ndt.hpp

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#ifndef PCL_REGISTRATION_NDT_IMPL_H_
#define PCL_REGISTRATION_NDT_IMPL_H_
 
namespace pcl {
 
template <typename PointSource, typename PointTarget>
NormalDistributionsTransform<PointSource, PointTarget>::NormalDistributionsTransform()
: target_cells_()
, resolution_(1.0f)
, step_size_(0.1)
, outlier_ratio_(0.55)
, gauss_d1_()
, gauss_d2_()
, trans_probability_()
{
  reg_name_ = "NormalDistributionsTransform";
 
  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
  const double gauss_c1 = 10.0 * (1 - outlier_ratio_);
  const double gauss_c2 = outlier_ratio_ / pow(resolution_, 3);
  const double gauss_d3 = -std::log(gauss_c2);
  gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
  gauss_d2_ =
      -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
                    gauss_d1_);
 
  transformation_epsilon_ = 0.1;
  max_iterations_ = 35;
}
 
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform<PointSource, PointTarget>::computeTransformation(
    PointCloudSource& output, const Eigen::Matrix4f& guess)
{
  nr_iterations_ = 0;
  converged_ = false;
 
  // Initializes the gaussian fitting parameters (eq. 6.8) [Magnusson 2009]
  const double gauss_c1 = 10 * (1 - outlier_ratio_);
  const double gauss_c2 = outlier_ratio_ / pow(resolution_, 3);
  const double gauss_d3 = -std::log(gauss_c2);
  gauss_d1_ = -std::log(gauss_c1 + gauss_c2) - gauss_d3;
  gauss_d2_ =
      -2 * std::log((-std::log(gauss_c1 * std::exp(-0.5) + gauss_c2) - gauss_d3) /
                    gauss_d1_);
 
  if (guess != Eigen::Matrix4f::Identity()) {
    // Initialise final transformation to the guessed one
    final_transformation_ = guess;
    // Apply guessed transformation prior to search for neighbours
    transformPointCloud(output, output, guess);
  }
 
  // Initialize Point Gradient and Hessian
  point_jacobian_.setZero();
  point_jacobian_.block<3, 3>(0, 0).setIdentity();
  point_hessian_.setZero();
 
  Eigen::Transform<float, 3, Eigen::Affine, Eigen::ColMajor> eig_transformation;
  eig_transformation.matrix() = final_transformation_;
 
  // Convert initial guess matrix to 6 element transformation vector
  Eigen::Matrix<double, 6, 1> transform, score_gradient;
  Eigen::Vector3f init_translation = eig_transformation.translation();
  Eigen::Vector3f init_rotation = eig_transformation.rotation().eulerAngles(0, 1, 2);
  transform << init_translation.cast<double>(), init_rotation.cast<double>();
 
  Eigen::Matrix<double, 6, 6> hessian;
 
  // Calculate derivates of initial transform vector, subsequent derivative calculations
  // are done in the step length determination.
  double score = computeDerivatives(score_gradient, hessian, output, transform);
 
  while (!converged_) {
    // Store previous transformation
    previous_transformation_ = transformation_;
 
    // Solve for decent direction using newton method, line 23 in Algorithm 2 [Magnusson
    // 2009]
    Eigen::JacobiSVD<Eigen::Matrix<double, 6, 6>> sv(
        hessian, Eigen::ComputeFullU | Eigen::ComputeFullV);
    // Negative for maximization as opposed to minimization
    Eigen::Matrix<double, 6, 1> delta = sv.solve(-score_gradient);
 
    // Calculate step length with guarnteed sufficient decrease [More, Thuente 1994]
    double delta_norm = delta.norm();
 
    if (delta_norm == 0 || std::isnan(delta_norm)) {
      trans_probability_ = score / static_cast<double>(input_->size());
      converged_ = delta_norm == 0;
      return;
    }
 
    delta /= delta_norm;
    delta_norm = computeStepLengthMT(transform,
                                     delta,
                                     delta_norm,
                                     step_size_,
                                     transformation_epsilon_ / 2,
                                     score,
                                     score_gradient,
                                     hessian,
                                     output);
    delta *= delta_norm;
 
