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 (at) 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 19 namespace 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 \\ 29 * u_{j+p} & = \frac{1}{p}\sum_{i=j}^{j+p-1}\bar{u}_i \quad\quad j=1,\hdots,n-p 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 typedef typename KnotVectorType::Scalar Scalar; 44 45 knots.resize(parameters.size()+degree+1); 46 47 for (DenseIndex j=1; j<parameters.size()-degree; ++j) 48 knots(j+degree) = parameters.segment(j,degree).mean(); 49 50 knots.segment(0,degree+1) = KnotVectorType::Zero(degree+1); 51 knots.segment(knots.size()-degree-1,degree+1) = KnotVectorType::Ones(degree+1); 52 } 53 54 /** 55 * \brief Computes chord length parameters which are required for spline interpolation. 56 * \ingroup Splines_Module 57 * 58 * \param[in] pts The data points to which a spline should be fit. 59 * \param[out] chord_lengths The resulting chord lenggth vector. 60 * 61 * \sa Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data 62 **/ 63 template <typename PointArrayType, typename KnotVectorType> 64 void ChordLengths(const PointArrayType& pts, KnotVectorType& chord_lengths) 65 { 66 typedef typename KnotVectorType::Scalar Scalar; 67 68 const DenseIndex n = pts.cols(); 69 70 // 1. compute the column-wise norms 71 chord_lengths.resize(pts.cols()); 72 chord_lengths[0] = 0; 73 chord_lengths.rightCols(n-1) = (pts.array().leftCols(n-1) - pts.array().rightCols(n-1)).matrix().colwise().norm(); 74 75 // 2. compute the partial sums 76 std::partial_sum(chord_lengths.data(), chord_lengths.data()+n, chord_lengths.data()); 77 78 // 3. normalize the data 79 chord_lengths /= chord_lengths(n-1); 80 chord_lengths(n-1) = Scalar(1); 81 } 82 83 /** 84 * \brief Spline fitting methods. 85 * \ingroup Splines_Module 86 **/ 87 template <typename SplineType> 88 struct SplineFitting 89 { 90 typedef typename SplineType::KnotVectorType KnotVectorType; 91 92 /** 93 * \brief Fits an interpolating Spline to the given data points. 94 * 95 * \param pts The points for which an interpolating spline will be computed. 96 * \param degree The degree of the interpolating spline. 97 * 98 * \returns A spline interpolating the initially provided points. 99 **/ 100 template <typename PointArrayType> 101 static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree); 102 103 /** 104 * \brief Fits an interpolating Spline to the given data points. 105 * 106 * \param pts The points for which an interpolating spline will be computed. 107 * \param degree The degree of the interpolating spline. 108 * \param knot_parameters The knot parameters for the interpolation. 109 * 110 * \returns A spline interpolating the initially provided points. 111 **/ 112 template <typename PointArrayType> 113 static SplineType Interpolate(const PointArrayType& pts, DenseIndex degree, const KnotVectorType& knot_parameters); 114 }; 115 116 template <typename SplineType> 117 template <typename PointArrayType> 118 SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree, const KnotVectorType& knot_parameters) 119 { 120 typedef typename SplineType::KnotVectorType::Scalar Scalar; 121 typedef typename SplineType::BasisVectorType BasisVectorType; 122 typedef typename SplineType::ControlPointVectorType ControlPointVectorType; 123 124 typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType; 125 126 KnotVectorType knots; 127 KnotAveraging(knot_parameters, degree, knots); 128 129 DenseIndex n = pts.cols(); 130 MatrixType A = MatrixType::Zero(n,n); 131 for (DenseIndex i=1; i<n-1; ++i) 132 { 133 const DenseIndex span = SplineType::Span(knot_parameters[i], degree, knots); 134 135 // The segment call should somehow be told the spline order at compile time. 136 A.row(i).segment(span-degree, degree+1) = SplineType::BasisFunctions(knot_parameters[i], degree, knots); 137 } 138 A(0,0) = 1.0; 139 A(n-1,n-1) = 1.0; 140 141 HouseholderQR<MatrixType> qr(A); 142 143 // Here, we are creating a temporary due to an Eigen issue. 144 ControlPointVectorType ctrls = qr.solve(MatrixType(pts.transpose())).transpose(); 145 146 return SplineType(knots, ctrls); 147 } 148 149 template <typename SplineType> 150 template <typename PointArrayType> 151 SplineType SplineFitting<SplineType>::Interpolate(const PointArrayType& pts, DenseIndex degree) 152 { 153 KnotVectorType chord_lengths; // knot parameters 154 ChordLengths(pts, chord_lengths); 155 return Interpolate(pts, degree, chord_lengths); 156 } 157 } 158 159 #endif // EIGEN_SPLINE_FITTING_H 160
