vatger/aman-es
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

add Eigen as a dependency

Browse Source
This commit is contained in:
Sven Czarnian
2021-12-16 15:59:56 +01:00
parent a08ac9b244
commit 27b422d806
479 changed files with 167893 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files

507
external/include/eigen3/unsupported/Eigen/src/Splines/Spline.h vendored Normal file
View File

@@ -0,0 +1,507 @@
// 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_H
#define EIGEN_SPLINE_H
#include "SplineFwd.h"
namespace Eigen
{
/**
* \ingroup Splines_Module
* \class Spline
* \brief A class representing multi-dimensional spline curves.
*
* The class represents B-splines with non-uniform knot vectors. Each control
* point of the B-spline is associated with a basis function
* \f{align*}
* C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
* \f}
*
* \tparam _Scalar The underlying data type (typically float or double)
* \tparam _Dim The curve dimension (e.g. 2 or 3)
* \tparam _Degree Per default set to Dynamic; could be set to the actual desired
* degree for optimization purposes (would result in stack allocation
* of several temporary variables).
**/
template <typename _Scalar, int _Dim, int _Degree>
class Spline
{
public:
typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
enum { Dimension = _Dim /*!< The spline curve's dimension. */ };
enum { Degree = _Degree /*!< The spline curve's degree. */ };
/** \brief The point type the spline is representing. */
typedef typename SplineTraits<Spline>::PointType PointType;
/** \brief The data type used to store knot vectors. */
typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;
/** \brief The data type used to store parameter vectors. */
typedef typename SplineTraits<Spline>::ParameterVectorType ParameterVectorType;
/** \brief The data type used to store non-zero basis functions. */
typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;
/** \brief The data type used to store the values of the basis function derivatives. */
typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;
/** \brief The data type representing the spline's control points. */
typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;
/**
* \brief Creates a (constant) zero spline.
* For Splines with dynamic degree, the resulting degree will be 0.
**/
Spline()
: m_knots(1, (Degree==Dynamic ? 2 : 2*Degree+2))
, m_ctrls(ControlPointVectorType::Zero(Dimension,(Degree==Dynamic ? 1 : Degree+1)))
{
// in theory this code can go to the initializer list but it will get pretty
// much unreadable ...
enum { MinDegree = (Degree==Dynamic ? 0 : Degree) };
m_knots.template segment<MinDegree+1>(0) = Array<Scalar,1,MinDegree+1>::Zero();
m_knots.template segment<MinDegree+1>(MinDegree+1) = Array<Scalar,1,MinDegree+1>::Ones();
}
/**
* \brief Creates a spline from a knot vector and control points.
* \param knots The spline's knot vector.
* \param ctrls The spline's control point vector.
**/
template <typename OtherVectorType, typename OtherArrayType>
Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {}
/**
* \brief Copy constructor for splines.
* \param spline The input spline.
**/
template <int OtherDegree>
Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) :
m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}
/**
* \brief Returns the knots of the underlying spline.
**/
const KnotVectorType& knots() const { return m_knots; }
/**
* \brief Returns the ctrls of the underlying spline.
**/
const ControlPointVectorType& ctrls() const { return m_ctrls; }
/**
* \brief Returns the spline value at a given site \f$u\f$.
*
* The function returns
* \f{align*}
* C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
* \f}
*
* \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
* \return The spline value at the given location \f$u\f$.
**/
PointType operator()(Scalar u) const;
/**
* \brief Evaluation of spline derivatives of up-to given order.
*
* The function returns
* \f{align*}
* \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
* \f}
* for i ranging between 0 and order.
*
* \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
* \param order The order up to which the derivatives are computed.
**/
typename SplineTraits<Spline>::DerivativeType
derivatives(Scalar u, DenseIndex order) const;
/**
* \copydoc Spline::derivatives
* Using the template version of this function is more efficieent since
* temporary objects are allocated on the stack whenever this is possible.
**/
template <int DerivativeOrder>
typename SplineTraits<Spline,DerivativeOrder>::DerivativeType
derivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
/**
* \brief Computes the non-zero basis functions at the given site.
*
* Splines have local support and a point from their image is defined
* by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
* spline degree.
*