    // Convert delta into matrix form
    convertTransform(delta, transformation_);
 
    transform += delta;
 
    // Update Visualizer (untested)
    if (update_visualizer_)
      update_visualizer_(output, pcl::Indices(), *target_, pcl::Indices());
 
    const double cos_angle =
        0.5 * (transformation_.template block<3, 3>(0, 0).trace() - 1);
    const double translation_sqr =
        transformation_.template block<3, 1>(0, 3).squaredNorm();
 
    nr_iterations_++;
 
    if (nr_iterations_ >= max_iterations_ ||
        ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
         (transformation_rotation_epsilon_ > 0 &&
          cos_angle >= transformation_rotation_epsilon_)) ||
        ((transformation_epsilon_ <= 0) &&
         (transformation_rotation_epsilon_ > 0 &&
          cos_angle >= transformation_rotation_epsilon_)) ||
        ((transformation_epsilon_ > 0 && translation_sqr <= transformation_epsilon_) &&
         (transformation_rotation_epsilon_ <= 0))) {
      converged_ = true;
    }
  }
 
  // Store transformation probability.  The realtive differences within each scan
  // registration are accurate but the normalization constants need to be modified for
  // it to be globally accurate
  trans_probability_ = score / static_cast<double>(input_->size());
}
 
template <typename PointSource, typename PointTarget>
double
NormalDistributionsTransform<PointSource, PointTarget>::computeDerivatives(
    Eigen::Matrix<double, 6, 1>& score_gradient,
    Eigen::Matrix<double, 6, 6>& hessian,
    const PointCloudSource& trans_cloud,
    const Eigen::Matrix<double, 6, 1>& transform,
    bool compute_hessian)
{
  score_gradient.setZero();
  hessian.setZero();
  double score = 0;
 
  // Precompute Angular Derivatives (eq. 6.19 and 6.21)[Magnusson 2009]
  computeAngleDerivatives(transform);
 
  // Update gradient and hessian for each point, line 17 in Algorithm 2 [Magnusson 2009]
  for (std::size_t idx = 0; idx < input_->size(); idx++) {
    // Transformed Point
    const auto& x_trans_pt = trans_cloud[idx];
 
    // Find neighbors (Radius search has been experimentally faster than direct neighbor
    // checking.
    std::vector<TargetGridLeafConstPtr> neighborhood;
    std::vector<float> distances;
    target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
 
    for (const auto& cell : neighborhood) {
      // Original Point
      const auto& x_pt = (*input_)[idx];
      const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
 
      // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
      const Eigen::Vector3d x_trans =
          x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
      // Inverse Covariance of Occupied Voxel
      // Uses precomputed covariance for speed.
      const Eigen::Matrix3d c_inv = cell->getInverseCov();
 
      // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
      // in Equations 6.18 and 6.20 [Magnusson 2009]
      computePointDerivatives(x);
      // Update score, gradient and hessian, lines 19-21 in Algorithm 2, according to
      // Equations 6.10, 6.12 and 6.13, respectively [Magnusson 2009]
      score +=
          updateDerivatives(score_gradient, hessian, x_trans, c_inv, compute_hessian);
    }
  }
  return score;
}
 
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform<PointSource, PointTarget>::computeAngleDerivatives(
    const Eigen::Matrix<double, 6, 1>& transform, bool compute_hessian)
{
  // Simplified math for near 0 angles
  const auto calculate_cos_sin = [](double angle, double& c, double& s) {
    if (std::abs(angle) < 10e-5) {
      c = 1.0;
      s = 0.0;
    }
    else {
      c = std::cos(angle);
      s = std::sin(angle);
    }
  };
 
  double cx, cy, cz, sx, sy, sz;
  calculate_cos_sin(transform(3), cx, sx);
  calculate_cos_sin(transform(4), cy, sy);
  calculate_cos_sin(transform(5), cz, sz);
 
