在之前的章节我已经讲过了G2O（General Graph Optimization）的相关原理以及应用，这次的BA优化问题呢，我们也可以转化为一个G2O来求解。
G2O（General Graph Optimization）———— 通用图优化。
G2O（General Graph Optimization）的核里带有种类多样的求解器，而它的顶点、边的类型也是多种多样。我们可以自己定义顶点和边。总的来说，如果一个优化问题能够表达成图（顶点与边），那么这个问题就可以用G2O（General Graph Optimization）去求解它。常见的，比如bundle adjustment（这里的BA优化），ICP，数据拟合，都可以用G2O（General Graph Optimization）来做。
如何使用图优化进行BA求解呢？
对于图优化问题，首先需要定义一个顶点和一个边。这里呢，我们使用顶点为第二个相机帧的位姿作为顶点，即节点。对于边呢，就是定义相机种的三维点在第二个相机帧的投影作为边。
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);// BA by g2o
typedef vector<eigen::vector2d, eigen::aligned_allocator<eigen::vector2d="">> VecVector2d;
typedef vector<eigen::vector3d, eigen::aligned_allocator<eigen::vector3d="">> VecVector3d;void bundleAdjustmentG2O(const VecVector3d &points_3d,const VecVector2d &points_2d,const Mat &K,Sophus::SE3d &pose
);// BA by gauss-newton
void bundleAdjustmentGaussNewton(const VecVector3d &points_3d,const VecVector2d &points_2d,const Mat &K,Sophus::SE3d &pose
);int main(int argc, char **argv) {if (argc != 5) {cout << "usage: pose_estimation_3d2d 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);assert(img_1.data && img_2.data && "Can not load images!");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 d1 = imread(argv[3], 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> pts_3d;vector<point2f> pts_2d;for (DMatch m:matches) {ushort d = d1.ptr<unsigned short="">(int(keypoints_1[m.queryIdx].pt.y))[int(keypoints_1[m.queryIdx].pt.x)];if (d == 0) // bad depthcontinue;float dd = d / 5000.0;Point2d p1 = pixel2cam(keypoints_1[m.queryIdx].pt, K);pts_3d.push_back(Point3f(p1.x * dd, p1.y * dd, dd));pts_2d.push_back(keypoints_ainIdx].pt);}cout << "3d-2d pairs: " << pts_3d.size() << endl;chrono::steady_clock::time_point t1 = chrono::steady_clock::now();Mat r, t;solvePnP(pts_3d, pts_2d, K, Mat(), r, t, false); // 调用OpenCV 的 PnP 求解，可选择EPNP，DLS等方法Mat R;cv::Rodrigues(r, R); // r为旋转向量形式，用Rodrigues公式转换为矩阵chrono::steady_clock::time_point t2 = chrono::steady_clock::now();chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);cout << "solve pnp in opencv cost time: " << unt() << " seconds." << endl;cout << "R=" << endl << R << endl;cout << "t=" << endl << t << endl;VecVector3d pts_3d_eigen;VecVector2d pts_2d_eigen;for (size_t i = 0; i < pts_3d.size(); ++i) {pts_3d_eigen.push_back(Eigen::Vector3d(pts_3d[i].x, pts_3d[i].y, pts_3d[i].z));pts_2d_eigen.push_back(Eigen::Vector2d(pts_2d[i].x, pts_2d[i].y));}cout << "calling bundle adjustment by gauss newton" << endl;Sophus::SE3d pose_gn;t1 = chrono::steady_clock::now();bundleAdjustmentGaussNewton(pts_3d_eigen, pts_2d_eigen, K, pose_gn);t2 = chrono::steady_clock::now();time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);cout << "solve pnp by gauss newton cost time: " << unt() << " seconds." << endl;cout << "calling bundle adjustment by g2o" << endl;Sophus::SE3d pose_g2o;t1 = chrono::steady_clock::now();bundleAdjustmentG2O(pts_3d_eigen, pts_2d_eigen, K, pose_g2o);t2 = chrono::steady_clock::now();time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);cout << "solve pnp by g2o cost time: " << unt() << " seconds." << endl;return 0;
