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ndt_2d.hpp
/*
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* Point Cloud Library (PCL) - www.pointclouds.org
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#ifndef PCL_NDT_2D_IMPL_H_
#define PCL_NDT_2D_IMPL_H_
#include <boost/core/noncopyable.hpp> // for boost::noncopyable
#include <Eigen/Eigenvalues> // for SelfAdjointEigenSolver, EigenSolver
#include <cmath>
#include <memory>
namespace pcl {
namespace ndt2d {
/** \brief Class to store vector value and first and second derivatives
* (grad vector and hessian matrix), so they can be returned easily from
* functions
*/
template <unsigned N = 3, typename T = double>
struct ValueAndDerivatives {
ValueAndDerivatives() : hessian(), grad(), value() {}
Eigen::Matrix<T, N, N> hessian;
Eigen::Matrix<T, N, 1> grad;
T value;
static ValueAndDerivatives<N, T>
Zero()
{
ValueAndDerivatives<N, T> r;
r.hessian = Eigen::Matrix<T, N, N>::Zero();
r.grad = Eigen::Matrix<T, N, 1>::Zero();
r.value = 0;
return r;
}
ValueAndDerivatives<N, T>&
operator+=(ValueAndDerivatives<N, T> const& r)
{
hessian += r.hessian;
grad += r.grad;
value += r.value;
return *this;
}
};
/** \brief A normal distribution estimation class.
*
* First the indices of of the points from a point cloud that should be
* modelled by the distribution are added with addIdx (...).
*
* Then estimateParams (...) uses the stored point indices to estimate the
* parameters of a normal distribution, and discards the stored indices.
*
* Finally the distriubution, and its derivatives, may be evaluated at any
* point using test (...).
*/
template <typename PointT>
class NormalDist {
using PointCloud = pcl::PointCloud<PointT>;
public:
NormalDist() = default;
/** \brief Store a point index to use later for estimating distribution parameters.
* \param[in] i Point index to store
*/
void
addIdx(std::size_t i)
{
pt_indices_.push_back(i);
}
/** \brief Estimate the normal distribution parameters given the point indices
* provided. Memory of point indices is cleared. \param[in] cloud Point cloud
* corresponding to indices passed to addIdx. \param[in] min_covar_eigvalue_mult Set
* the smallest eigenvalue to this times the largest.
*/
void
estimateParams(const PointCloud& cloud, double min_covar_eigvalue_mult = 0.001)
{
Eigen::Vector2d sx = Eigen::Vector2d::Zero();
Eigen::Matrix2d sxx = Eigen::Matrix2d::Zero();
for (const auto& pt_index : pt_indices_) {
Eigen::Vector2d p(cloud[pt_index].x, cloud[pt_index].y);
sx += p;
sxx += p * p.transpose();
}
n_ = pt_indices_.size();
if (n_ >= min_n_) {
mean_ = sx / static_cast<double>(n_);
// Using maximum likelihood estimation as in the original paper
Eigen::Matrix2d covar =
(sxx - 2 * (sx * mean_.transpose())) / static_cast<double>(n_) +
mean_ * mean_.transpose();
Eigen::SelfAdjointEigenSolver<Eigen::Matrix2d> solver(covar);
if (solver.eigenvalues()[0] < min_covar_eigvalue_mult * solver.eigenvalues()[1]) {
PCL_DEBUG("[pcl::NormalDist::estimateParams] NDT normal fit: adjusting "
"eigenvalue %f\n",
solver.eigenvalues()[0]);
Eigen::Matrix2d l = solver.eigenvalues().asDiagonal();
Eigen::Matrix2d q = solver.eigenvectors();
// set minimum smallest eigenvalue:
l(0, 0) = l(1, 1) * min_covar_eigvalue_mult;
covar = q * l * q.transpose();
}
covar_inv_ = covar.inverse();
}
pt_indices_.clear();
}
/** \brief Return the 'score' (denormalised likelihood) and derivatives of score of
* the point p given this distribution. \param[in] transformed_pt Location to
* evaluate at. \param[in] cos_theta sin(theta) of the current rotation angle
* of rigid transformation: to avoid repeated evaluation \param[in] sin_theta
* cos(theta) of the current rotation angle of rigid transformation: to avoid repeated
* evaluation estimateParams must have been called after at least three points were
* provided, or this will return zero.
