GlobalOptimization.cpp

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// ----------------------------------------------------------------------------
// -                        CloudViewer: www.cloudViewer.org                  -
// ----------------------------------------------------------------------------
// Copyright (c) 2018-2024 www.cloudViewer.org
// SPDX-License-Identifier: MIT
// ----------------------------------------------------------------------------
 
#include "pipelines/registration/GlobalOptimization.h"
 
#include <Eigen.h>
#include <Logging.h>
#include <Timer.h>
 
#include <Eigen/Dense>
#include <Eigen/Sparse>
#include <tuple>
#include <vector>
 
#include "pipelines/registration/GlobalOptimizationConvergenceCriteria.h"
#include "pipelines/registration/GlobalOptimizationMethod.h"
#include "pipelines/registration/PoseGraph.h"
 
namespace cloudViewer {
namespace pipelines {
namespace registration {
 
 
// clang-format off
const std::vector<Eigen::Matrix4d, utility::Matrix4d_allocator>
        jacobian_operator = {
                // for alpha
                (Eigen::Matrix4d() << 0, 0, 0, 0,
                                      0, 0, -1, 0,
                                      0, 1, 0, 0,
                                      0, 0, 0, 0).finished(),
                // for beta
                (Eigen::Matrix4d() << 0, 0, 1, 0,
                                      0, 0, 0, 0,
                                      -1, 0, 0, 0,
                                      0, 0, 0, 0).finished(),
                // for gamma
                (Eigen::Matrix4d() << 0, -1, 0, 0,
                                      1, 0, 0, 0,
                                      0, 0, 0, 0,
                                      0, 0, 0, 0).finished(),
                // for a
                (Eigen::Matrix4d() << 0, 0, 0, 1,
                                      0, 0, 0, 0,
                                      0, 0, 0, 0,
                                      0, 0, 0, 0).finished(),
                // for b
                (Eigen::Matrix4d() << 0, 0, 0, 0,
                                      0, 0, 0, 1,
                                      0, 0, 0, 0,
                                      0, 0, 0, 0).finished(),
                // for c
                (Eigen::Matrix4d() << 0, 0, 0, 0,
                                      0, 0, 0, 0,
                                      0, 0, 0, 1,
                                      0, 0, 0, 0).finished()};
// clang-format on
 
inline Eigen::Vector6d GetLinearized6DVector(const Eigen::Matrix4d &input) {
    Eigen::Vector6d output;
    output(0) = (-input(1, 2) + input(2, 1)) / 2.0;
    output(1) = (-input(2, 0) + input(0, 2)) / 2.0;
    output(2) = (-input(0, 1) + input(1, 0)) / 2.0;
    output.block<3, 1>(3, 0) = input.block<3, 1>(0, 3);
    return output;
}
 
inline Eigen::Vector6d GetMisalignmentVector(const Eigen::Matrix4d &X_inv,
                                             const Eigen::Matrix4d &Ts,
                                             const Eigen::Matrix4d &Tt_inv) {
    Eigen::Matrix4d temp;
    temp.noalias() = X_inv * Tt_inv * Ts;
    return GetLinearized6DVector(temp);
}
 
inline std::tuple<Eigen::Matrix4d, Eigen::Matrix4d, Eigen::Matrix4d>
GetRelativePoses(const PoseGraph &pose_graph, int edge_id) {
    const PoseGraphEdge &te = pose_graph.edges_[edge_id];
    const PoseGraphNode &ts = pose_graph.nodes_[te.source_node_id_];
    const PoseGraphNode &tt = pose_graph.nodes_[te.target_node_id_];
    Eigen::Matrix4d X_inv = te.transformation_.inverse();
    Eigen::Matrix4d Ts = ts.pose_;
    Eigen::Matrix4d Tt_inv = tt.pose_.inverse();
    return std::make_tuple(std::move(X_inv), std::move(Ts), std::move(Tt_inv));
}
 
