unsupported/Eigen/src/Splines/Spline.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_SPLINE_H
#define EIGEN_SPLINE_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

#include "SplineFwd.h"

namespace Eigen {
/**
 * \ingroup Splines_Module
 * \class Spline
 * \brief A class representing multi-dimensional spline curves.
 *
 * The class represents B-splines with non-uniform knot vectors. Each control
 * point of the B-spline is associated with a basis function
 * \f{align*}
 *   C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
 * \f}
 *
 * \tparam Scalar_ The underlying data type (typically float or double)
 * \tparam Dim_ The curve dimension (e.g. 2 or 3)
 * \tparam Degree_ Per default set to Dynamic; could be set to the actual desired
 *                degree for optimization purposes (would result in stack allocation
 *                of several temporary variables).
 **/
template <typename Scalar_, int Dim_, int Degree_>
class Spline {
 public:
  typedef Scalar_ Scalar; /*!< The spline curve's scalar type. */
  enum { Dimension = Dim_ /*!< The spline curve's dimension. */ };
  enum { Degree = Degree_ /*!< The spline curve's degree. */ };

  /** \brief The point type the spline is representing. */
  typedef typename SplineTraits<Spline>::PointType PointType;

  /** \brief The data type used to store knot vectors. */
  typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;

  /** \brief The data type used to store parameter vectors. */
  typedef typename SplineTraits<Spline>::ParameterVectorType ParameterVectorType;

  /** \brief The data type used to store non-zero basis functions. */
  typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;

  /** \brief The data type used to store the values of the basis function derivatives. */
  typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;

  /** \brief The data type representing the spline's control points. */
  typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;

  /**
   * \brief Creates a (constant) zero spline.
   * For Splines with dynamic degree, the resulting degree will be 0.
   **/
  Spline()
      : m_knots(1, (Degree == Dynamic ? 2 : 2 * Degree + 2)),
        m_ctrls(ControlPointVectorType::Zero(Dimension, (Degree == Dynamic ? 1 : Degree + 1))) {
    // in theory this code can go to the initializer list but it will get pretty
    // much unreadable ...
    enum { MinDegree = (Degree == Dynamic ? 0 : Degree) };
    m_knots.template segment<MinDegree + 1>(0) = Array<Scalar, 1, MinDegree + 1>::Zero();
    m_knots.template segment<MinDegree + 1>(MinDegree + 1) = Array<Scalar, 1, MinDegree + 1>::Ones();
  }

  /**
   * \brief Creates a spline from a knot vector and control points.
   * \param knots The spline's knot vector.
   * \param ctrls The spline's control point vector.
   **/
  template <typename OtherVectorType, typename OtherArrayType>
  Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {}

  /**
   * \brief Copy constructor for splines.
   * \param spline The input spline.
   **/
  template <int OtherDegree>
  Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) : m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}

  /**
   * \brief Returns the knots of the underlying spline.
   **/
  const KnotVectorType& knots() const { return m_knots; }

  /**
   * \brief Returns the ctrls of the underlying spline.
   **/
  const ControlPointVectorType& ctrls() const { return m_ctrls; }

  /**
   * \brief Returns the spline value at a given site \f$u\f$.
   *
   * The function returns
   * \f{align*}
   *   C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
   * \f}
   *
   * \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
   * \return The spline value at the given location \f$u\f$.
   **/
  PointType operator()(Scalar u) const;

  /**
   * \brief Evaluation of spline derivatives of up-to given order.
   *
   * The function returns
   * \f{align*}
   *   \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
   * \f}
   * for i ranging between 0 and order.
   *
   * \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
   * \param order The order up to which the derivatives are computed.
   **/
  typename SplineTraits<Spline>::DerivativeType derivatives(Scalar u, DenseIndex order) const;

  /**
   * \copydoc Spline::derivatives
   * Using the template version of this function is more efficieent since
   * temporary objects are allocated on the stack whenever this is possible.
   **/
  template <int DerivativeOrder>
  typename SplineTraits<Spline, DerivativeOrder>::DerivativeType derivatives(Scalar u,
                                                                             DenseIndex order = DerivativeOrder) const;

