三维坐标点)
int N = pts1.size();
for ( int i=0; i<N; i++ )
{
p1 += pts1[i];
p2 += pts2[i];
}
p1 = Point3f( Vec3f(p1) / N); //Vec3f指的是图像3通道
p2 = Point3f( Vec3f(p2) / N);
// vector<Point3f>向量，向量的每个分量都是3D点
vector<Point3f> q1 ( N ), q2 ( N ); // remove the center去质心后的点
for ( int i=0; i<N; i++ )
{
q1[i] = pts1[i] - p1;
q2[i] = pts2[i] - p2;
}
// 2 计算W=sum(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;
// 3 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;
// 4. R=U*V^T
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 ) );
}
///
BA解决ICP问题
自定义边EdgeProjectXYZRGBDPoseOnly类（3D-3D的边 ,g2o中无3d-3d的边。
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 );
/**********************************************************************************************************/
注解：
Vector3D map(const Vector3D & xyz) const
{
return _r*xyz + _t;
}
}
传入一个3D点，返回r*p+t，很明显就是求变换后点的坐标。
在g2o中，用SE3Quat类型表示变换T，此类型中有个成员函数就是map()，作用为对一个3D点进行坐标变换，
/**********************************************************************************************************/
virtual void linearizeOplus() //雅克比矩阵
{
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;
};
//BA求解ICP
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);
}
//执行优化
optimizer.setVerbose( true );
optimizer.initializeOptimization();
optimizer.optimize(10);
cout<<endl<<"after optimization:"<<endl;
cout<<"T="<<endl<<Eigen::Isometry3d( pose->estimate() ).matrix()<<endl;
}
结果
[ INFO:0] Initialize OpenCL runtime...
-- Max dist : 95.000000
-- Min dist : 7.000000
一共找到了81组匹配点
3d-3d pairs: 75
W= 11.8688 -0.717698 1.89486
-1.88065 3.83391 -5.78219
3.25846 -5.82734 9.65267
U= 0.592295 -0.805658 -0.0101195
-0.418171 -0.318113 0.850845
0.688709 0.499719 0.525319
V= 0.64736 -0.761401 -0.0345329
-0.388765 -0.368829 0.844291
0.655581 0.533135 0.534772
ICP via SVD results:
R = [0.9972065647956201, 0.05834273442898391, -0.04663895869192625;
-0.05787745545449197, 0.998260122172866, 0.01126626067193237;
0.04721511705620757, -0.008535444848613793, 0.9988482762174666]
t = [0.1379879629890433;
-0.06551699470729988;
-0.02981649388290575]
R_inv = [0.9972065647956201, -0.05787745545449197, 0.04721511705620757;
0.05834273442898391, 0.998260122172866, -0.008535444848613793;
-0.04663895869192625, 0.01126626067193237, 0.9988482762174666]
t_inv = [-0.1399866702492459;
0.05709791102272541;
0.03695589996443214]
calling bundle adjustment
iteration= 0 chi2= 18157.747747 time= 5.0389e-05 cumTime= 5.0389e-05 edges= 75 schur= 0
iteration= 1 chi2= 18151.933202 time= 1.3988e-05 cumTime= 6.4377e-05 edges= 75 schur= 0
iteration= 2 chi2= 18151.932131 time= 1.3026e-05 cumTime= 7.7403e-05 edges= 75 schur= 0
iteration= 3 chi2= 18151.932130 time= 1.2161e-05 cumTime= 8.95641e-05 edges= 75 schur= 0
iteration= 4 chi2= 18151.932130 time= 1.1238e-05 cumTime= 0.000100802 edges= 75 schur= 0
iteration= 5 chi2= 18151.932130 time= 1.3357e-05 cumTime= 0.000114159 edges= 75 schur= 0
iteration= 6 chi2= 18151.932130 time= 1.1731e-05 cumTime= 0.00012589 edges= 75 schur= 0
iteration= 7 chi2= 18151.932130 time= 1.1751e-05 cumTime= 0.000137641 edges= 75 schur= 0
iteration= 8 chi2= 18151.932130 time= 1.2621e-05 cumTime= 0.000150262 edges= 75 schur= 0
iteration= 9 chi2= 18151.932130 time= 1.1748e-05 cumTime= 0.00016201 edges= 75 schur= 0
optimization costs time: 0.000471091 seconds.
after optimization:
T=
0.997207 0.0583427 -0.046639 0.137988
-0.0578775 0.99826 0.0112663 -0.065517
0.0472151 -0.00853546 0.998848 -0.0298169
0 0 0 1
p1 = [-0.0374123, -0.830816, 2.7448]
p2 = [-0.0111479, -0.746763, 2.7652]
(R*p2+t) = [-0.04566300488482969;
-0.7791822151698653;
2.738046267661636]
p1 = [-0.243698, -0.117719, 1.5848]
p2 = [-0.299118, -0.0975683, 1.6558]
(R*p2+t) = [-0.243212054430598;
-0.1269486029625337;
1.610786345002579]
p1 = [-0.627753, 0.160186, 1.3396]
p2 = [-0.709645, 0.159033, 1.4212]
(R*p2+t) = [-0.6266796777024644;
0.1503229238263245;
1.354883323538178]
p1 = [-0.323443, 0.104873, 1.4266]
p2 = [-0.399079, 0.12047, 1.4838]
(R*p2+t) = [-0.3221508525590339;
0.09455772952719482;
1.432403794814274]
p1 = [-0.627221, 0.101454, 1.3116]
p2 = [-0.709709, 0.100216, 1.3998]
(R*p2+t) = [-0.6291763602679332;
0.09137127679150184;
1.334006907588644]
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