  // Precomputed angular gradient components. Letters correspond to Equation 6.19
  // [Magnusson 2009]
  angular_jacobian_.setZero();
  angular_jacobian_.row(0).noalias() = Eigen::Vector4d(
      (-sx * sz + cx * sy * cz), (-sx * cz - cx * sy * sz), (-cx * cy), 1.0); // a
  angular_jacobian_.row(1).noalias() = Eigen::Vector4d(
      (cx * sz + sx * sy * cz), (cx * cz - sx * sy * sz), (-sx * cy), 1.0); // b
  angular_jacobian_.row(2).noalias() =
      Eigen::Vector4d((-sy * cz), sy * sz, cy, 1.0); // c
  angular_jacobian_.row(3).noalias() =
      Eigen::Vector4d(sx * cy * cz, (-sx * cy * sz), sx * sy, 1.0); // d
  angular_jacobian_.row(4).noalias() =
      Eigen::Vector4d((-cx * cy * cz), cx * cy * sz, (-cx * sy), 1.0); // e
  angular_jacobian_.row(5).noalias() =
      Eigen::Vector4d((-cy * sz), (-cy * cz), 0, 1.0); // f
  angular_jacobian_.row(6).noalias() =
      Eigen::Vector4d((cx * cz - sx * sy * sz), (-cx * sz - sx * sy * cz), 0, 1.0); // g
  angular_jacobian_.row(7).noalias() =
      Eigen::Vector4d((sx * cz + cx * sy * sz), (cx * sy * cz - sx * sz), 0, 1.0); // h
 
  if (compute_hessian) {
    // Precomputed angular hessian components. Letters correspond to Equation 6.21 and
    // numbers correspond to row index [Magnusson 2009]
    angular_hessian_.setZero();
    angular_hessian_.row(0).noalias() = Eigen::Vector4d(
        (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), sx * cy, 0.0f); // a2
    angular_hessian_.row(1).noalias() = Eigen::Vector4d(
        (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), (-cx * cy), 0.0f); // a3
 
    angular_hessian_.row(2).noalias() =
        Eigen::Vector4d((cx * cy * cz), (-cx * cy * sz), (cx * sy), 0.0f); // b2
    angular_hessian_.row(3).noalias() =
        Eigen::Vector4d((sx * cy * cz), (-sx * cy * sz), (sx * sy), 0.0f); // b3
 
    // The sign of 'sx * sz' in c2 is incorrect in the thesis, and is fixed here.
    angular_hessian_.row(4).noalias() = Eigen::Vector4d(
        (-sx * cz - cx * sy * sz), (sx * sz - cx * sy * cz), 0, 0.0f); // c2
    angular_hessian_.row(5).noalias() = Eigen::Vector4d(
        (cx * cz - sx * sy * sz), (-sx * sy * cz - cx * sz), 0, 0.0f); // c3
 
    angular_hessian_.row(6).noalias() =
        Eigen::Vector4d((-cy * cz), (cy * sz), (-sy), 0.0f); // d1
    angular_hessian_.row(7).noalias() =
        Eigen::Vector4d((-sx * sy * cz), (sx * sy * sz), (sx * cy), 0.0f); // d2
    angular_hessian_.row(8).noalias() =
        Eigen::Vector4d((cx * sy * cz), (-cx * sy * sz), (-cx * cy), 0.0f); // d3
 
    angular_hessian_.row(9).noalias() =
        Eigen::Vector4d((sy * sz), (sy * cz), 0, 0.0f); // e1
    angular_hessian_.row(10).noalias() =
        Eigen::Vector4d((-sx * cy * sz), (-sx * cy * cz), 0, 0.0f); // e2
    angular_hessian_.row(11).noalias() =
        Eigen::Vector4d((cx * cy * sz), (cx * cy * cz), 0, 0.0f); // e3
 
    angular_hessian_.row(12).noalias() =
        Eigen::Vector4d((-cy * cz), (cy * sz), 0, 0.0f); // f1
    angular_hessian_.row(13).noalias() = Eigen::Vector4d(
        (-cx * sz - sx * sy * cz), (-cx * cz + sx * sy * sz), 0, 0.0f); // f2
    angular_hessian_.row(14).noalias() = Eigen::Vector4d(
        (-sx * sz + cx * sy * cz), (-cx * sy * sz - sx * cz), 0, 0.0f); // f3
  }
}
 