*
*/
ValueAndDerivatives<3, double>
test(const PointT& transformed_pt,
const double& cos_theta,
const double& sin_theta) const
{
if (n_ < min_n_)
return ValueAndDerivatives<3, double>::Zero();
ValueAndDerivatives<3, double> r;
const double x = transformed_pt.x;
const double y = transformed_pt.y;
const Eigen::Vector2d p_xy(transformed_pt.x, transformed_pt.y);
const Eigen::Vector2d q = p_xy - mean_;
const Eigen::RowVector2d qt_cvi(q.transpose() * covar_inv_);
const double exp_qt_cvi_q = std::exp(-0.5 * static_cast<double>(qt_cvi * q));
r.value = -exp_qt_cvi_q;
Eigen::Matrix<double, 2, 3> jacobian;
jacobian << 1, 0, -(x * sin_theta + y * cos_theta), 0, 1,
x * cos_theta - y * sin_theta;
for (std::size_t i = 0; i < 3; i++)
r.grad[i] = static_cast<double>(qt_cvi * jacobian.col(i)) * exp_qt_cvi_q;
// second derivative only for i == j == 2:
const Eigen::Vector2d d2q_didj(y * sin_theta - x * cos_theta,
-(x * sin_theta + y * cos_theta));
for (std::size_t i = 0; i < 3; i++)
for (std::size_t j = 0; j < 3; j++)
r.hessian(i, j) =
-exp_qt_cvi_q *
(static_cast<double>(-qt_cvi * jacobian.col(i)) *
static_cast<double>(-qt_cvi * jacobian.col(j)) +
(-qt_cvi * ((i == 2 && j == 2) ? d2q_didj : Eigen::Vector2d::Zero())) +
(-jacobian.col(j).transpose() * covar_inv_ * jacobian.col(i)));
return r;
}
protected:
const std::size_t min_n_{3};
std::size_t n_{0};
std::vector<std::size_t> pt_indices_;
Eigen::Vector2d mean_;
Eigen::Matrix2d covar_inv_;
};
/** \brief Build a set of normal distributions modelling a 2D point cloud,
* and provide the value and derivatives of the model at any point via the
* test (...) function.
*/
template <typename PointT>
class NDTSingleGrid : public boost::noncopyable {
using PointCloud = pcl::PointCloud<PointT>;
using PointCloudConstPtr = typename PointCloud::ConstPtr;
using NormalDist = pcl::ndt2d::NormalDist<PointT>;
public:
NDTSingleGrid(PointCloudConstPtr cloud,
const Eigen::Vector2f& about,
const Eigen::Vector2f& extent,
const Eigen::Vector2f& step)
: min_(about - extent)
, max_(min_ + 2 * extent)
, step_(step)
, cells_((max_[0] - min_[0]) / step_[0], (max_[1] - min_[1]) / step_[1])
, normal_distributions_(cells_[0], cells_[1])
{
// sort through all points, assigning them to distributions:
std::size_t used_points = 0;
for (std::size_t i = 0; i < cloud->size(); i++)
if (NormalDist* n = normalDistForPoint(cloud->at(i))) {
n->addIdx(i);
used_points++;
}
PCL_DEBUG("[pcl::NDTSingleGrid] NDT single grid %dx%d using %d/%d points\n",
cells_[0],
cells_[1],
used_points,
cloud->size());
// then bake the distributions such that they approximate the
// points (and throw away memory of the points)
for (int x = 0; x < cells_[0]; x++)
for (int y = 0; y < cells_[1]; y++)
normal_distributions_.coeffRef(x, y).estimateParams(*cloud);
}
/** \brief Return the 'score' (denormalised likelihood) and derivatives of score of
* the point p given this distribution. \param[in] transformed_pt Location to
* evaluate at. \param[in] cos_theta sin(theta) of the current rotation angle
* of rigid transformation: to avoid repeated evaluation \param[in] sin_theta
* cos(theta) of the current rotation angle of rigid transformation: to avoid repeated
* evaluation
*/
ValueAndDerivatives<3, double>
test(const PointT& transformed_pt,
const double& cos_theta,
const double& sin_theta) const
{
const NormalDist* n = normalDistForPoint(transformed_pt);
// index is in grid, return score from the normal distribution from
// the correct part of the grid:
if (n)
return n->test(transformed_pt, cos_theta, sin_theta);
return ValueAndDerivatives<3, double>::Zero();
}
protected:
/** \brief Return the normal distribution covering the location of point p
* \param[in] p a point
*/
NormalDist*
normalDistForPoint(PointT const& p) const
{
// this would be neater in 3d...