std::tuple<Eigen::Matrix6d, Eigen::Matrix6d> GetJacobian(
        const Eigen::Matrix4d &X_inv,
        const Eigen::Matrix4d &Ts,
        const Eigen::Matrix4d &Tt_inv) {
    Eigen::Matrix6d Js = Eigen::Matrix6d::Zero();
    for (int i = 0; i < 6; i++) {
        Eigen::Matrix4d temp = X_inv * Tt_inv * jacobian_operator[i] * Ts;
        Js.block<6, 1>(0, i) = GetLinearized6DVector(temp);
    }
    Eigen::Matrix6d Jt = Eigen::Matrix6d::Zero();
    for (int i = 0; i < 6; i++) {
        Eigen::Matrix4d temp = X_inv * Tt_inv * -jacobian_operator[i] * Ts;
        Jt.block<6, 1>(0, i) = GetLinearized6DVector(temp);
    }
    return std::make_tuple(std::move(Js), std::move(Jt));
}
 
int UpdateConfidence(PoseGraph &pose_graph,
                     const Eigen::VectorXd &zeta,
                     const double line_process_weight,
                     const GlobalOptimizationOption &option) {
    int n_edges = (int)pose_graph.edges_.size();
    int valid_edges_num = 0;
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        PoseGraphEdge &t = pose_graph.edges_[iter_edge];
        if (t.uncertain_) {
            Eigen::Vector6d e = zeta.block<6, 1>(iter_edge * 6, 0);
            double residual_square = e.transpose() * t.information_ * e;
            double temp = line_process_weight /
                          (line_process_weight + residual_square);
            double temp2 = temp * temp;
            t.confidence_ = temp2;
            if (temp2 > option.edge_prune_threshold_) valid_edges_num++;
        }
    }
    return valid_edges_num;
}
 
double ComputeResidual(const PoseGraph &pose_graph,
                       const Eigen::VectorXd &zeta,
                       const double line_process_weight,
                       const GlobalOptimizationOption &option) {
    int n_edges = (int)pose_graph.edges_.size();
    double residual = 0.0;
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        const PoseGraphEdge &te = pose_graph.edges_[iter_edge];
        double line_process_iter = te.confidence_;
        Eigen::Vector6d e = zeta.block<6, 1>(iter_edge * 6, 0);
        residual += line_process_iter * e.transpose() * te.information_ * e +
                    line_process_weight * pow(sqrt(line_process_iter) - 1, 2.0);
    }
    return residual;
}
 
Eigen::VectorXd ComputeZeta(const PoseGraph &pose_graph) {
    int n_edges = (int)pose_graph.edges_.size();
    Eigen::VectorXd output(n_edges * 6);
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        Eigen::Matrix4d X_inv, Ts, Tt_inv;
        std::tie(X_inv, Ts, Tt_inv) = GetRelativePoses(pose_graph, iter_edge);
        Eigen::Vector6d e = GetMisalignmentVector(X_inv, Ts, Tt_inv);
        output.block<6, 1>(iter_edge * 6, 0) = e;
    }
    return output;
}
 
std::tuple<Eigen::MatrixXd, Eigen::VectorXd> ComputeLinearSystem(
        const PoseGraph &pose_graph, const Eigen::VectorXd &zeta) {
    int n_nodes = (int)pose_graph.nodes_.size();
    int n_edges = (int)pose_graph.edges_.size();
    Eigen::MatrixXd H(n_nodes * 6, n_nodes * 6);
    Eigen::VectorXd b(n_nodes * 6);
    H.setZero();
    b.setZero();
 
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        const PoseGraphEdge &t = pose_graph.edges_[iter_edge];
        Eigen::Vector6d e = zeta.block<6, 1>(iter_edge * 6, 0);
 
        Eigen::Matrix4d X_inv, Ts, Tt_inv;
        std::tie(X_inv, Ts, Tt_inv) = GetRelativePoses(pose_graph, iter_edge);
 
