/* <![CDATA[ */
function get_sym_list(){return [["Macro","xm",[["EIGEN_SPLINE_FITTING_H",11]]],["Namespace","xn",[["Eigen",19]]],["Struct","xs",[["SplineFitting",86]]],["Typedef","xt",[["ControlPointVectorType",119],["KnotVectorType",88],["MatrixType",121],["Scalar",64],["Scalar",118]]],["Function","xf",[["ChordLengths",62],["Interpolate",116],["Interpolate",148],["KnotAveraging",41]]]];} /* ]]> */1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
5//
6// This Source Code Form is subject to the terms of the Mozilla
7// Public License v. 2.0. If a copy of the MPL was not distributed
8// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10#ifndef EIGEN_SPLINE_FITTING_H
11#define EIGEN_SPLINE_FITTING_H
12
13#include <numeric>
14
15#include "SplineFwd.h"
16
17#include <Eigen/QR>
18
19namespace Eigen
20{
21  /**
22   * \brief Computes knot averages.
23   * \ingroup Splines_Module
24   *
25   * The knots are computed as
26   * \f{align*}
27   *  u_0 & = \hdots = u_p = 0 \\
28   *  u_{m-p} & = \hdots = u_{m} = 1 \\
30   * \f}
31   * where \f$p\f$ is the degree and \f$m+1\f$ the number knots
32   * of the desired interpolating spline.
33   *
34   * \param[in] parameters The input parameters. During interpolation one for each data point.
35   * \param[in] degree The spline degree which is used during the interpolation.
36   * \param[out] knots The output knot vector.
37   *
38   * \sa Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data
39   **/
40  template <typename KnotVectorType>
41  void KnotAveraging(const KnotVectorType& parameters, DenseIndex degree, KnotVectorType& knots)
42  {
43    knots.resize(parameters.size()+degree+1);
44
45    for (DenseIndex j=1; j<parameters.size()-degree; ++j)
46      knots(j+degree) = parameters.segment(j,degree).mean();
47
48    knots.segment(0,degree+1) = KnotVectorType::Zero(degree+1);
49    knots.segment(knots.size()-degree-1,degree+1) = KnotVectorType::Ones(degree+1);
50  }
51
52  /**
53   * \brief Computes chord length parameters which are required for spline interpolation.
54   * \ingroup Splines_Module
55   *
56   * \param[in] pts The data points to which a spline should be fit.
57   * \param[out] chord_lengths The resulting chord lenggth vector.
58   *
59   * \sa Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data
60   **/
61  template <typename PointArrayType, typename KnotVectorType>
62  void ChordLengths(const PointArrayType& pts, KnotVectorType& chord_lengths)
63  {
64    typedef typename KnotVectorType::Scalar Scalar;
65
66    const DenseIndex n = pts.cols();
67
68    // 1. compute the column-wise norms
69    chord_lengths.resize(pts.cols());
70    chord_lengths[0] = 0;
71    chord_lengths.rightCols(n-1) = (pts.array().leftCols(n-1) - pts.array().rightCols(n-1)).matrix().colwise().norm();
72
73    // 2. compute the partial sums
74    std::partial_sum(chord_lengths.data(), chord_lengths.data()+n, chord_lengths.data());
75
76    // 3. normalize the data
77    chord_lengths /= chord_lengths(n-1);
78    chord_lengths(n-1) = Scalar(1);
79  }
80
81  /**
82   * \brief Spline fitting methods.
83   * \ingroup Splines_Module
84   **/
85  template <typename SplineType>
86  struct SplineFitting
87  {
88    typedef typename SplineType::KnotVectorType KnotVectorType;
89
90    /**
91     * \brief Fits an interpolating Spline to the given data points.
92     *
93     * \param pts The points for which an interpolating spline will be computed.
94     * \param degree The degree of the interpolating spline.
95     *
96     * \returns A spline interpolating the initially provided points.
97     **/
98    template <typename PointArrayType>
99    static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree);
100
101    /**
102     * \brief Fits an interpolating Spline to the given data points.
103     *
104     * \param pts The points for which an interpolating spline will be computed.
105     * \param degree The degree of the interpolating spline.
106     * \param knot_parameters The knot parameters for the interpolation.
107     *
108     * \returns A spline interpolating the initially provided points.
109     **/
110    template <typename PointArrayType>
111    static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree, const KnotVectorType& knot_parameters);
112  };
113
114  template <typename SplineType>
115  template <typename PointArrayType>
116  SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree, const KnotVectorType& knot_parameters)
117  {
118    typedef typename SplineType::KnotVectorType::Scalar Scalar;
119    typedef typename SplineType::ControlPointVectorType ControlPointVectorType;
120
121    typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType;
122
123    KnotVectorType knots;
124    KnotAveraging(knot_parameters, degree, knots);
125
126    DenseIndex n = pts.cols();
127    MatrixType A = MatrixType::Zero(n,n);
128    for (DenseIndex i=1; i<n-1; ++i)
129    {
130      const DenseIndex span = SplineType::Span(knot_parameters[i], degree, knots);
131
132      // The segment call should somehow be told the spline order at compile time.
133      A.row(i).segment(span-degree, degree+1) = SplineType::BasisFunctions(knot_parameters[i], degree, knots);
134    }
135    A(0,0) = 1.0;
136    A(n-1,n-1) = 1.0;
137
138    HouseholderQR<MatrixType> qr(A);
139
140    // Here, we are creating a temporary due to an Eigen issue.
141    ControlPointVectorType ctrls = qr.solve(MatrixType(pts.transpose())).transpose();
142
143    return SplineType(knots, ctrls);
144  }
145
146  template <typename SplineType>
147  template <typename PointArrayType>
148  SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree)
149  {
150    KnotVectorType chord_lengths; // knot parameters
151    ChordLengths(pts, chord_lengths);
152    return Interpolate(pts, degree, chord_lengths);
153  }
154}
155
156#endif // EIGEN_SPLINE_FITTING_H
157