* This function computes the \f$p+1\f$ non-zero basis function values
* for a given parameter value \f$u\f$. It returns
* \f{align*}{
* N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
* \f}
*
* \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions
* are computed.
**/
typename SplineTraits<Spline>::BasisVectorType
basisFunctions(Scalar u) const;
/**
* \brief Computes the non-zero spline basis function derivatives up to given order.
*
* The function computes
* \f{align*}{
* \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
* \f}
* with i ranging from 0 up to the specified order.
*
* \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
* derivatives are computed.
* \param order The order up to which the basis function derivatives are computes.
**/
typename SplineTraits<Spline>::BasisDerivativeType
basisFunctionDerivatives(Scalar u, DenseIndex order) const;
/**
* \copydoc Spline::basisFunctionDerivatives
* Using the template version of this function is more efficieent since
* temporary objects are allocated on the stack whenever this is possible.
**/
template <int DerivativeOrder>
typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType
basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
/**
* \brief Returns the spline degree.
**/
DenseIndex degree() const;
/**
* \brief Returns the span within the knot vector in which u is falling.
* \param u The site for which the span is determined.
**/
DenseIndex span(Scalar u) const;
/**
* \brief Computes the spang within the provided knot vector in which u is falling.
**/
static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots);
/**
* \brief Returns the spline's non-zero basis functions.
*
* The function computes and returns
* \f{align*}{
* N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
* \f}
*
* \param u The site at which the basis functions are computed.
* \param degree The degree of the underlying spline.
* \param knots The underlying spline's knot vector.
**/
static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);
/**
* \copydoc Spline::basisFunctionDerivatives
* \param degree The degree of the underlying spline
* \param knots The underlying spline's knot vector.
**/
static BasisDerivativeType BasisFunctionDerivatives(
const Scalar u, const DenseIndex order, const DenseIndex degree, const KnotVectorType& knots);
private:
KnotVectorType m_knots; /*!< Knot vector. */
ControlPointVectorType m_ctrls; /*!< Control points. */
template <typename DerivativeType>
static void BasisFunctionDerivativesImpl(
const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
const DenseIndex order,
const DenseIndex p,
const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
DerivativeType& N_);
};
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::Span(
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::Scalar u,
DenseIndex degree,
const typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::KnotVectorType& knots)
{
// Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
if (u <= knots(0)) return degree;
const Scalar* pos = std::upper_bound(knots.data()+degree-1, knots.data()+knots.size()-degree-1, u);
return static_cast<DenseIndex>( std::distance(knots.data(), pos) - 1 );
}
template <typename _Scalar, int _Dim, int _Degree>
typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType
Spline<_Scalar, _Dim, _Degree>::BasisFunctions(
typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
DenseIndex degree,
const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
{
const DenseIndex p = degree;
const DenseIndex i = Spline::Span(u, degree, knots);
const KnotVectorType& U = knots;
BasisVectorType left(p+1); left(0) = Scalar(0);
BasisVectorType right(p+1); right(0) = Scalar(0);
VectorBlock<BasisVectorType,Degree>(left,1,p) = u - VectorBlock<const KnotVectorType,Degree>(U,i+1-p,p).reverse();
VectorBlock<BasisVectorType,Degree>(right,1,p) = VectorBlock<const KnotVectorType,Degree>(U,i+1,p) - u;
BasisVectorType N(1,p+1);
N(0) = Scalar(1);
for (DenseIndex j=1; j<=p; ++j)
{
Scalar saved = Scalar(0);
for (DenseIndex r=0; r<j; r++)
{
const Scalar tmp = N(r)/(right(r+1)+left(j-r));
N[r] = saved + right(r+1)*tmp;
saved = left(j-r)*tmp;
}
N(j) = saved;
}
return N;
}
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const
{
if (_Degree == Dynamic)
return m_knots.size() - m_ctrls.cols() - 1;
else
return _Degree;
}
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const
{
return Spline::Span(u, degree(), knots());
}
template <typename _Scalar, int _Dim, int _Degree>
typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const
{
enum { Order = SplineTraits<Spline>::OrderAtCompileTime };
const DenseIndex span = this->span(u);
const DenseIndex p = degree();
const BasisVectorType basis_funcs = basisFunctions(u);
const Replicate<BasisVectorType,Dimension,1> ctrl_weights(basis_funcs);
const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(ctrls(),0,span-p,Dimension,p+1);