        Eigen::Matrix6d Js, Jt;
        std::tie(Js, Jt) = GetJacobian(X_inv, Ts, Tt_inv);
        Eigen::Matrix6d JsT_Info = Js.transpose() * t.information_;
        Eigen::Matrix6d JtT_Info = Jt.transpose() * t.information_;
        Eigen::Vector6d eT_Info = e.transpose() * t.information_;
        double line_process_iter = t.confidence_;
 
        int id_i = t.source_node_id_ * 6;
        int id_j = t.target_node_id_ * 6;
        H.block<6, 6>(id_i, id_i).noalias() +=
                line_process_iter * JsT_Info * Js;
        H.block<6, 6>(id_i, id_j).noalias() +=
                line_process_iter * JsT_Info * Jt;
        H.block<6, 6>(id_j, id_i).noalias() +=
                line_process_iter * JtT_Info * Js;
        H.block<6, 6>(id_j, id_j).noalias() +=
                line_process_iter * JtT_Info * Jt;
        b.block<6, 1>(id_i, 0).noalias() -=
                line_process_iter * eT_Info.transpose() * Js;
        b.block<6, 1>(id_j, 0).noalias() -=
                line_process_iter * eT_Info.transpose() * Jt;
    }
    return std::make_tuple(std::move(H), std::move(b));
}
 
Eigen::VectorXd UpdatePoseVector(const PoseGraph &pose_graph) {
    int n_nodes = (int)pose_graph.nodes_.size();
    Eigen::VectorXd output(n_nodes * 6);
    for (int iter_node = 0; iter_node < n_nodes; iter_node++) {
        Eigen::Vector6d output_iter = utility::TransformMatrix4dToVector6d(
                pose_graph.nodes_[iter_node].pose_);
        output.block<6, 1>(iter_node * 6, 0) = output_iter;
    }
    return output;
}
 
std::shared_ptr<PoseGraph> UpdatePoseGraph(const PoseGraph &pose_graph,
                                           const Eigen::VectorXd delta) {
    std::shared_ptr<PoseGraph> pose_graph_updated =
            std::make_shared<PoseGraph>();
    *pose_graph_updated = pose_graph;
    int n_nodes = (int)pose_graph.nodes_.size();
    for (int iter_node = 0; iter_node < n_nodes; iter_node++) {
        Eigen::Vector6d delta_iter = delta.block<6, 1>(iter_node * 6, 0);
        pose_graph_updated->nodes_[iter_node].pose_ =
                utility::TransformVector6dToMatrix4d(delta_iter) *
                pose_graph_updated->nodes_[iter_node].pose_;
    }
    return pose_graph_updated;
}
 
bool CheckRightTerm(const Eigen::VectorXd &right_term,
                    const GlobalOptimizationConvergenceCriteria &criteria) {
    if (right_term.maxCoeff() < criteria.min_right_term_) {
        utility::LogDebug("Maximum coefficient of right term < {:e}",
                          criteria.min_right_term_);
        return true;
    }
    return false;
}
 
bool CheckRelativeIncrement(
        const Eigen::VectorXd &delta,
        const Eigen::VectorXd &x,
        const GlobalOptimizationConvergenceCriteria &criteria) {
    if (delta.norm() < criteria.min_relative_increment_ *
                               (x.norm() + criteria.min_relative_increment_)) {
        utility::LogDebug("Delta.norm() < {:e} * (x.norm() + {:e})",
                          criteria.min_relative_increment_,
                          criteria.min_relative_increment_);
        return true;
    }
    return false;
}
 
bool CheckRelativeResidualIncrement(
        double current_residual,
        double new_residual,
        const GlobalOptimizationConvergenceCriteria &criteria) {
    if (current_residual - new_residual <
        criteria.min_relative_residual_increment_ * current_residual) {
        utility::LogDebug(
                "Current_residual - new_residual < {:e} * current_residual",
                criteria.min_relative_residual_increment_);
        return true;
    }
    return false;
}
 
bool CheckResidual(double residual,
                   const GlobalOptimizationConvergenceCriteria &criteria) {
    if (residual < criteria.min_residual_) {
        utility::LogDebug("Current_residual < {:e}", criteria.min_residual_);
        return true;
    }
    return false;
}
 