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform<PointSource, PointTarget>::computePointDerivatives(
    const Eigen::Vector3d& x, bool compute_hessian)
{
  // Calculate first derivative of Transformation Equation 6.17 w.r.t. transform vector.
  // Derivative w.r.t. ith element of transform vector corresponds to column i,
  // Equation 6.18 and 6.19 [Magnusson 2009]
  Eigen::Matrix<double, 8, 1> point_angular_jacobian =
      angular_jacobian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
  point_jacobian_(1, 3) = point_angular_jacobian[0];
  point_jacobian_(2, 3) = point_angular_jacobian[1];
  point_jacobian_(0, 4) = point_angular_jacobian[2];
  point_jacobian_(1, 4) = point_angular_jacobian[3];
  point_jacobian_(2, 4) = point_angular_jacobian[4];
  point_jacobian_(0, 5) = point_angular_jacobian[5];
  point_jacobian_(1, 5) = point_angular_jacobian[6];
  point_jacobian_(2, 5) = point_angular_jacobian[7];
 
  if (compute_hessian) {
    Eigen::Matrix<double, 15, 1> point_angular_hessian =
        angular_hessian_ * Eigen::Vector4d(x[0], x[1], x[2], 0.0);
 
    // Vectors from Equation 6.21 [Magnusson 2009]
    const Eigen::Vector3d a(0, point_angular_hessian[0], point_angular_hessian[1]);
    const Eigen::Vector3d b(0, point_angular_hessian[2], point_angular_hessian[3]);
    const Eigen::Vector3d c(0, point_angular_hessian[4], point_angular_hessian[5]);
    const Eigen::Vector3d d = point_angular_hessian.block<3, 1>(6, 0);
    const Eigen::Vector3d e = point_angular_hessian.block<3, 1>(9, 0);
    const Eigen::Vector3d f = point_angular_hessian.block<3, 1>(12, 0);
 
    // Calculate second derivative of Transformation Equation 6.17 w.r.t. transform
    // vector. Derivative w.r.t. ith and jth elements of transform vector corresponds to
    // the 3x1 block matrix starting at (3i,j), Equation 6.20 and 6.21 [Magnusson 2009]
    point_hessian_.block<3, 1>(9, 3) = a;
    point_hessian_.block<3, 1>(12, 3) = b;
    point_hessian_.block<3, 1>(15, 3) = c;
    point_hessian_.block<3, 1>(9, 4) = b;
    point_hessian_.block<3, 1>(12, 4) = d;
    point_hessian_.block<3, 1>(15, 4) = e;
    point_hessian_.block<3, 1>(9, 5) = c;
    point_hessian_.block<3, 1>(12, 5) = e;
    point_hessian_.block<3, 1>(15, 5) = f;
  }
}
 
template <typename PointSource, typename PointTarget>
double
NormalDistributionsTransform<PointSource, PointTarget>::updateDerivatives(
    Eigen::Matrix<double, 6, 1>& score_gradient,
    Eigen::Matrix<double, 6, 6>& hessian,
    const Eigen::Vector3d& x_trans,
    const Eigen::Matrix3d& c_inv,
    bool compute_hessian) const
{
  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
  double e_x_cov_x = std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
  // Calculate probability of transformed points existence, Equation 6.9 [Magnusson
  // 2009]
  const double score_inc = -gauss_d1_ * e_x_cov_x;
 
  e_x_cov_x = gauss_d2_ * e_x_cov_x;
 