}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 bundleAdjustmentGaussNewton(const VecVector3d &points_3d,const VecVector2d &points_2d,const Mat &K,Sophus::SE3d &pose) {typedef Eigen::Matrix<double, 6,="" 1=""> Vector6d;const int iterations = 10;double cost = 0, lastCost = 0;double fx = K.at<double>(0, 0);double fy = K.at<double>(1, 1);double cx = K.at<double>(0, 2);double cy = K.at<double>(1, 2);for (int iter = 0; iter < iterations; iter++) {Eigen::Matrix<double, 6,="" 6=""> H = Eigen::Matrix<double, 6,="" 6="">::Zero();Vector6d b = Vector6d::Zero();cost = 0;// compute costfor (int i = 0; i < points_3d.size(); i++) {Eigen::Vector3d pc = pose * points_3d[i];double inv_z = 1.0 / pc[2];double inv_z2 = inv_z * inv_z;Eigen::Vector2d proj(fx * pc[0] / pc[2] + cx, fy * pc[1] / pc[2] + cy);Eigen::Vector2d e = points_2d[i] - proj;cost += e.squaredNorm();Eigen::Matrix<double, 2,="" 6=""> J;J << -fx * inv_z,0,fx * pc[0] * inv_z2,fx * pc[0] * pc[1] * inv_z2,-fx - fx * pc[0] * pc[0] * inv_z2,fx * pc[1] * inv_z,0,-fy * inv_z,fy * pc[1] * inv_z2,fy + fy * pc[1] * pc[1] * inv_z2,-fy * pc[0] * pc[1] * inv_z2,-fy * pc[0] * inv_z;H += J.transpose() * J;b += -J.transpose() * e;}Vector6d dx;dx = H.ldlt().solve(b);if (isnan(dx[0])) {cout << "result is nan!" << endl;break;}if (iter > 0 && cost >= lastCost) {// cost increase, up<ickey>date is not goodcout << "cost: " << cost << ", last cost: " << lastCost << endl;break;}// precision(12) << cost << endl;if (dx.norm() < 1e-6) {// convergebreak;}}cout << "pose by g-n: n" << pose.matrix() << endl;
}/// vertex and edges used in g2o ba
class VertexPose : public g2o::BaseVertex<6, Sophus::SE3d> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;virtual void setToOriginImpl() override {_estimate = Sophus::SE3d();}/// left multiplication on SE3virtual void oplusImpl(const double *update) override {Eigen::Matrix<double, 6,="" 1=""> update_eigen;update_eigen << update[0], update[1], update[2], update[3], update[4], update[5];_estimate = Sophus::SE3d::exp(update_eigen) * _estimate;}virtual bool read(istream &in) override {}virtual bool write(ostream &out) const override {}
};class EdgeProjection : public g2o::BaseUnaryEdge<2, Eigen::Vector2d, VertexPose> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;EdgeProjection(const Eigen::Vector3d &pos, const Eigen::Matrix3d &K) : _pos3d(pos), _K(K) {}virtual void computeError() override {const VertexPose *v = static_cast<vertexpose *=""> (_vertices[0]);Sophus::SE3d T = v->estimate();Eigen::Vector3d pos_pixel = _K * (T * _pos3d);pos_pixel /= pos_pixel[2];_error = _measurement - pos_pixel.head<2>();}virtual void linearizeOplus() override {const VertexPose *v = static_cast<vertexpose *=""> (_vertices[0]);Sophus::SE3d T = v->estimate();Eigen::Vector3d pos_cam = T * _pos3d;double fx = _K(0, 0);double fy = _K(1, 1);double cx = _K(0, 2);double cy = _K(1, 2);double X = pos_cam[0];double Y = pos_cam[1];double Z = pos_cam[2];double Z2 = Z * Z;_jacobianOplusXi<< -fx / Z, 0, fx * X / Z2, fx * X * Y / Z2, -fx - fx * X * X / Z2, fx * Y / Z,0, -fy / Z, fy * Y / (Z * Z), fy + fy * Y * Y / Z2, -fy * X * Y / Z2, -fy * X / Z;}virtual bool read(istream &in) override {}virtual bool write(ostream &out) const override {}private:Eigen::Vector3d _pos3d;Eigen::Matrix3d _K;
};void bundleAdjustmentG2O(const VecVector3d &points_3d,const VecVector2d &points_2d,const Mat &K,Sophus::SE3d &pose) {// 构建图优化，先设定g2otypedef g2o::BlockSolver<g2o::blocksolvertraits<6, 3="">> BlockSolverType; // pose is 6, landmark is 3typedef g2o::LinearSolverDense<blocksolvertype::posematrixtype> LinearSolverType; // 线性求解器类型// 梯度下降方法，可以从GN, LM, DogLeg 中选auto solver = new g2o::OptimizationAlgorithmGaussNewton(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// vertexVertexPose *vertex_pose = new VertexPose(); // camera vertex_posevertex_pose->setId(0);vertex_pose->setEstimate(Sophus::SE3d());optimizer.addVertex(vertex_pose);// KEigen::Matrix3d K_eigen;K_eigen <<K.at<double>(0, 0), K.at<double>(0, 1), K.at<double>(0, 2),K.at<double>(1, 0), K.at<double>(1, 1), K.at<double>(1, 2),K.at<double>(2, 0), K.at<double>(2, 1), K.at<double>(2, 2);// edgesint index = 1;for (size_t i = 0; i < points_2d.size(); ++i) {auto p2d = points_2d[i];auto p3d = points_3d[i];EdgeProjection *edge = new EdgeProjection(p3d, K_eigen);edge->setId(index);edge->setVertex(0, vertex_pose);edge->setMeasurement(p2d);edge->setInformation(Eigen::Matrix2d::Identity());optimizer.addEdge(edge);index++;}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: " << unt() << " seconds." << endl;cout << "pose estimated by g2o =n" << vertex_pose->estimate().matrix() << endl;pose = vertex_pose->estimate();
}ICP（Iterative Closest Point）———— 迭代最近点。这是一个3D-3D点的优化问题，具体就是在三维点知道了如何匹配特征点，通过三维特征点匹配出相关的位姿优化问题。所以ICP（Iterative Closest Point）可以是经常在激光雷达的SLAM构建中经常遇到。
对于ICP（Iterative Closest Point）问题的求解，在学术领域主要分为2种类型：第一：使用奇异值分解求解；第二：使用非线性优化求解。
> ICP算法流程
首先对于一幅点云中的每个点，在另一幅点云中计算匹配点（最近点）
极小化匹配点间的匹配误差，计算位姿
然后将计算的位姿作用于点云
再重新计算匹配点
如此迭代，直到迭代次数达到阈值，或者极小化的能量函数变化量小于设定阈值
对于使用奇异值求解的办法，即SVD方法。首先对于已经知道的2个三维点进行投影误差的计算，构建最小二乘问题。分别求取旋转矩阵，在求取平移矩阵。
我们依然构建三维点误差的最小二乘估计，通过之前介绍的PNP类似的办法求取优化。如果使用李群李代数，则在求导时用李群李代数的扰动模型。
/// g2o edge
class EdgeProjectXYZRGBDPoseOnly : public g2o::BaseUnaryEdge<3, Eigen::Vector3d, VertexPose> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;EdgeProjectXYZRGBDPoseOnly(const Eigen::Vector3d &point) : _point(point) {}virtual void computeError() override {const VertexPose *pose = static_cast<const vertexpose="" *=""> ( _vertices[0] );_error = _measurement - pose->estimate() * _point;}virtual void linearizeOplus() override {VertexPose *pose = static_cast<vertexpose *="">(_vertices[0]);Sophus::SE3d T = pose->estimate();Eigen::Vector3d xyz_trans = T * _point;_jacobianOplusXi.block<3, 3>(0, 0) = -Eigen::Matrix3d::Identity();_jacobianOplusXi.block<3, 3>(0, 3) = Sophus::SO3d::hat(xyz_trans);}bool read(istream &in) {}bool write(ostream &out) const {}protected:Eigen::Vector3d _point;