  /**
   * \brief Computes the non-zero basis functions at the given site.
   *
   * Splines have local support and a point from their image is defined
   * by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
   * spline degree.
   *
   * This function computes the \f$p+1\f$ non-zero basis function values
   * for a given parameter value \f$u\f$. It returns
   * \f{align*}{
   *   N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
   * \f}
   *
   * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions
   *          are computed.
   **/
  typename SplineTraits<Spline>::BasisVectorType basisFunctions(Scalar u) const;

  /**
   * \brief Computes the non-zero spline basis function derivatives up to given order.
   *
   * The function computes
   * \f{align*}{
   *   \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
   * \f}
   * with i ranging from 0 up to the specified order.
   *
   * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
   *          derivatives are computed.
   * \param order The order up to which the basis function derivatives are computes.
   **/
  typename SplineTraits<Spline>::BasisDerivativeType basisFunctionDerivatives(Scalar u, DenseIndex order) const;

  /**
   * \copydoc Spline::basisFunctionDerivatives
   * Using the template version of this function is more efficieent since
   * temporary objects are allocated on the stack whenever this is possible.
   **/
  template <int DerivativeOrder>
  typename SplineTraits<Spline, DerivativeOrder>::BasisDerivativeType basisFunctionDerivatives(
      Scalar u, DenseIndex order = DerivativeOrder) const;

  /**
   * \brief Returns the spline degree.
   **/
  DenseIndex degree() const;

  /**
   * \brief Returns the span within the knot vector in which u is falling.
   * \param u The site for which the span is determined.
   **/
  DenseIndex span(Scalar u) const;

  /**
   * \brief Computes the span within the provided knot vector in which u is falling.
   **/
  static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree,
                         const typename SplineTraits<Spline>::KnotVectorType& knots);

  /**
   * \brief Returns the spline's non-zero basis functions.
   *
   * The function computes and returns
   * \f{align*}{
   *   N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
   * \f}
   *
   * \param u The site at which the basis functions are computed.
   * \param degree The degree of the underlying spline.
   * \param knots The underlying spline's knot vector.
   **/
  static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);

  /**
   * \copydoc Spline::basisFunctionDerivatives
   * \param degree The degree of the underlying spline
   * \param knots The underlying spline's knot vector.
   **/
  static BasisDerivativeType BasisFunctionDerivatives(const Scalar u, const DenseIndex order, const DenseIndex degree,
                                                      const KnotVectorType& knots);

 private:
  KnotVectorType m_knots;         /*!< Knot vector. */
  ControlPointVectorType m_ctrls; /*!< Control points. */

  template <typename DerivativeType>
  static void BasisFunctionDerivativesImpl(const typename Spline<Scalar_, Dim_, Degree_>::Scalar u,
                                           const DenseIndex order, const DenseIndex p,
                                           const typename Spline<Scalar_, Dim_, Degree_>::KnotVectorType& U,
                                           DerivativeType& N_);
};

template <typename Scalar_, int Dim_, int Degree_>
DenseIndex Spline<Scalar_, Dim_, Degree_>::Span(
    typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::Scalar u, DenseIndex degree,
    const typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::KnotVectorType& knots) {
  // Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
  if (u <= knots(0)) return degree;
  const Scalar* pos = std::upper_bound(knots.data() + degree - 1, knots.data() + knots.size() - degree - 1, u);
  return static_cast<DenseIndex>(std::distance(knots.data(), pos) - 1);
}

template <typename Scalar_, int Dim_, int Degree_>
typename Spline<Scalar_, Dim_, Degree_>::BasisVectorType Spline<Scalar_, Dim_, Degree_>::BasisFunctions(
    typename Spline<Scalar_, Dim_, Degree_>::Scalar u, DenseIndex degree,
    const typename Spline<Scalar_, Dim_, Degree_>::KnotVectorType& knots) {
  const DenseIndex p = degree;
  const DenseIndex i = Spline::Span(u, degree, knots);

  const KnotVectorType& U = knots;

  BasisVectorType left(p + 1);
  left(0) = Scalar(0);
  BasisVectorType right(p + 1);
  right(0) = Scalar(0);

  VectorBlock<BasisVectorType, Degree>(left, 1, p) =
      u - VectorBlock<const KnotVectorType, Degree>(U, i + 1 - p, p).reverse();
  VectorBlock<BasisVectorType, Degree>(right, 1, p) = VectorBlock<const KnotVectorType, Degree>(U, i + 1, p) - u;