Eigen::Vector2f idxf;
for (std::size_t i = 0; i < 2; i++)
idxf[i] = (p.getVector3fMap()[i] - min_[i]) / step_[i];
Eigen::Vector2i idxi = idxf.cast<int>();
for (std::size_t i = 0; i < 2; i++)
if (idxi[i] >= cells_[i] || idxi[i] < 0)
return nullptr;
// const cast to avoid duplicating this function in const and
// non-const variants...
return const_cast<NormalDist*>(&normal_distributions_.coeffRef(idxi[0], idxi[1]));
}
Eigen::Vector2f min_;
Eigen::Vector2f max_;
Eigen::Vector2f step_;
Eigen::Vector2i cells_;
Eigen::Matrix<NormalDist, Eigen::Dynamic, Eigen::Dynamic> normal_distributions_;
};
/** \brief Build a Normal Distributions Transform of a 2D point cloud. This
* consists of the sum of four overlapping models of the original points
* with normal distributions.
* The value and derivatives of the model at any point can be evaluated
* with the test (...) function.
*/
template <typename PointT>
class NDT2D : public boost::noncopyable {
using PointCloud = pcl::PointCloud<PointT>;
using PointCloudConstPtr = typename PointCloud::ConstPtr;
using SingleGrid = NDTSingleGrid<PointT>;
public:
/** \brief
* \param[in] cloud the input point cloud
* \param[in] about Centre of the grid for normal distributions model
* \param[in] extent Extent of grid for normal distributions model
* \param[in] step Size of region that each normal distribution will model
*/
NDT2D(PointCloudConstPtr cloud,
const Eigen::Vector2f& about,
const Eigen::Vector2f& extent,
const Eigen::Vector2f& step)
{
Eigen::Vector2f dx(step[0] / 2, 0);
Eigen::Vector2f dy(0, step[1] / 2);
single_grids_[0].reset(new SingleGrid(cloud, about, extent, step));
single_grids_[1].reset(new SingleGrid(cloud, about + dx, extent, step));
single_grids_[2].reset(new SingleGrid(cloud, about + dy, extent, step));
single_grids_[3].reset(new SingleGrid(cloud, about + dx + dy, extent, step));
}
/** \brief Return the 'score' (denormalised likelihood) and derivatives of score of
* the point p given this distribution. \param[in] transformed_pt Location to
* evaluate at. \param[in] cos_theta sin(theta) of the current rotation angle
* of rigid transformation: to avoid repeated evaluation \param[in] sin_theta
* cos(theta) of the current rotation angle of rigid transformation: to avoid repeated
* evaluation
*/
ValueAndDerivatives<3, double>
test(const PointT& transformed_pt,
const double& cos_theta,
const double& sin_theta) const
{
ValueAndDerivatives<3, double> r = ValueAndDerivatives<3, double>::Zero();
for (const auto& single_grid : single_grids_)
r += single_grid->test(transformed_pt, cos_theta, sin_theta);
return r;
}
protected:
std::shared_ptr<SingleGrid> single_grids_[4];
};
} // namespace ndt2d
} // namespace pcl
namespace Eigen {
/* This NumTraits specialisation is necessary because NormalDist is used as
* the element type of an Eigen Matrix.