bool CheckMaxIteration(int iteration,
                       const GlobalOptimizationConvergenceCriteria &criteria) {
    if (iteration >= criteria.max_iteration_) {
        utility::LogDebug("Reached maximum number of iterations ({:d})",
                          criteria.max_iteration_);
        return true;
    }
    return false;
}
 
bool CheckMaxIterationLM(
        int iteration, const GlobalOptimizationConvergenceCriteria &criteria) {
    if (iteration >= criteria.max_iteration_lm_) {
        utility::LogDebug("Reached maximum number of iterations ({:d})",
                          criteria.max_iteration_lm_);
        return true;
    }
    return false;
}
 
double ComputeLineProcessWeight(const PoseGraph &pose_graph,
                                const GlobalOptimizationOption &option) {
    int n_edges = (int)pose_graph.edges_.size();
    double average_number_of_correspondences = 0.0;
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        double number_of_correspondences =
                pose_graph.edges_[iter_edge].information_(5, 5);
        average_number_of_correspondences += number_of_correspondences;
    }
    if (n_edges > 0) {
        // see Section 5 in [Choi et al 2015]
        average_number_of_correspondences /= (double)n_edges;
        double line_process_weight =
                option.preference_loop_closure_ *
                pow(option.max_correspondence_distance_, 2) *
                average_number_of_correspondences;
        return line_process_weight;
    } else {
        return 0.0;
    }
}
 
void CompensateReferencePoseGraphNode(PoseGraph &pose_graph_new,
                                      const PoseGraph &pose_graph_orig,
                                      int reference_node) {
    utility::LogDebug("CompensateReferencePoseGraphNode : reference : {:d}",
                      reference_node);
    int n_nodes = (int)pose_graph_new.nodes_.size();
    if (reference_node < 0 || reference_node >= n_nodes) {
        return;
    } else {
        Eigen::Matrix4d compensation =
                pose_graph_orig.nodes_[reference_node].pose_ *
                pose_graph_new.nodes_[reference_node].pose_.inverse();
        for (int i = 0; i < n_nodes; i++) {
            pose_graph_new.nodes_[i].pose_ =
                    compensation * pose_graph_new.nodes_[i].pose_;
        }
    }
}
 
bool ValidatePoseGraphConnectivity(const PoseGraph &pose_graph,
                                   bool ignore_uncertain_edges = false) {
    size_t n_nodes = pose_graph.nodes_.size();
    size_t n_edges = pose_graph.edges_.size();
 
    // Test if the connected component containing the first node is the entire
    // graph
    std::vector<int> nodes_to_explore{};
    std::vector<int> component{};
    if (n_nodes > 0) {
        nodes_to_explore.push_back(0);
        component.push_back(0);
    }
    while (!nodes_to_explore.empty()) {
        int i = nodes_to_explore.back();
        nodes_to_explore.pop_back();
        for (size_t j = 0; j < n_edges; j++) {
            const PoseGraphEdge &t = pose_graph.edges_[j];
            if (ignore_uncertain_edges && t.uncertain_) {
                continue;
            }
            int adjacent_node{-1};
            if (t.source_node_id_ == i) {
                adjacent_node = t.target_node_id_;
            } else if (t.target_node_id_ == i) {
                adjacent_node = t.source_node_id_;
            }
            if (adjacent_node != -1) {
                auto find_result = std::find(component.begin(), component.end(),
                                             adjacent_node);
                if (find_result == component.end()) {
                    nodes_to_explore.push_back(adjacent_node);
                    component.push_back(adjacent_node);
                }
            }
        }
    }
    return component.size() == n_nodes;
}
 
bool ValidatePoseGraph(const PoseGraph &pose_graph) {
    int n_nodes = (int)pose_graph.nodes_.size();
    int n_edges = (int)pose_graph.edges_.size();
 
    if (!ValidatePoseGraphConnectivity(pose_graph, false)) {
        utility::LogWarning("Invalid PoseGraph - graph is not connected.");
        return false;
    }
 