  BasisVectorType N(1, p + 1);
  N(0) = Scalar(1);
  for (DenseIndex j = 1; j <= p; ++j) {
    Scalar saved = Scalar(0);
    for (DenseIndex r = 0; r < j; r++) {
      const Scalar tmp = N(r) / (right(r + 1) + left(j - r));
      N[r] = saved + right(r + 1) * tmp;
      saved = left(j - r) * tmp;
    }
    N(j) = saved;
  }
  return N;
}

template <typename Scalar_, int Dim_, int Degree_>
DenseIndex Spline<Scalar_, Dim_, Degree_>::degree() const {
  if (Degree_ == Dynamic)
    return m_knots.size() - m_ctrls.cols() - 1;
  else
    return Degree_;
}

template <typename Scalar_, int Dim_, int Degree_>
DenseIndex Spline<Scalar_, Dim_, Degree_>::span(Scalar u) const {
  return Spline::Span(u, degree(), knots());
}

template <typename Scalar_, int Dim_, int Degree_>
typename Spline<Scalar_, Dim_, Degree_>::PointType Spline<Scalar_, Dim_, Degree_>::operator()(Scalar u) const {
  enum { Order = SplineTraits<Spline>::OrderAtCompileTime };

  const DenseIndex span = this->span(u);
  const DenseIndex p = degree();
  const BasisVectorType basis_funcs = basisFunctions(u);

  const Replicate<BasisVectorType, Dimension, 1> ctrl_weights(basis_funcs);
  const Block<const ControlPointVectorType, Dimension, Order> ctrl_pts(ctrls(), 0, span - p, Dimension, p + 1);
  return (ctrl_weights * ctrl_pts).rowwise().sum();
}

/* --------------------------------------------------------------------------------------------- */

template <typename SplineType, typename DerivativeType>
void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der) {
  enum { Dimension = SplineTraits<SplineType>::Dimension };
  enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
  enum { DerivativeOrder = DerivativeType::ColsAtCompileTime };

  typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
  typedef typename SplineTraits<SplineType, DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
  typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;

  const DenseIndex p = spline.degree();
  const DenseIndex span = spline.span(u);

  const DenseIndex n = (std::min)(p, order);

  der.resize(Dimension, n + 1);

  // Retrieve the basis function derivatives up to the desired order...
  const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n + 1);

  // ... and perform the linear combinations of the control points.
  for (DenseIndex der_order = 0; der_order < n + 1; ++der_order) {
    const Replicate<BasisDerivativeRowXpr, Dimension, 1> ctrl_weights(basis_func_ders.row(der_order));
    const Block<const ControlPointVectorType, Dimension, Order> ctrl_pts(spline.ctrls(), 0, span - p, Dimension, p + 1);
    der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
  }
}

template <typename Scalar_, int Dim_, int Degree_>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::DerivativeType Spline<Scalar_, Dim_, Degree_>::derivatives(
    Scalar u, DenseIndex order) const {
  typename SplineTraits<Spline>::DerivativeType res;
  derivativesImpl(*this, u, order, res);
  return res;
}

template <typename Scalar_, int Dim_, int Degree_>
template <int DerivativeOrder>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_>, DerivativeOrder>::DerivativeType
Spline<Scalar_, Dim_, Degree_>::derivatives(Scalar u, DenseIndex order) const {
  typename SplineTraits<Spline, DerivativeOrder>::DerivativeType res;
  derivativesImpl(*this, u, order, res);
  return res;
}

template <typename Scalar_, int Dim_, int Degree_>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::BasisVectorType Spline<Scalar_, Dim_, Degree_>::basisFunctions(
    Scalar u) const {
  return Spline::BasisFunctions(u, degree(), knots());
}

/* --------------------------------------------------------------------------------------------- */

template <typename Scalar_, int Dim_, int Degree_>
template <typename DerivativeType>
void Spline<Scalar_, Dim_, Degree_>::BasisFunctionDerivativesImpl(
    const typename Spline<Scalar_, Dim_, Degree_>::Scalar u, const DenseIndex order, const DenseIndex p,
    const typename Spline<Scalar_, Dim_, Degree_>::KnotVectorType& U, DerivativeType& N_) {
  typedef Spline<Scalar_, Dim_, Degree_> SplineType;
  enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };

  const DenseIndex span = SplineType::Span(u, p, U);

  const DenseIndex n = (std::min)(p, order);