  // Error checking for invalid values.
  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
    return 0;
  }
 
  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
  e_x_cov_x *= gauss_d1_;
 
  for (int i = 0; i < 6; i++) {
    // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
    // 2009]
    const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
 
    // Update gradient, Equation 6.12 [Magnusson 2009]
    score_gradient(i) += x_trans.dot(cov_dxd_pi) * e_x_cov_x;
 
    if (compute_hessian) {
      for (Eigen::Index j = 0; j < hessian.cols(); j++) {
        // Update hessian, Equation 6.13 [Magnusson 2009]
        hessian(i, j) +=
            e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
                             x_trans.dot(c_inv * point_jacobian_.col(j)) +
                         x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
                         point_jacobian_.col(j).dot(cov_dxd_pi));
      }
    }
  }
 
  return score_inc;
}
 
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform<PointSource, PointTarget>::computeHessian(
    Eigen::Matrix<double, 6, 6>& hessian, const PointCloudSource& trans_cloud)
{
  hessian.setZero();
 
  // Precompute Angular Derivatives unessisary because only used after regular
  // derivative calculation Update hessian for each point, line 17 in Algorithm 2
  // [Magnusson 2009]
  for (std::size_t idx = 0; idx < input_->size(); idx++) {
    // Transformed Point
    const auto& x_trans_pt = trans_cloud[idx];
 
    // Find nieghbors (Radius search has been experimentally faster than direct neighbor
    // checking.
    std::vector<TargetGridLeafConstPtr> neighborhood;
    std::vector<float> distances;
    target_cells_.radiusSearch(x_trans_pt, resolution_, neighborhood, distances);
 
    for (const auto& cell : neighborhood) {
      // Original Point
      const auto& x_pt = (*input_)[idx];
      const Eigen::Vector3d x = x_pt.getVector3fMap().template cast<double>();
 
      // Denorm point, x_k' in Equations 6.12 and 6.13 [Magnusson 2009]
      const Eigen::Vector3d x_trans =
          x_trans_pt.getVector3fMap().template cast<double>() - cell->getMean();
      // Inverse Covariance of Occupied Voxel
      // Uses precomputed covariance for speed.
      const Eigen::Matrix3d c_inv = cell->getInverseCov();
 
      // Compute derivative of transform function w.r.t. transform vector, J_E and H_E
      // in Equations 6.18 and 6.20 [Magnusson 2009]
      computePointDerivatives(x);
      // Update hessian, lines 21 in Algorithm 2, according to Equations 6.10, 6.12
      // and 6.13, respectively [Magnusson 2009]
      updateHessian(hessian, x_trans, c_inv);
    }
  }
}
 
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform<PointSource, PointTarget>::updateHessian(
    Eigen::Matrix<double, 6, 6>& hessian,
    const Eigen::Vector3d& x_trans,
    const Eigen::Matrix3d& c_inv) const
{
  // e^(-d_2/2 * (x_k - mu_k)^T Sigma_k^-1 (x_k - mu_k)) Equation 6.9 [Magnusson 2009]
  double e_x_cov_x =
      gauss_d2_ * std::exp(-gauss_d2_ * x_trans.dot(c_inv * x_trans) / 2);
 
  // Error checking for invalid values.
  if (e_x_cov_x > 1 || e_x_cov_x < 0 || std::isnan(e_x_cov_x)) {
    return;
  }
 