    if (!ValidatePoseGraphConnectivity(pose_graph, true)) {
        utility::LogWarning(
                "Certain-edge subset of PoseGraph is not connected.");
    }
 
    for (int j = 0; j < n_edges; j++) {
        bool valid = false;
        const PoseGraphEdge &t = pose_graph.edges_[j];
        if (t.source_node_id_ >= 0 && t.source_node_id_ < n_nodes &&
            t.target_node_id_ >= 0 && t.target_node_id_ < n_nodes)
            valid = true;
        if (!valid) {
            utility::LogWarning(
                    "Invalid PoseGraph - an edge references an invalide "
                    "node.");
            return false;
        }
    }
    for (int j = 0; j < n_edges; j++) {
        const PoseGraphEdge &t = pose_graph.edges_[j];
        if (!t.uncertain_ && t.confidence_ != 1.0) {
            utility::LogWarning(
                    "Invalid PoseGraph - the certain edge does not have 1.0 as "
                    "a confidence.");
            return false;
        }
    }
    utility::LogDebug("Validating PoseGraph - finished.");
    return true;
}
 
std::shared_ptr<PoseGraph> CreatePoseGraphWithoutInvalidEdges(
        const PoseGraph &pose_graph, const GlobalOptimizationOption &option) {
    std::shared_ptr<PoseGraph> pose_graph_pruned =
            std::make_shared<PoseGraph>();
 
    int n_nodes = (int)pose_graph.nodes_.size();
    for (int iter_node = 0; iter_node < n_nodes; iter_node++) {
        const PoseGraphNode &t = pose_graph.nodes_[iter_node];
        pose_graph_pruned->nodes_.push_back(t);
    }
    int n_edges = (int)pose_graph.edges_.size();
    for (int iter_edge = 0; iter_edge < n_edges; iter_edge++) {
        const PoseGraphEdge &t = pose_graph.edges_[iter_edge];
        if (t.uncertain_) {
            if (t.confidence_ > option.edge_prune_threshold_) {
                pose_graph_pruned->edges_.push_back(t);
            }
        } else {
            pose_graph_pruned->edges_.push_back(t);
        }
    }
    return pose_graph_pruned;
}
 
void GlobalOptimizationGaussNewton::OptimizePoseGraph(
        PoseGraph &pose_graph,
        const GlobalOptimizationConvergenceCriteria &criteria,
        const GlobalOptimizationOption &option) const {
    int n_nodes = (int)pose_graph.nodes_.size();
    int n_edges = (int)pose_graph.edges_.size();
    double line_process_weight = ComputeLineProcessWeight(pose_graph, option);
 
    utility::LogDebug(
            "[GlobalOptimizationGaussNewton] Optimizing PoseGraph having {:d} "
            "nodes and {:d} edges.",
            n_nodes, n_edges);
    utility::LogDebug("Line process weight : {:f}", line_process_weight);
 
    Eigen::VectorXd zeta = ComputeZeta(pose_graph);
    double current_residual, new_residual;
    new_residual =
            ComputeResidual(pose_graph, zeta, line_process_weight, option);
    current_residual = new_residual;
 
    int valid_edges_num;
    valid_edges_num =
            UpdateConfidence(pose_graph, zeta, line_process_weight, option);
 
    Eigen::MatrixXd H;
    Eigen::VectorXd b;
    Eigen::VectorXd x = UpdatePoseVector(pose_graph);
 
    std::tie(H, b) = ComputeLinearSystem(pose_graph, zeta);
 
    utility::LogDebug("[Initial     ] residual : {:e}", current_residual);
 
    bool stop = false;
    if (CheckRightTerm(b, criteria)) return;
 
    utility::Timer timer_overall;
    timer_overall.Start();
    int iter;
    for (iter = 0; !stop; iter++) {
        utility::Timer timer_iter;
        timer_iter.Start();
 
        Eigen::VectorXd delta(H.cols());
        bool solver_success = false;
 