return (ctrl_weights * ctrl_pts).rowwise().sum();
}
/* --------------------------------------------------------------------------------------------- */
template <typename SplineType, typename DerivativeType>
void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der)
{
enum { Dimension = SplineTraits<SplineType>::Dimension };
enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
enum { DerivativeOrder = DerivativeType::ColsAtCompileTime };
typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
typedef typename SplineTraits<SplineType,DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;
const DenseIndex p = spline.degree();
const DenseIndex span = spline.span(u);
const DenseIndex n = (std::min)(p, order);
der.resize(Dimension,n+1);
// Retrieve the basis function derivatives up to the desired order...
const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n+1);
// ... and perform the linear combinations of the control points.
for (DenseIndex der_order=0; der_order<n+1; ++der_order)
{
const Replicate<BasisDerivativeRowXpr,Dimension,1> ctrl_weights( basis_func_ders.row(der_order) );
const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(spline.ctrls(),0,span-p,Dimension,p+1);
der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
}
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::DerivativeType
Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline >::DerivativeType res;
derivativesImpl(*this, u, order, res);
return res;
}
template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::DerivativeType
Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline, DerivativeOrder >::DerivativeType res;
derivativesImpl(*this, u, order, res);
return res;
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisVectorType
Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const
{
return Spline::BasisFunctions(u, degree(), knots());
}
/* --------------------------------------------------------------------------------------------- */
template <typename _Scalar, int _Dim, int _Degree>
template <typename DerivativeType>
void Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivativesImpl(
const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
const DenseIndex order,
const DenseIndex p,
const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
DerivativeType& N_)
{
typedef Spline<_Scalar, _Dim, _Degree> SplineType;
enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
const DenseIndex span = SplineType::Span(u, p, U);
const DenseIndex n = (std::min)(p, order);
N_.resize(n+1, p+1);
BasisVectorType left = BasisVectorType::Zero(p+1);
BasisVectorType right = BasisVectorType::Zero(p+1);
Matrix<Scalar,Order,Order> ndu(p+1,p+1);
Scalar saved, temp; // FIXME These were double instead of Scalar. Was there a reason for that?
ndu(0,0) = 1.0;
DenseIndex j;
for (j=1; j<=p; ++j)
{
left[j] = u-U[span+1-j];
right[j] = U[span+j]-u;
saved = 0.0;
for (DenseIndex r=0; r<j; ++r)
{
/* Lower triangle */
ndu(j,r) = right[r+1]+left[j-r];
temp = ndu(r,j-1)/ndu(j,r);
/* Upper triangle */
ndu(r,j) = static_cast<Scalar>(saved+right[r+1] * temp);
saved = left[j-r] * temp;
}
ndu(j,j) = static_cast<Scalar>(saved);
}
for (j = p; j>=0; --j)
N_(0,j) = ndu(j,p);
// Compute the derivatives
DerivativeType a(n+1,p+1);
DenseIndex r=0;
for (; r<=p; ++r)
{
DenseIndex s1,s2;
s1 = 0; s2 = 1; // alternate rows in array a
a(0,0) = 1.0;
// Compute the k-th derivative
for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
{
Scalar d = 0.0;
DenseIndex rk,pk,j1,j2;
rk = r-k; pk = p-k;
if (r>=k)
{
a(s2,0) = a(s1,0)/ndu(pk+1,rk);
d = a(s2,0)*ndu(rk,pk);
}
if (rk>=-1) j1 = 1;
else j1 = -rk;
if (r-1 <= pk) j2 = k-1;
else j2 = p-r;
for (j=j1; j<=j2; ++j)
{
a(s2,j) = (a(s1,j)-a(s1,j-1))/ndu(pk+1,rk+j);
d += a(s2,j)*ndu(rk+j,pk);
}
if (r<=pk)
{
a(s2,k) = -a(s1,k-1)/ndu(pk+1,r);
d += a(s2,k)*ndu(r,pk);
}
N_(k,r) = static_cast<Scalar>(d);
j = s1; s1 = s2; s2 = j; // Switch rows
}
}
/* Multiply through by the correct factors */
/* (Eq. [2.9]) */
r = p;
for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
{
for (j=p; j>=0; --j) N_(k,j) *= r;
r *= p-k;
}
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits<Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType der;
BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
return der;
}
template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType der;
BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
return der;
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits<Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivatives(
const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
const DenseIndex order,
const DenseIndex degree,
const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
{
typename SplineTraits<Spline>::BasisDerivativeType der;
BasisFunctionDerivativesImpl(u, order, degree, knots, der);
return der;
}
}
#endif // EIGEN_SPLINE_H