};int main(int argc, char **argv) {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_ainIdx].pt.y))[int(keypoints_ainIdx].pt.x)];}cout << "3d-3d pairs: " << pts1.size() << endl;Mat R, t;pose_estimation_3d3d(pts1, pts2, R, t);cout << "calling bundle adjustment" << endl;bundleAdjustment(pts1, pts2, R, t);}
}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 OpenCV3Ptr<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 < ws; 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 < ws; i++) {if (match[i].distance <= max(2 * min_dist, 30.0)) {matches.push_back(match[i]);}}
}void pose_estimation_3d3d(const vector<point3f> &pts1,const vector<point3f> &pts2,Mat &R, Mat &t) {Point3f p1, p2; // center of massint 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 centerfor (int i = 0; i < N; i++) {q1[i] = pts1[i] - p1;q2[i] = pts2[i] - p2;}// compute q1*q2^TEigen::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 WEigen::JacobiSVD<eigen::matrix3d> svd(W, Eigen::ComputeFullU | Eigen::ComputeFullV);Eigen::Matrix3d U = svd.matrixU();Eigen::Matrix3d V = svd.matrixV();cout << "U=" << U << endl;cout << "V=" << V << endl;Eigen::Matrix3d R_ = U * (V.transpose());if (R_.determinant() < 0) {R_ = -R_;}Eigen::Vector3d t_ = Eigen::Vector3d(p1.x, p1.y, p1.z) - R_ * Eigen::Vector3d(p2.x, p2.y, p2.z);// convert to cv::MatR = (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) {// 构建图优化，先设定g2otypedef g2o::BlockSolverX BlockSolverType;typedef g2o::LinearSolverDense<blocksolvertype::posematrixtype> LinearSolverType; // 线性求解器类型// 梯度下降方法，可以从GN, LM, DogLeg 中选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// vertexVertexPose *pose = new VertexPose(); // camera posepose->setId(0);pose->setEstimate(Sophus::SE3d());optimizer.addVertex(pose);// edgesfor (size_t i = 0; i < pts1.size(); i++) {EdgeProjectXYZRGBDPoseOnly *edge = new EdgeProjectXYZRGBDPoseOnly(Eigen::Vector3d(pts2[i].x, pts2[i].y, pts2[i].z));edge->setVertex(0, pose);edge->setMeasurement(Eigen::Vector3d(pts1[i].x, pts1[i].y, pts1[i].z));edge->setInformation(Eigen::Matrix3d::Identity());optimizer.addEdge(edge);}chrono::steady_clock::time_point t1 = chrono::steady_clock::now();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: " << unt() << " seconds." << endl;cout << endl << "after optimization:" << endl;cout << "T=n" << pose->estimate().matrix() << endl;// convert to cv::MatEigen::Matrix3d R_ = pose->estimate().rotationMatrix();Eigen::Vector3d t_ = pose->estimate().translation();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));
}到这里，我们已经介绍完了2D-2D,3D-2D(PNP),3D-3D问题的求解，对于前端的位姿估计想必也有了更深的理解，这里其实还有很多问题，在实际运行的机器人上还会存在各种各样的情况与误差，我在未来会继续讲解这一部分的问题和相关求解。
> 参考资料：视觉SLAM十四讲
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