        // Solve H_LM @ delta == b using a sparse solver
        std::tie(solver_success, delta) = utility::SolveLinearSystemPSD(
                H, b, /*prefer_sparse=*/true, /*check_symmetric=*/false,
                /*check_det=*/false, /*check_psd=*/false);
 
        stop = stop || CheckRelativeIncrement(delta, x, criteria);
        if (stop) {
            break;
        } else {
            std::shared_ptr<PoseGraph> pose_graph_new =
                    UpdatePoseGraph(pose_graph, delta);
 
            Eigen::VectorXd zeta_new;
            zeta_new = ComputeZeta(*pose_graph_new);
            new_residual = ComputeResidual(pose_graph, zeta_new,
                                           line_process_weight, option);
            stop = stop || CheckRelativeResidualIncrement(
                                   current_residual, new_residual, criteria);
            if (stop) break;
            current_residual = new_residual;
 
            zeta = zeta_new;
            pose_graph = *pose_graph_new;
            x = UpdatePoseVector(pose_graph);
            valid_edges_num = UpdateConfidence(pose_graph, zeta,
                                               line_process_weight, option);
            std::tie(H, b) = ComputeLinearSystem(pose_graph, zeta);
 
            stop = stop || CheckRightTerm(b, criteria);
            if (stop) break;
        }
        timer_iter.Stop();
        utility::LogDebug(
                "[Iteration {:02d}] residual : {:e}, valid edges : {:d}, time "
                ": {:.3f} "
                "sec.",
                iter, current_residual, valid_edges_num,
                timer_iter.GetDurationInMillisecond() / 1000.0);
        stop = stop || CheckResidual(current_residual, criteria) ||
               CheckMaxIteration(iter, criteria);
    }  // end for
    timer_overall.Stop();
    utility::LogDebug(
            "[GlobalOptimizationGaussNewton] total time : {:.3f} sec.",
            timer_overall.GetDurationInMillisecond() / 1000.0);
}
 
void GlobalOptimizationLevenbergMarquardt::OptimizePoseGraph(
        PoseGraph &pose_graph,
        const GlobalOptimizationConvergenceCriteria &criteria,
        const GlobalOptimizationOption &option) const {
    int n_nodes = (int)pose_graph.nodes_.size();
    int n_edges = (int)pose_graph.edges_.size();
    double line_process_weight = ComputeLineProcessWeight(pose_graph, option);
 
    utility::LogDebug(
            "[GlobalOptimizationLM] Optimizing PoseGraph having {:d} nodes and "
            "{:d} edges.",
            n_nodes, n_edges);
    utility::LogDebug("Line process weight : {:f}", line_process_weight);
 
    Eigen::VectorXd zeta = ComputeZeta(pose_graph);
    double current_residual, new_residual;
    new_residual =
            ComputeResidual(pose_graph, zeta, line_process_weight, option);
    current_residual = new_residual;
 
    int valid_edges_num =
            UpdateConfidence(pose_graph, zeta, line_process_weight, option);
 
    Eigen::MatrixXd H_I = Eigen::MatrixXd::Identity(n_nodes * 6, n_nodes * 6);
    Eigen::MatrixXd H;
    Eigen::VectorXd b;
    Eigen::VectorXd x = UpdatePoseVector(pose_graph);
 
    std::tie(H, b) = ComputeLinearSystem(pose_graph, zeta);
 
    Eigen::VectorXd H_diag = H.diagonal();
    double tau = 1e-5;
    double current_lambda = tau * H_diag.maxCoeff();
    double ni = 2.0;
    double rho = 0.0;
 
    utility::LogDebug("[Initial     ] residual : {:e}, lambda : {:e}",
                      current_residual, current_lambda);
 
    bool stop = false;
    stop = stop || CheckRightTerm(b, criteria);
    if (stop) return;
 
    utility::Timer timer_overall;
    timer_overall.Start();
    for (int iter = 0; !stop; iter++) {
        utility::Timer timer_iter;
        timer_iter.Start();
        int lm_count = 0;
        do {
            Eigen::MatrixXd H_LM = H + current_lambda * H_I;
            Eigen::VectorXd delta(H_LM.cols());
            bool solver_success = false;
 
