unsupported/Eigen/src/Splines/SplineFitting.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_FITTING_H
#define EIGEN_SPLINE_FITTING_H

#include <algorithm>
#include <functional>
#include <numeric>
#include <vector>

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

#include "SplineFwd.h"

#include "../../../../Eigen/LU"
#include "../../../../Eigen/QR"

namespace Eigen {
/**
 * \brief Computes knot averages.
 * \ingroup Splines_Module
 *
 * The knots are computed as
 * \f{align*}
 *  u_0 & = \hdots = u_p = 0 \\
 *  u_{m-p} & = \hdots = u_{m} = 1 \\
 *  u_{j+p} & = \frac{1}{p}\sum_{i=j}^{j+p-1}\bar{u}_i \quad\quad j=1,\hdots,n-p
 * \f}
 * where \f$p\f$ is the degree and \f$m+1\f$ the number knots
 * of the desired interpolating spline.
 *
 * \param[in] parameters The input parameters. During interpolation one for each data point.
 * \param[in] degree The spline degree which is used during the interpolation.
 * \param[out] knots The output knot vector.
 *
 * \sa Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data
 **/
template <typename KnotVectorType>
void KnotAveraging(const KnotVectorType& parameters, DenseIndex degree, KnotVectorType& knots) {
  knots.resize(parameters.size() + degree + 1);

  for (DenseIndex j = 1; j < parameters.size() - degree; ++j) knots(j + degree) = parameters.segment(j, degree).mean();

  knots.segment(0, degree + 1) = KnotVectorType::Zero(degree + 1);
  knots.segment(knots.size() - degree - 1, degree + 1) = KnotVectorType::Ones(degree + 1);
}

/**
 * \brief Computes knot averages when derivative constraints are present.
 * Note that this is a technical interpretation of the referenced article
 * since the algorithm contained therein is incorrect as written.
 * \ingroup Splines_Module
 *
 * \param[in] parameters The parameters at which the interpolation B-Spline
 *            will intersect the given interpolation points. The parameters
 *            are assumed to be a non-decreasing sequence.
 * \param[in] degree The degree of the interpolating B-Spline. This must be
 *            greater than zero.
 * \param[in] derivativeIndices The indices corresponding to parameters at
 *            which there are derivative constraints. The indices are assumed
 *            to be a non-decreasing sequence.
 * \param[out] knots The calculated knot vector. These will be returned as a
 *             non-decreasing sequence
 *
 * \sa Les A. Piegl, Khairan Rajab, Volha Smarodzinana. 2008.
 * Curve interpolation with directional constraints for engineering design.
 * Engineering with Computers
 **/
template <typename KnotVectorType, typename ParameterVectorType, typename IndexArray>
void KnotAveragingWithDerivatives(const ParameterVectorType& parameters, const unsigned int degree,
                                  const IndexArray& derivativeIndices, KnotVectorType& knots) {
  typedef typename ParameterVectorType::Scalar Scalar;

  DenseIndex numParameters = parameters.size();
  DenseIndex numDerivatives = derivativeIndices.size();

  if (numDerivatives < 1) {
    KnotAveraging(parameters, degree, knots);
    return;
  }

  DenseIndex startIndex;
  DenseIndex endIndex;

  DenseIndex numInternalDerivatives = numDerivatives;

  if (derivativeIndices[0] == 0) {
    startIndex = 0;
    --numInternalDerivatives;
  } else {
    startIndex = 1;
  }
  if (derivativeIndices[numDerivatives - 1] == numParameters - 1) {
    endIndex = numParameters - degree;
    --numInternalDerivatives;
  } else {
    endIndex = numParameters - degree - 1;
  }

  // There are (endIndex - startIndex + 1) knots obtained from the averaging
  // and 2 for the first and last parameters.
  DenseIndex numAverageKnots = endIndex - startIndex + 3;
  KnotVectorType averageKnots(numAverageKnots);
  averageKnots[0] = parameters[0];

  int newKnotIndex = 0;
  for (DenseIndex i = startIndex; i <= endIndex; ++i)
    averageKnots[++newKnotIndex] = parameters.segment(i, degree).mean();
  averageKnots[++newKnotIndex] = parameters[numParameters - 1];

  newKnotIndex = -1;

  ParameterVectorType temporaryParameters(numParameters + 1);
  KnotVectorType derivativeKnots(numInternalDerivatives);
  for (DenseIndex i = 0; i < numAverageKnots - 1; ++i) {
    temporaryParameters[0] = averageKnots[i];
    ParameterVectorType parameterIndices(numParameters);
    int temporaryParameterIndex = 1;
    for (DenseIndex j = 0; j < numParameters; ++j) {
      Scalar parameter = parameters[j];
      if (parameter >= averageKnots[i] && parameter < averageKnots[i + 1]) {
        parameterIndices[temporaryParameterIndex] = j;
        temporaryParameters[temporaryParameterIndex++] = parameter;
      }
    }
    temporaryParameters[temporaryParameterIndex] = averageKnots[i + 1];

    for (int j = 0; j <= temporaryParameterIndex - 2; ++j) {
      for (DenseIndex k = 0; k < derivativeIndices.size(); ++k) {
        if (parameterIndices[j + 1] == derivativeIndices[k] && parameterIndices[j + 1] != 0 &&
            parameterIndices[j + 1] != numParameters - 1) {
          derivativeKnots[++newKnotIndex] = temporaryParameters.segment(j, 3).mean();
          break;
        }
      }
    }
  }

