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 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