            // Solve H_LM @ delta == b using a sparse solver
            std::tie(solver_success, delta) = utility::SolveLinearSystemPSD(
                    H_LM, b, /*prefer_sparse=*/true, /*check_symmetric=*/false,
                    /*check_det=*/false, /*check_psd=*/false);
 
            stop = stop || CheckRelativeIncrement(delta, x, criteria);
            if (!stop) {
                std::shared_ptr<PoseGraph> pose_graph_new =
                        UpdatePoseGraph(pose_graph, delta);
 
                Eigen::VectorXd zeta_new;
                zeta_new = ComputeZeta(*pose_graph_new);
                new_residual = ComputeResidual(pose_graph, zeta_new,
                                               line_process_weight, option);
                rho = (current_residual - new_residual) /
                      (delta.dot(current_lambda * delta + b) + 1e-3);
                if (rho > 0) {
                    stop = stop ||
                           CheckRelativeResidualIncrement(
                                   current_residual, new_residual, criteria);
                    if (stop) break;
                    double alpha = 1. - pow((2 * rho - 1), 3);
                    alpha = (std::min)(alpha, criteria.upper_scale_factor_);
                    double scaleFactor =
                            (std::max)(criteria.lower_scale_factor_, alpha);
                    current_lambda *= scaleFactor;
                    ni = 2;
                    current_residual = new_residual;
 
                    zeta = zeta_new;
                    pose_graph = *pose_graph_new;
                    x = UpdatePoseVector(pose_graph);
                    valid_edges_num = UpdateConfidence(
                            pose_graph, zeta, line_process_weight, option);
                    std::tie(H, b) = ComputeLinearSystem(pose_graph, zeta);
 
                    stop = stop || CheckRightTerm(b, criteria);
                    if (stop) break;
                } else {
                    current_lambda *= ni;
                    ni *= 2;
                }
            }
            lm_count++;
            stop = stop || CheckMaxIterationLM(lm_count, criteria);
        } while (!((rho > 0) || stop));
        timer_iter.Stop();
        if (!stop) {
            utility::LogDebug(
                    "[Iteration {:02d}] residual : {:e}, valid edges : {:d}, "
                    "time : "
                    "{:.3f} sec.",
                    iter, current_residual, valid_edges_num,
                    timer_iter.GetDurationInMillisecond() / 1000.0);
        }
        stop = stop || CheckResidual(current_residual, criteria) ||
               CheckMaxIteration(iter, criteria);
    }  // end for
    timer_overall.Stop();
    utility::LogDebug("[GlobalOptimizationLM] total time : {:.3f} sec.",
                      timer_overall.GetDurationInMillisecond() / 1000.0);
}
 
void GlobalOptimization(PoseGraph &pose_graph,
                        const GlobalOptimizationMethod &method
                        /* = GlobalOptimizationLevenbergMarquardt() */,
                        const GlobalOptimizationConvergenceCriteria &criteria
                        /* = GlobalOptimizationConvergenceCriteria() */,
                        const GlobalOptimizationOption &option
                        /* = GlobalOptimizationOption() */) {
    if (!ValidatePoseGraph(pose_graph)) return;
    std::shared_ptr<PoseGraph> pose_graph_pre = std::make_shared<PoseGraph>();
    *pose_graph_pre = pose_graph;
    method.OptimizePoseGraph(*pose_graph_pre, criteria, option);
    auto pose_graph_pre_pruned =
            CreatePoseGraphWithoutInvalidEdges(*pose_graph_pre, option);
    method.OptimizePoseGraph(*pose_graph_pre_pruned, criteria, option);
    auto pose_graph_pre_pruned_2 =
            CreatePoseGraphWithoutInvalidEdges(*pose_graph_pre_pruned, option);
    CompensateReferencePoseGraphNode(*pose_graph_pre_pruned_2, pose_graph,
                                     option.reference_node_);
    pose_graph = *pose_graph_pre_pruned_2;
}
 
}  // namespace registration
}  // namespace pipelines
}  // namespace cloudViewer