*/
template <typename PointT>
struct NumTraits<pcl::ndt2d::NormalDist<PointT>> {
using Real = double;
using Literal = double;
static Real
dummy_precision()
{
return 1.0;
}
enum {
IsComplex = 0,
IsInteger = 0,
IsSigned = 0,
RequireInitialization = 1,
ReadCost = 1,
AddCost = 1,
MulCost = 1
};
};
} // namespace Eigen
namespace pcl {
template <typename PointSource, typename PointTarget>
void
NormalDistributionsTransform2D<PointSource, PointTarget>::computeTransformation(
PointCloudSource& output, const Eigen::Matrix4f& guess)
{
PointCloudSource intm_cloud = output;
nr_iterations_ = 0;
converged_ = false;
if (guess != Eigen::Matrix4f::Identity()) {
transformation_ = guess;
transformPointCloud(output, intm_cloud, transformation_);
}
// build Normal Distribution Transform of target cloud:
ndt2d::NDT2D<PointTarget> target_ndt(target_, grid_centre_, grid_extent_, grid_step_);
// can't seem to use .block<> () member function on transformation_
// directly... gcc bug?
Eigen::Matrix4f& transformation = transformation_;
// work with x translation, y translation and z rotation: extending to 3D
// would be some tricky maths, but not impossible.
const Eigen::Matrix3f initial_rot(transformation.block<3, 3>(0, 0));
const Eigen::Vector3f rot_x(initial_rot * Eigen::Vector3f::UnitX());
const double z_rotation = std::atan2(rot_x[1], rot_x[0]);
Eigen::Vector3d xytheta_transformation(
transformation(0, 3), transformation(1, 3), z_rotation);
while (!converged_) {
const double cos_theta = std::cos(xytheta_transformation[2]);
const double sin_theta = std::sin(xytheta_transformation[2]);
previous_transformation_ = transformation;
ndt2d::ValueAndDerivatives<3, double> score =
ndt2d::ValueAndDerivatives<3, double>::Zero();
for (std::size_t i = 0; i < intm_cloud.size(); i++)
score += target_ndt.test(intm_cloud[i], cos_theta, sin_theta);
PCL_DEBUG("[pcl::NormalDistributionsTransform2D::computeTransformation] NDT score "
"%f (x=%f,y=%f,r=%f)\n",
float(score.value),
xytheta_transformation[0],
xytheta_transformation[1],
xytheta_transformation[2]);
if (score.value != 0) {
// test for positive definiteness, and adjust to ensure it if necessary:
Eigen::EigenSolver<Eigen::Matrix3d> solver;
solver.compute(score.hessian, false);
double min_eigenvalue = 0;
for (int i = 0; i < 3; i++)
if (solver.eigenvalues()[i].real() < min_eigenvalue)
min_eigenvalue = solver.eigenvalues()[i].real();
// ensure "safe" positive definiteness: this is a detail missing
// from the original paper
if (min_eigenvalue < 0) {
double lambda = 1.1 * min_eigenvalue - 1;
score.hessian += Eigen::Vector3d(-lambda, -lambda, -lambda).asDiagonal();
solver.compute(score.hessian, false);
PCL_DEBUG("[pcl::NormalDistributionsTransform2D::computeTransformation] adjust "
"hessian: %f: new eigenvalues:%f %f %f\n",
float(lambda),
solver.eigenvalues()[0].real(),
solver.eigenvalues()[1].real(),
solver.eigenvalues()[2].real());
}
assert(solver.eigenvalues()[0].real() >= 0 &&
solver.eigenvalues()[1].real() >= 0 &&
solver.eigenvalues()[2].real() >= 0);
Eigen::Vector3d delta_transformation(-score.hessian.inverse() * score.grad);
Eigen::Vector3d new_transformation =
xytheta_transformation + newton_lambda_.cwiseProduct(delta_transformation);