  // Reusable portion of Equation 6.12 and 6.13 [Magnusson 2009]
  e_x_cov_x *= gauss_d1_;
 
  for (int i = 0; i < 6; i++) {
    // Sigma_k^-1 d(T(x,p))/dpi, Reusable portion of Equation 6.12 and 6.13 [Magnusson
    // 2009]
    const Eigen::Vector3d cov_dxd_pi = c_inv * point_jacobian_.col(i);
 
    for (Eigen::Index j = 0; j < hessian.cols(); j++) {
      // Update hessian, Equation 6.13 [Magnusson 2009]
      hessian(i, j) +=
          e_x_cov_x * (-gauss_d2_ * x_trans.dot(cov_dxd_pi) *
                           x_trans.dot(c_inv * point_jacobian_.col(j)) +
                       x_trans.dot(c_inv * point_hessian_.block<3, 1>(3 * i, j)) +
                       point_jacobian_.col(j).dot(cov_dxd_pi));
    }
  }
}
 
template <typename PointSource, typename PointTarget>
bool
NormalDistributionsTransform<PointSource, PointTarget>::updateIntervalMT(
    double& a_l,
    double& f_l,
    double& g_l,
    double& a_u,
    double& f_u,
    double& g_u,
    double a_t,
    double f_t,
    double g_t) const
{
  // Case U1 in Update Algorithm and Case a in Modified Update Algorithm [More, Thuente
  // 1994]
  if (f_t > f_l) {
    a_u = a_t;
    f_u = f_t;
    g_u = g_t;
    return false;
  }
  // Case U2 in Update Algorithm and Case b in Modified Update Algorithm [More, Thuente
  // 1994]
  if (g_t * (a_l - a_t) > 0) {
    a_l = a_t;
    f_l = f_t;
    g_l = g_t;
    return false;
  }
  // Case U3 in Update Algorithm and Case c in Modified Update Algorithm [More, Thuente
  // 1994]
  if (g_t * (a_l - a_t) < 0) {
    a_u = a_l;
    f_u = f_l;
    g_u = g_l;
 
    a_l = a_t;
    f_l = f_t;
    g_l = g_t;
    return false;
  }
  // Interval Converged
  return true;
}
 
template <typename PointSource, typename PointTarget>
double
NormalDistributionsTransform<PointSource, PointTarget>::trialValueSelectionMT(
    double a_l,
    double f_l,
    double g_l,
    double a_u,
    double f_u,
    double g_u,
    double a_t,
    double f_t,
    double g_t) const
{
  if (a_t == a_l && a_t == a_u) {
    return a_t;
  }
 
  // Endpoints condition check [More, Thuente 1994], p.299 - 300
  enum class EndpointsCondition { Case1, Case2, Case3, Case4 };
  EndpointsCondition condition;
 
  if (a_t == a_l) {
    condition = EndpointsCondition::Case4;
  }
  else if (f_t > f_l) {
    condition = EndpointsCondition::Case1;
  }
  else if (g_t * g_l < 0) {
    condition = EndpointsCondition::Case2;
  }
  else if (std::fabs(g_t) <= std::fabs(g_l)) {
    condition = EndpointsCondition::Case3;
  }
  else {
    condition = EndpointsCondition::Case4;
  }
 
  switch (condition) {
  case EndpointsCondition::Case1: {
    // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
    // Equation 2.4.52 [Sun, Yuan 2006]
    const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
    const double w = std::sqrt(z * z - g_t * g_l);
    // Equation 2.4.56 [Sun, Yuan 2006]
    const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
 
    // Calculate the minimizer of the quadratic that interpolates f_l, f_t and g_l
    // Equation 2.4.2 [Sun, Yuan 2006]
    const double a_q =
        a_l - 0.5 * (a_l - a_t) * g_l / (g_l - (f_l - f_t) / (a_l - a_t));
 
    if (std::fabs(a_c - a_l) < std::fabs(a_q - a_l)) {
      return a_c;
    }
    return 0.5 * (a_q + a_c);
  }
 
  case EndpointsCondition::Case2: {
    // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
    // Equation 2.4.52 [Sun, Yuan 2006]
    const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
    const double w = std::sqrt(z * z - g_t * g_l);
    // Equation 2.4.56 [Sun, Yuan 2006]
    const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
 
