ndt_2d.hpp

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#ifndef PCL_NDT_2D_IMPL_H_
#define PCL_NDT_2D_IMPL_H_
 
#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() : min_n_(3), n_(0) {}
 
  /** \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 (auto i = pt_indices_.cbegin(); i != pt_indices_.cend(); i++) {
      Eigen::Vector2d p(cloud[*i].x, cloud[*i].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 * 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] = 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 *
            (double(-qt_cvi * jacobian.col(i)) * 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_;
 
  std::size_t n_;
  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_