  KnotVectorType temporaryKnots(averageKnots.size() + derivativeKnots.size());

  std::merge(averageKnots.data(), averageKnots.data() + averageKnots.size(), derivativeKnots.data(),
             derivativeKnots.data() + derivativeKnots.size(), temporaryKnots.data());

  // Number of knots (one for each point and derivative) plus spline order.
  DenseIndex numKnots = numParameters + numDerivatives + degree + 1;
  knots.resize(numKnots);

  knots.head(degree).fill(temporaryKnots[0]);
  knots.tail(degree).fill(temporaryKnots.template tail<1>()[0]);
  knots.segment(degree, temporaryKnots.size()) = temporaryKnots;
}

/**
 * \brief Computes chord length parameters which are required for spline interpolation.
 * \ingroup Splines_Module
 *
 * \param[in] pts The data points to which a spline should be fit.
 * \param[out] chord_lengths The resulting chord length vector.
 *
 * \sa Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data
 **/
template <typename PointArrayType, typename KnotVectorType>
void ChordLengths(const PointArrayType& pts, KnotVectorType& chord_lengths) {
  typedef typename KnotVectorType::Scalar Scalar;

  const DenseIndex n = pts.cols();

  // 1. compute the column-wise norms
  chord_lengths.resize(pts.cols());
  chord_lengths[0] = 0;
  chord_lengths.rightCols(n - 1) =
      (pts.array().leftCols(n - 1) - pts.array().rightCols(n - 1)).matrix().colwise().norm();

  // 2. compute the partial sums
  std::partial_sum(chord_lengths.data(), chord_lengths.data() + n, chord_lengths.data());

  // 3. normalize the data
  chord_lengths /= chord_lengths(n - 1);
  chord_lengths(n - 1) = Scalar(1);
}

/**
 * \brief Spline fitting methods.
 * \ingroup Splines_Module
 **/
template <typename SplineType>
struct SplineFitting {
  typedef typename SplineType::KnotVectorType KnotVectorType;
  typedef typename SplineType::ParameterVectorType ParameterVectorType;

  /**
   * \brief Fits an interpolating Spline to the given data points.
   *
   * \param pts The points for which an interpolating spline will be computed.
   * \param degree The degree of the interpolating spline.
   *
   * \returns A spline interpolating the initially provided points.
   **/
  template <typename PointArrayType>
  static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree);

  /**
   * \brief Fits an interpolating Spline to the given data points.
   *
   * \param pts The points for which an interpolating spline will be computed.
   * \param degree The degree of the interpolating spline.
   * \param knot_parameters The knot parameters for the interpolation.
   *
   * \returns A spline interpolating the initially provided points.
   **/
  template <typename PointArrayType>
  static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree, const KnotVectorType& knot_parameters);

  /**
   * \brief Fits an interpolating spline to the given data points and
   * derivatives.
   *
   * \param points The points for which an interpolating spline will be computed.
   * \param derivatives The desired derivatives of the interpolating spline at interpolation
   *                    points.
   * \param derivativeIndices An array indicating which point each derivative belongs to. This
   *                          must be the same size as @a derivatives.
   * \param degree The degree of the interpolating spline.
   *
   * \returns A spline interpolating @a points with @a derivatives at those points.
   *
   * \sa Les A. Piegl, Khairan Rajab, Volha Smarodzinana. 2008.
   * Curve interpolation with directional constraints for engineering design.
   * Engineering with Computers
   **/
  template <typename PointArrayType, typename IndexArray>
  static SplineType InterpolateWithDerivatives(const PointArrayType& points, const PointArrayType& derivatives,
                                               const IndexArray& derivativeIndices, const unsigned int degree);

  /**
   * \brief Fits an interpolating spline to the given data points and derivatives.
   *
   * \param points The points for which an interpolating spline will be computed.
   * \param derivatives The desired derivatives of the interpolating spline at interpolation points.
   * \param derivativeIndices An array indicating which point each derivative belongs to. This
   *                          must be the same size as @a derivatives.
   * \param degree The degree of the interpolating spline.
   * \param parameters The parameters corresponding to the interpolation points.
   *
   * \returns A spline interpolating @a points with @a derivatives at those points.
   *
   * \sa Les A. Piegl, Khairan Rajab, Volha Smarodzinana. 2008.
   * Curve interpolation with directional constraints for engineering design.
   * Engineering with Computers
   */
  template <typename PointArrayType, typename IndexArray>
  static SplineType InterpolateWithDerivatives(const PointArrayType& points, const PointArrayType& derivatives,
                                               const IndexArray& derivativeIndices, const unsigned int degree,
                                               const ParameterVectorType& parameters);
};

template <typename SplineType>
template <typename PointArrayType>
SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree,
                                                  const KnotVectorType& knot_parameters) {
  typedef typename SplineType::KnotVectorType::Scalar Scalar;
  typedef typename SplineType::ControlPointVectorType ControlPointVectorType;

  typedef Matrix<Scalar, Dynamic, Dynamic> MatrixType;