    // Calculate the minimizer of the quadratic that interpolates f_l, g_l and g_t
    // Equation 2.4.5 [Sun, Yuan 2006]
    const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
 
    if (std::fabs(a_c - a_t) >= std::fabs(a_s - a_t)) {
      return a_c;
    }
    return a_s;
  }
 
  case EndpointsCondition::Case3: {
    // Calculate the minimizer of the cubic that interpolates f_l, f_t, g_l and g_t
    // Equation 2.4.52 [Sun, Yuan 2006]
    const double z = 3 * (f_t - f_l) / (a_t - a_l) - g_t - g_l;
    const double w = std::sqrt(z * z - g_t * g_l);
    const double a_c = a_l + (a_t - a_l) * (w - g_l - z) / (g_t - g_l + 2 * w);
 
    // Calculate the minimizer of the quadratic that interpolates g_l and g_t
    // Equation 2.4.5 [Sun, Yuan 2006]
    const double a_s = a_l - (a_l - a_t) / (g_l - g_t) * g_l;
 
    double a_t_next;
 
    if (std::fabs(a_c - a_t) < std::fabs(a_s - a_t)) {
      a_t_next = a_c;
    }
    else {
      a_t_next = a_s;
    }
 
    if (a_t > a_l) {
      return std::min(a_t + 0.66 * (a_u - a_t), a_t_next);
    }
    return std::max(a_t + 0.66 * (a_u - a_t), a_t_next);
  }
 
  default:
  case EndpointsCondition::Case4: {
    // Calculate the minimizer of the cubic that interpolates f_u, f_t, g_u and g_t
    // Equation 2.4.52 [Sun, Yuan 2006]
    const double z = 3 * (f_t - f_u) / (a_t - a_u) - g_t - g_u;
    const double w = std::sqrt(z * z - g_t * g_u);
    // Equation 2.4.56 [Sun, Yuan 2006]
    return a_u + (a_t - a_u) * (w - g_u - z) / (g_t - g_u + 2 * w);
  }
  }
}
 
template <typename PointSource, typename PointTarget>
double
NormalDistributionsTransform<PointSource, PointTarget>::computeStepLengthMT(
    const Eigen::Matrix<double, 6, 1>& x,
    Eigen::Matrix<double, 6, 1>& step_dir,
    double step_init,
    double step_max,
    double step_min,
    double& score,
    Eigen::Matrix<double, 6, 1>& score_gradient,
    Eigen::Matrix<double, 6, 6>& hessian,
    PointCloudSource& trans_cloud)
{
  // Set the value of phi(0), Equation 1.3 [More, Thuente 1994]
  const double phi_0 = -score;
  // Set the value of phi'(0), Equation 1.3 [More, Thuente 1994]
  double d_phi_0 = -(score_gradient.dot(step_dir));
 
  if (d_phi_0 >= 0) {
    // Not a decent direction
    if (d_phi_0 == 0) {
      return 0;
    }
    // Reverse step direction and calculate optimal step.
    d_phi_0 *= -1;
    step_dir *= -1;
  }
 
  // The Search Algorithm for T(mu) [More, Thuente 1994]
 
  const int max_step_iterations = 10;
  int step_iterations = 0;
 
  // Sufficient decreace constant, Equation 1.1 [More, Thuete 1994]
  const double mu = 1.e-4;
  // Curvature condition constant, Equation 1.2 [More, Thuete 1994]
  const double nu = 0.9;
 
  // Initial endpoints of Interval I,
  double a_l = 0, a_u = 0;
 
  // Auxiliary function psi is used until I is determined ot be a closed interval,
  // Equation 2.1 [More, Thuente 1994]
  double f_l = auxilaryFunction_PsiMT(a_l, phi_0, phi_0, d_phi_0, mu);
  double g_l = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
 
  double f_u = auxilaryFunction_PsiMT(a_u, phi_0, phi_0, d_phi_0, mu);
  double g_u = auxilaryFunction_dPsiMT(d_phi_0, d_phi_0, mu);
 