430
external/include/eigen3/unsupported/Eigen/src/Splines/SplineFitting.h vendored Normal file
View File

@@ -0,0 +1,430 @@
// 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>
#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 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;
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 (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(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);
}
}
#endif // EIGEN_SPLINE_FITTING_H

93
external/include/eigen3/unsupported/Eigen/src/Splines/SplineFwd.h vendored Normal file
View File

@@ -0,0 +1,93 @@
// 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_SPLINES_FWD_H
#define EIGEN_SPLINES_FWD_H
#include <Eigen/Core>
namespace Eigen
{
template <typename Scalar, int Dim, int Degree = Dynamic> class Spline;
template < typename SplineType, int DerivativeOrder = Dynamic > struct SplineTraits {};
/**
* \ingroup Splines_Module
* \brief Compile-time attributes of the Spline class for Dynamic degree.
**/
template <typename _Scalar, int _Dim, int _Degree>
struct SplineTraits< Spline<_Scalar, _Dim, _Degree>, Dynamic >
{
typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
enum { Dimension = _Dim /*!< The spline curve's dimension. */ };
enum { Degree = _Degree /*!< The spline curve's degree. */ };
enum { OrderAtCompileTime = _Degree==Dynamic ? Dynamic : _Degree+1 /*!< The spline curve's order at compile-time. */ };
enum { NumOfDerivativesAtCompileTime = OrderAtCompileTime /*!< The number of derivatives defined for the current spline. */ };
enum { DerivativeMemoryLayout = Dimension==1 ? RowMajor : ColMajor /*!< The derivative type's memory layout. */ };
/** \brief The data type used to store non-zero basis functions. */
typedef Array<Scalar,1,OrderAtCompileTime> BasisVectorType;
/** \brief The data type used to store the values of the basis function derivatives. */
typedef Array<Scalar,Dynamic,Dynamic,RowMajor,NumOfDerivativesAtCompileTime,OrderAtCompileTime> BasisDerivativeType;
/** \brief The data type used to store the spline's derivative values. */
typedef Array<Scalar,Dimension,Dynamic,DerivativeMemoryLayout,Dimension,NumOfDerivativesAtCompileTime> DerivativeType;
/** \brief The point type the spline is representing. */
typedef Array<Scalar,Dimension,1> PointType;
/** \brief The data type used to store knot vectors. */
typedef Array<Scalar,1,Dynamic> KnotVectorType;
/** \brief The data type used to store parameter vectors. */
typedef Array<Scalar,1,Dynamic> ParameterVectorType;
/** \brief The data type representing the spline's control points. */
typedef Array<Scalar,Dimension,Dynamic> ControlPointVectorType;
};
/**
* \ingroup Splines_Module
* \brief Compile-time attributes of the Spline class for fixed degree.
*
* The traits class inherits all attributes from the SplineTraits of Dynamic degree.
**/
template < typename _Scalar, int _Dim, int _Degree, int _DerivativeOrder >
struct SplineTraits< Spline<_Scalar, _Dim, _Degree>, _DerivativeOrder > : public SplineTraits< Spline<_Scalar, _Dim, _Degree> >
{
enum { OrderAtCompileTime = _Degree==Dynamic ? Dynamic : _Degree+1 /*!< The spline curve's order at compile-time. */ };
enum { NumOfDerivativesAtCompileTime = _DerivativeOrder==Dynamic ? Dynamic : _DerivativeOrder+1 /*!< The number of derivatives defined for the current spline. */ };
enum { DerivativeMemoryLayout = _Dim==1 ? RowMajor : ColMajor /*!< The derivative type's memory layout. */ };
/** \brief The data type used to store the values of the basis function derivatives. */
typedef Array<_Scalar,Dynamic,Dynamic,RowMajor,NumOfDerivativesAtCompileTime,OrderAtCompileTime> BasisDerivativeType;
/** \brief The data type used to store the spline's derivative values. */
typedef Array<_Scalar,_Dim,Dynamic,DerivativeMemoryLayout,_Dim,NumOfDerivativesAtCompileTime> DerivativeType;
};
/** \brief 2D float B-spline with dynamic degree. */
typedef Spline<float,2> Spline2f;
/** \brief 3D float B-spline with dynamic degree. */
typedef Spline<float,3> Spline3f;
/** \brief 2D double B-spline with dynamic degree. */
typedef Spline<double,2> Spline2d;
/** \brief 3D double B-spline with dynamic degree. */
typedef Spline<double,3> Spline3d;
}
#endif // EIGEN_SPLINES_FWD_H