  KnotVectorType knots;
  KnotAveraging(knot_parameters, degree, knots);

  DenseIndex n = pts.cols();
  MatrixType A = MatrixType::Zero(n, n);
  for (DenseIndex i = 1; i < n - 1; ++i) {
    const DenseIndex span = SplineType::Span(knot_parameters[i], degree, knots);

    // The segment call should somehow be told the spline order at compile time.
    A.row(i).segment(span - degree, degree + 1) = SplineType::BasisFunctions(knot_parameters[i], degree, knots);
  }
  A(0, 0) = 1.0;
  A(n - 1, n - 1) = 1.0;

  HouseholderQR<MatrixType> qr(A);

  // Here, we are creating a temporary due to an Eigen issue.
  ControlPointVectorType ctrls = qr.solve(MatrixType(pts.transpose())).transpose();

  return SplineType(knots, ctrls);
}

template <typename SplineType>
template <typename PointArrayType>
SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree) {
  KnotVectorType chord_lengths;  // knot parameters
  ChordLengths(pts, chord_lengths);
  return Interpolate(pts, degree, chord_lengths);
}

template <typename SplineType>
template <typename PointArrayType, typename IndexArray>
SplineType SplineFitting<SplineType>::InterpolateWithDerivatives(const PointArrayType& points,
                                                                 const PointArrayType& derivatives,
                                                                 const IndexArray& derivativeIndices,
                                                                 const unsigned int degree,
                                                                 const ParameterVectorType& parameters) {
  typedef typename SplineType::KnotVectorType::Scalar Scalar;
  typedef typename SplineType::ControlPointVectorType ControlPointVectorType;

  typedef Matrix<Scalar, Dynamic, Dynamic> MatrixType;

  const DenseIndex n = points.cols() + derivatives.cols();

  KnotVectorType knots;

  KnotAveragingWithDerivatives(parameters, degree, derivativeIndices, knots);

  // fill matrix
  MatrixType A = MatrixType::Zero(n, n);

  // Use these dimensions for quicker populating, then transpose for solving.
  MatrixType b(points.rows(), n);

  DenseIndex startRow;
  DenseIndex derivativeStart;

  // End derivatives.
  if (derivativeIndices[0] == 0) {
    A.template block<1, 2>(1, 0) << -1, 1;

    Scalar y = (knots(degree + 1) - knots(0)) / degree;
    b.col(1) = y * derivatives.col(0);

    startRow = 2;
    derivativeStart = 1;
  } else {
    startRow = 1;
    derivativeStart = 0;
  }
  if (derivativeIndices[derivatives.cols() - 1] == points.cols() - 1) {
    A.template block<1, 2>(n - 2, n - 2) << -1, 1;

    Scalar y = (knots(knots.size() - 1) - knots(knots.size() - (degree + 2))) / degree;
    b.col(b.cols() - 2) = y * derivatives.col(derivatives.cols() - 1);
  }

  DenseIndex row = startRow;
  DenseIndex derivativeIndex = derivativeStart;
  for (DenseIndex i = 1; i < parameters.size() - 1; ++i) {
    const DenseIndex span = SplineType::Span(parameters[i], degree, knots);

    if (derivativeIndex < derivativeIndices.size() && derivativeIndices[derivativeIndex] == i) {
      A.block(row, span - degree, 2, degree + 1) =
          SplineType::BasisFunctionDerivatives(parameters[i], 1, degree, knots);

      b.col(row++) = points.col(i);
      b.col(row++) = derivatives.col(derivativeIndex++);
    } else {
      A.row(row).segment(span - degree, degree + 1) = SplineType::BasisFunctions(parameters[i], degree, knots);
      b.col(row++) = points.col(i);
    }
  }
  b.col(0) = points.col(0);
  b.col(b.cols() - 1) = points.col(points.cols() - 1);
  A(0, 0) = 1;
  A(n - 1, n - 1) = 1;

  // Solve
  FullPivLU<MatrixType> lu(A);
  ControlPointVectorType controlPoints = lu.solve(MatrixType(b.transpose())).transpose();

  SplineType spline(knots, controlPoints);

  return spline;
}

template <typename SplineType>
template <typename PointArrayType, typename IndexArray>
SplineType SplineFitting<SplineType>::InterpolateWithDerivatives(const PointArrayType& points,
                                                                 const PointArrayType& derivatives,
                                                                 const IndexArray& derivativeIndices,
                                                                 const unsigned int degree) {
  ParameterVectorType parameters;
  ChordLengths(points, parameters);
  return InterpolateWithDerivatives(points, derivatives, derivativeIndices, degree, parameters);
}
}  // namespace Eigen

#endif  // EIGEN_SPLINE_FITTING_H