  // Check used to allow More-Thuente step length calculation to be skipped by making
  // step_min == step_max
  bool interval_converged = (step_max - step_min) < 0, open_interval = true;
 
  double a_t = step_init;
  a_t = std::min(a_t, step_max);
  a_t = std::max(a_t, step_min);
 
  Eigen::Matrix<double, 6, 1> x_t = x + step_dir * a_t;
 
  // Convert x_t into matrix form
  convertTransform(x_t, final_transformation_);
 
  // New transformed point cloud
  transformPointCloud(*input_, trans_cloud, final_transformation_);
 
  // Updates score, gradient and hessian.  Hessian calculation is unessisary but testing
  // showed that most step calculations use the initial step suggestion and
  // recalculation the reusable portions of the hessian would intail more computation
  // time.
  score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, true);
 
  // Calculate phi(alpha_t)
  double phi_t = -score;
  // Calculate phi'(alpha_t)
  double d_phi_t = -(score_gradient.dot(step_dir));
 
  // Calculate psi(alpha_t)
  double psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
  // Calculate psi'(alpha_t)
  double d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
 
  // Iterate until max number of iterations, interval convergance or a value satisfies
  // the sufficient decrease, Equation 1.1, and curvature condition, Equation 1.2 [More,
  // Thuente 1994]
  while (!interval_converged && step_iterations < max_step_iterations &&
         !(psi_t <= 0 /*Sufficient Decrease*/ &&
           d_phi_t <= -nu * d_phi_0 /*Curvature Condition*/)) {
    // Use auxiliary function if interval I is not closed
    if (open_interval) {
      a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
    }
    else {
      a_t = trialValueSelectionMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
    }
 
    a_t = std::min(a_t, step_max);
    a_t = std::max(a_t, step_min);
 
    x_t = x + step_dir * a_t;
 
    // Convert x_t into matrix form
    convertTransform(x_t, final_transformation_);
 
    // New transformed point cloud
    // Done on final cloud to prevent wasted computation
    transformPointCloud(*input_, trans_cloud, final_transformation_);
 
    // Updates score, gradient. Values stored to prevent wasted computation.
    score = computeDerivatives(score_gradient, hessian, trans_cloud, x_t, false);
 
    // Calculate phi(alpha_t+)
    phi_t = -score;
    // Calculate phi'(alpha_t+)
    d_phi_t = -(score_gradient.dot(step_dir));
 
    // Calculate psi(alpha_t+)
    psi_t = auxilaryFunction_PsiMT(a_t, phi_t, phi_0, d_phi_0, mu);
    // Calculate psi'(alpha_t+)
    d_psi_t = auxilaryFunction_dPsiMT(d_phi_t, d_phi_0, mu);
 
    // Check if I is now a closed interval
    if (open_interval && (psi_t <= 0 && d_psi_t >= 0)) {
      open_interval = false;
 
      // Converts f_l and g_l from psi to phi
      f_l += phi_0 - mu * d_phi_0 * a_l;
      g_l += mu * d_phi_0;
 
      // Converts f_u and g_u from psi to phi
      f_u += phi_0 - mu * d_phi_0 * a_u;
      g_u += mu * d_phi_0;
    }
 
    if (open_interval) {
      // Update interval end points using Updating Algorithm [More, Thuente 1994]
      interval_converged =
          updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, psi_t, d_psi_t);
    }
    else {
      // Update interval end points using Modified Updating Algorithm [More, Thuente
      // 1994]
      interval_converged =
          updateIntervalMT(a_l, f_l, g_l, a_u, f_u, g_u, a_t, phi_t, d_phi_t);
    }
 
    step_iterations++;
  }
 
  // If inner loop was run then hessian needs to be calculated.
  // Hessian is unnessisary for step length determination but gradients are required
  // so derivative and transform data is stored for the next iteration.
  if (step_iterations) {
    computeHessian(hessian, trans_cloud);
  }
 
  return a_t;
}
 
} // namespace pcl
 
#endif // PCL_REGISTRATION_NDT_IMPL_H_