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// 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 <numeric>
#include "SplineFwd.h"
#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 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 lenggth 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;
/**
* \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);
};
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);
}
}
#endif // EIGEN_SPLINE_FITTING_H
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