xytheta_transformation = new_transformation;
// update transformation matrix from x, y, theta:
transformation.block<3, 3>(0, 0).matrix() = Eigen::Matrix3f(Eigen::AngleAxisf(
static_cast<float>(xytheta_transformation[2]), Eigen::Vector3f::UnitZ()));
transformation.block<3, 1>(0, 3).matrix() =
Eigen::Vector3f(static_cast<float>(xytheta_transformation[0]),
static_cast<float>(xytheta_transformation[1]),
0.0f);
// std::cout << "new transformation:\n" << transformation << std::endl;
}
else {
PCL_ERROR("[pcl::NormalDistributionsTransform2D::computeTransformation] no "
"overlap: try increasing the size or reducing the step of the grid\n");
break;
}
transformPointCloud(output, intm_cloud, transformation);
nr_iterations_++;
if (update_visualizer_)
update_visualizer_(output, *indices_, *target_, *indices_);
// std::cout << "eps=" << std::abs ((transformation - previous_transformation_).sum
// ()) << std::endl;
Eigen::Matrix4f transformation_delta =
transformation.inverse() * previous_transformation_;
double cos_angle =
0.5 * (transformation_delta.coeff(0, 0) + transformation_delta.coeff(1, 1) +
transformation_delta.coeff(2, 2) - 1);
double translation_sqr =
transformation_delta.coeff(0, 3) * transformation_delta.coeff(0, 3) +
transformation_delta.coeff(1, 3) * transformation_delta.coeff(1, 3) +
transformation_delta.coeff(2, 3) * transformation_delta.coeff(2, 3);
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;
}
}
final_transformation_ = transformation;
output = intm_cloud;
}
} // namespace pcl
#endif // PCL_NDT_2D_IMPL_H_
pcl::NormalDistributionsTransform2D::computeTransformation
void computeTransformation(PointCloudSource &output, const Eigen::Matrix4f &guess) override
Rigid transformation computation method with initial guess.
Definition ndt_2d.hpp:392
pcl::PointCloud
PointCloud represents the base class in PCL for storing collections of 3D points.
Definition point_cloud.h:173
pcl::PointCloud::ConstPtr
shared_ptr< const PointCloud< PointT > > ConstPtr
Definition point_cloud.h:414
pcl::ndt2d::NDT2D
Build a Normal Distributions Transform of a 2D point cloud.
Definition ndt_2d.hpp:311
pcl::ndt2d::NDT2D::test
ValueAndDerivatives< 3, double > test(const PointT &transformed_pt, const double &cos_theta, const double &sin_theta) const
Return the 'score' (denormalised likelihood) and derivatives of score of the point p given this distr...
Definition ndt_2d.hpp:344
pcl::ndt2d::NDT2D::single_grids_
std::shared_ptr< SingleGrid > single_grids_[4]
Definition ndt_2d.hpp:355
pcl::ndt2d::NDT2D::NDT2D
NDT2D(PointCloudConstPtr cloud, const Eigen::Vector2f &about, const Eigen::Vector2f &extent, const Eigen::Vector2f &step)
Definition ndt_2d.hpp:323
pcl::ndt2d::NDTSingleGrid
Build a set of normal distributions modelling a 2D point cloud, and provide the value and derivatives...
Definition ndt_2d.hpp:219
pcl::ndt2d::NDTSingleGrid::min_
Eigen::Vector2f min_
Definition ndt_2d.hpp:296
pcl::ndt2d::NDTSingleGrid::max_
Eigen::Vector2f max_
Definition ndt_2d.hpp:297
pcl::ndt2d::NDTSingleGrid::normalDistForPoint
NormalDist * normalDistForPoint(PointT const &p) const
Return the normal distribution covering the location of point p.
