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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, typename Scalar>
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::
NormalDistributionsTransform()
: target_cells_()
{
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, typename Scalar>
void
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::computeTransformation(
PointCloudSource& output, const Matrix4& guess)
{
nr_iterations_ = 0;
converged_ = false;
if (target_cells_.getCentroids()->empty()) {
PCL_ERROR("[%s::computeTransformation] Voxel grid is not searchable!\n",
getClassName().c_str());
return;
}
// 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 != Matrix4::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<Scalar, 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;
Vector3 init_translation = eig_transformation.translation();
Vector3 init_rotation = eig_transformation.rotation().eulerAngles(0, 1, 2);
transform << init_translation.template cast<double>(),
init_rotation.template 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 guaranteed sufficient decrease [More, Thuente 1994]
double delta_norm = delta.norm();
if (delta_norm == 0 || std::isnan(delta_norm)) {
trans_likelihood_ = 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 likelihood. The relative differences within each scan
// registration are accurate but the normalization constants need to be modified for
// it to be globally accurate
trans_likelihood_ = score / static_cast<double>(input_->size());
}
template <typename PointSource, typename PointTarget, typename Scalar>
double
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
void
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
void
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
double
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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 likelihood 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, typename Scalar>
void
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::computeHessian(
Eigen::Matrix<double, 6, 6>& hessian, const PointCloudSource& trans_cloud)
{
hessian.setZero();
// Precompute Angular Derivatives unnecessary 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 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 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, typename Scalar>
void
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
bool
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
double
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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, typename Scalar>
double
NormalDistributionsTransform<PointSource, PointTarget, Scalar>::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]
constexpr int max_step_iterations = 10;
int step_iterations = 0;
// Sufficient decrease constant, Equation 1.1 [More, Thuete 1994]
constexpr double mu = 1.e-4;
// Curvature condition constant, Equation 1.2 [More, Thuete 1994]
constexpr 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 unnecessary but
// testing showed that most step calculations use the initial step suggestion and
// recalculation the reusable portions of the hessian would entail 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 convergence 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 unnecessary 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_
pcl::NormalDistributionsTransform::computePointDerivatives
void computePointDerivatives(const Eigen::Vector3d &x, bool compute_hessian=true)
Compute point derivatives.
Definition ndt.hpp:321
pcl::NormalDistributionsTransform::gauss_d2_
double gauss_d2_
Definition ndt.h:597
pcl::NormalDistributionsTransform::computeTransformation
virtual void computeTransformation(PointCloudSource &output)
Estimate the transformation and returns the transformed source (input) as output.
Definition ndt.h:315
pcl::NormalDistributionsTransform::updateDerivatives
double 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=true) const
Compute individual point contributions to derivatives of likelihood function w.r.t.
Definition ndt.hpp:367
pcl::NormalDistributionsTransform::computeDerivatives
double 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=true)
Compute derivatives of likelihood function w.r.t.
Definition ndt.hpp:185
pcl::NormalDistributionsTransform::computeAngleDerivatives
void computeAngleDerivatives(const Eigen::Matrix< double, 6, 1 > &transform, bool compute_hessian=true)
Precompute angular components of derivatives.
Definition ndt.hpp:236
pcl::NormalDistributionsTransform::PointCloudSource
typename Registration< PointSource, PointTarget, Scalar >::PointCloudSource PointCloudSource
Definition ndt.h:70
pcl::NormalDistributionsTransform::NormalDistributionsTransform
NormalDistributionsTransform()
Constructor.
Definition ndt.hpp:48
pcl::NormalDistributionsTransform::updateIntervalMT
bool 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
Update interval of possible step lengths for More-Thuente method, in More-Thuente (1994)
Definition ndt.hpp:492
pcl::NormalDistributionsTransform::computeHessian
void computeHessian(Eigen::Matrix< double, 6, 6 > &hessian, const PointCloudSource &trans_cloud)
Compute hessian of likelihood function w.r.t.
Definition ndt.hpp:415
pcl::NormalDistributionsTransform::Vector3
typename Eigen::Matrix< Scalar, 3, 1 > Vector3
Definition ndt.h:97
pcl::NormalDistributionsTransform::resolution_
float resolution_
The side length of voxels.
Definition ndt.h:586
pcl::NormalDistributionsTransform::Matrix4
typename Registration< PointSource, PointTarget, Scalar >::Matrix4 Matrix4
Definition ndt.h:98
pcl::NormalDistributionsTransform::outlier_ratio_
double outlier_ratio_
The ratio of outliers of points w.r.t.
Definition ndt.h:593
pcl::NormalDistributionsTransform::updateHessian
void updateHessian(Eigen::Matrix< double, 6, 6 > &hessian, const Eigen::Vector3d &x_trans, const Eigen::Matrix3d &c_inv) const
Compute individual point contributions to hessian of likelihood function w.r.t.
Definition ndt.hpp:457
pcl::NormalDistributionsTransform::gauss_d1_
double gauss_d1_
The normalization constants used fit the point distribution to a normal distribution,...
Definition ndt.h:597
pcl::NormalDistributionsTransform::trialValueSelectionMT
double 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
Select new trial value for More-Thuente method.
Definition ndt.hpp:537
pcl::NormalDistributionsTransform::computeStepLengthMT
double computeStepLengthMT(const Eigen::Matrix< double, 6, 1 > &transform, 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)
Compute line search step length and update transform and likelihood derivatives using More-Thuente me...
Definition ndt.hpp:650
pcl::Registration::reg_name_
std::string reg_name_
The registration method name.
Definition registration.h:548
pcl::Registration::max_iterations_
int max_iterations_
The maximum number of iterations the internal optimization should run for.
Definition registration.h:563
pcl::Registration::transformation_epsilon_
double transformation_epsilon_
The maximum difference between two consecutive transformations in order to consider convergence (user...
Definition registration.h:585
pcl::transformPointCloud
void transformPointCloud(const pcl::PointCloud< PointT > &cloud_in, pcl::PointCloud< PointT > &cloud_out, const Eigen::Matrix< Scalar, 4, 4 > &transform, bool copy_all_fields)
Apply a rigid transform defined by a 4x4 matrix.
Definition transforms.hpp:221
pcl
Definition convolution.h:46
pcl::Indices
IndicesAllocator<> Indices
Type used for indices in PCL.
Definition types.h:133