  N_.resize(n + 1, p + 1);

  BasisVectorType left = BasisVectorType::Zero(p + 1);
  BasisVectorType right = BasisVectorType::Zero(p + 1);

  Matrix<Scalar, Order, Order> ndu(p + 1, p + 1);

  Scalar saved, temp;  // FIXME These were double instead of Scalar. Was there a reason for that?

  ndu(0, 0) = 1.0;

  DenseIndex j;
  for (j = 1; j <= p; ++j) {
    left[j] = u - U[span + 1 - j];
    right[j] = U[span + j] - u;
    saved = 0.0;

    for (DenseIndex r = 0; r < j; ++r) {
      /* Lower triangle */
      ndu(j, r) = right[r + 1] + left[j - r];
      temp = ndu(r, j - 1) / ndu(j, r);
      /* Upper triangle */
      ndu(r, j) = static_cast<Scalar>(saved + right[r + 1] * temp);
      saved = left[j - r] * temp;
    }

    ndu(j, j) = static_cast<Scalar>(saved);
  }

  for (j = p; j >= 0; --j) N_(0, j) = ndu(j, p);

  // Compute the derivatives
  DerivativeType a(n + 1, p + 1);
  DenseIndex r = 0;
  for (; r <= p; ++r) {
    DenseIndex s1, s2;
    s1 = 0;
    s2 = 1;  // alternate rows in array a
    a(0, 0) = 1.0;

    // Compute the k-th derivative
    for (DenseIndex k = 1; k <= static_cast<DenseIndex>(n); ++k) {
      Scalar d = 0.0;
      DenseIndex rk, pk, j1, j2;
      rk = r - k;
      pk = p - k;

      if (r >= k) {
        a(s2, 0) = a(s1, 0) / ndu(pk + 1, rk);
        d = a(s2, 0) * ndu(rk, pk);
      }

      if (rk >= -1)
        j1 = 1;
      else
        j1 = -rk;

      if (r - 1 <= pk)
        j2 = k - 1;
      else
        j2 = p - r;

      for (j = j1; j <= j2; ++j) {
        a(s2, j) = (a(s1, j) - a(s1, j - 1)) / ndu(pk + 1, rk + j);
        d += a(s2, j) * ndu(rk + j, pk);
      }

      if (r <= pk) {
        a(s2, k) = -a(s1, k - 1) / ndu(pk + 1, r);
        d += a(s2, k) * ndu(r, pk);
      }

      N_(k, r) = static_cast<Scalar>(d);
      j = s1;
      s1 = s2;
      s2 = j;  // Switch rows
    }
  }

  /* Multiply through by the correct factors */
  /* (Eq. [2.9])                             */
  r = p;
  for (DenseIndex k = 1; k <= static_cast<DenseIndex>(n); ++k) {
    for (j = p; j >= 0; --j) N_(k, j) *= r;
    r *= p - k;
  }
}

template <typename Scalar_, int Dim_, int Degree_>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::BasisDerivativeType
Spline<Scalar_, Dim_, Degree_>::basisFunctionDerivatives(Scalar u, DenseIndex order) const {
  typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::BasisDerivativeType der;
  BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
  return der;
}

template <typename Scalar_, int Dim_, int Degree_>
template <int DerivativeOrder>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_>, DerivativeOrder>::BasisDerivativeType
Spline<Scalar_, Dim_, Degree_>::basisFunctionDerivatives(Scalar u, DenseIndex order) const {
  typename SplineTraits<Spline<Scalar_, Dim_, Degree_>, DerivativeOrder>::BasisDerivativeType der;
  BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
  return der;
}

template <typename Scalar_, int Dim_, int Degree_>
typename SplineTraits<Spline<Scalar_, Dim_, Degree_> >::BasisDerivativeType
Spline<Scalar_, Dim_, Degree_>::BasisFunctionDerivatives(
    const typename Spline<Scalar_, Dim_, Degree_>::Scalar u, const DenseIndex order, const DenseIndex degree,
    const typename Spline<Scalar_, Dim_, Degree_>::KnotVectorType& knots) {
  typename SplineTraits<Spline>::BasisDerivativeType der;
  BasisFunctionDerivativesImpl(u, order, degree, knots, der);
  return der;
}
}  // namespace Eigen

#endif  // EIGEN_SPLINE_H