Definition ndt_2d.hpp:281
pcl::ndt2d::NDTSingleGrid::NDTSingleGrid
NDTSingleGrid(PointCloudConstPtr cloud, const Eigen::Vector2f &about, const Eigen::Vector2f &extent, const Eigen::Vector2f &step)
Definition ndt_2d.hpp:225
pcl::ndt2d::NDTSingleGrid::normal_distributions_
Eigen::Matrix< NormalDist, Eigen::Dynamic, Eigen::Dynamic > normal_distributions_
Definition ndt_2d.hpp:301
pcl::ndt2d::NDTSingleGrid::cells_
Eigen::Vector2i cells_
Definition ndt_2d.hpp:299
pcl::ndt2d::NDTSingleGrid::step_
Eigen::Vector2f step_
Definition ndt_2d.hpp:298
pcl::ndt2d::NDTSingleGrid::test
ValueAndDerivatives< 3, double > test(const PointT &transformed_pt, const double &cos_theta, const double &sin_theta) const
Return the 'score' (denormalised likelihood) and derivatives of score of the point p given this distr...
Definition ndt_2d.hpp:264
pcl::ndt2d::NormalDist
A normal distribution estimation class.
Definition ndt_2d.hpp:98
pcl::ndt2d::NormalDist::pt_indices_
std::vector< std::size_t > pt_indices_
Definition ndt_2d.hpp:209
pcl::ndt2d::NormalDist::n_
std::size_t n_
Definition ndt_2d.hpp:208
pcl::ndt2d::NormalDist::min_n_
const std::size_t min_n_
Definition ndt_2d.hpp:206
pcl::ndt2d::NormalDist::NormalDist
NormalDist()=default
pcl::ndt2d::NormalDist::addIdx
void addIdx(std::size_t i)
Store a point index to use later for estimating distribution parameters.
Definition ndt_2d.hpp:108
pcl::ndt2d::NormalDist::estimateParams
void estimateParams(const PointCloud &cloud, double min_covar_eigvalue_mult=0.001)
Estimate the normal distribution parameters given the point indices provided.
Definition ndt_2d.hpp:119
pcl::ndt2d::NormalDist::test
ValueAndDerivatives< 3, double > test(const PointT &transformed_pt, const double &cos_theta, const double &sin_theta) const
Return the 'score' (denormalised likelihood) and derivatives of score of the point p given this distr...
Definition ndt_2d.hpp:166
pcl::ndt2d::NormalDist::mean_
Eigen::Vector2d mean_
Definition ndt_2d.hpp:210
pcl::ndt2d::NormalDist::covar_inv_
Eigen::Matrix2d covar_inv_
Definition ndt_2d.hpp:211
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
Eigen
Definition bfgs.h:10
pcl
Definition convolution.h:46
Eigen::NumTraits< pcl::ndt2d::NormalDist< PointT > >::dummy_precision
static Real dummy_precision()
Definition ndt_2d.hpp:371
Eigen::NumTraits< pcl::ndt2d::NormalDist< PointT > >::Literal
double Literal
Definition ndt_2d.hpp:369
Eigen::NumTraits< pcl::ndt2d::NormalDist< PointT > >::Real
double Real
Definition ndt_2d.hpp:368
pcl::PointXYZRGB
A point structure representing Euclidean xyz coordinates, and the RGB color.
Definition point_types.hpp:594
pcl::ndt2d::ValueAndDerivatives
Class to store vector value and first and second derivatives (grad vector and hessian matrix),...
Definition ndt_2d.hpp:59
pcl::ndt2d::ValueAndDerivatives::hessian
Eigen::Matrix< T, N, N > hessian
Definition ndt_2d.hpp:62
pcl::ndt2d::ValueAndDerivatives::value
T value
Definition ndt_2d.hpp:64
pcl::ndt2d::ValueAndDerivatives::Zero
static ValueAndDerivatives< N, T > Zero()
Definition ndt_2d.hpp:67
pcl::ndt2d::ValueAndDerivatives::ValueAndDerivatives
ValueAndDerivatives()
Definition ndt_2d.hpp:60
pcl::ndt2d::ValueAndDerivatives::operator+=
ValueAndDerivatives< N, T > & operator+=(ValueAndDerivatives< N, T > const &r)
Definition ndt_2d.hpp:77
pcl::ndt2d::ValueAndDerivatives::grad
Eigen::Matrix< T, N, 1 > grad
Definition ndt_2d.hpp:63