FunctionLibrary.h

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// -*- LSST-C++ -*-
 
/*
 * LSST Data Management System
 * Copyright 2008, 2009, 2010 LSST Corporation.
 *
 * This product includes software developed by the
 * LSST Project (http://www.lsst.org/).
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the LSST License Statement and
 * the GNU General Public License along with this program.  If not,
 * see <http://www.lsstcorp.org/LegalNotices/>.
 */
 
#ifndef LSST_AFW_MATH_FUNCTIONLIBRARY_H
#define LSST_AFW_MATH_FUNCTIONLIBRARY_H
/*
 * Define a collection of useful Functions.
 */
#include <algorithm>
#include <cmath>
 
#include "lsst/geom.h"
#include "lsst/afw/math/Function.h"
#include "lsst/geom/Angle.h"
 
namespace lsst {
namespace afw {
namespace math {
 
template <typename ReturnT>
class IntegerDeltaFunction1 : public Function1<ReturnT> {
public:
    explicit IntegerDeltaFunction1(double xo) : Function1<ReturnT>(0), _xo(xo) {}
 
    IntegerDeltaFunction1(IntegerDeltaFunction1 const&) = default;
    IntegerDeltaFunction1(IntegerDeltaFunction1&&) = default;
    IntegerDeltaFunction1& operator=(IntegerDeltaFunction1 const&) = default;
    IntegerDeltaFunction1& operator=(IntegerDeltaFunction1&&) = default;
    ~IntegerDeltaFunction1() noexcept override = default;
 
    std::shared_ptr<Function1<ReturnT>> clone() const override {
        return std::shared_ptr<Function1<ReturnT>>(new IntegerDeltaFunction1(_xo));
    }
 
    ReturnT operator()(double x) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        return static_cast<ReturnT>(x == _xo);
    }
 
    std::string toString(std::string const& prefix = "") const override {
        std::ostringstream os;
        os << "IntegerDeltaFunction1 [" << _xo << "]: ";
        os << Function1<ReturnT>::toString(prefix);
        return os.str();
    };
 
private:
    double _xo;
 
protected:
    /* Default constructor: intended only for serialization */
    explicit IntegerDeltaFunction1() : Function1<ReturnT>(0), _xo(0.0) {}
};
 
template <typename ReturnT>
class IntegerDeltaFunction2 : public Function2<ReturnT> {
public:
    explicit IntegerDeltaFunction2(double xo, double yo) : Function2<ReturnT>(0), _xo(xo), _yo(yo) {}
 
    IntegerDeltaFunction2(IntegerDeltaFunction2 const&) = default;
    IntegerDeltaFunction2(IntegerDeltaFunction2&&) = default;
    IntegerDeltaFunction2& operator=(IntegerDeltaFunction2 const&) = default;
    IntegerDeltaFunction2& operator=(IntegerDeltaFunction2&&) = default;
    ~IntegerDeltaFunction2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(new IntegerDeltaFunction2(_xo, _yo));
    }
 
    ReturnT operator()(double x, double y) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        return static_cast<ReturnT>((x == _xo) && (y == _yo));
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "IntegerDeltaFunction2 [" << _xo << ", " << _yo << "]: ";
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
private:
    double _xo;
    double _yo;
 
protected:
    /* Default constructor: intended only for serialization */
    explicit IntegerDeltaFunction2() : Function2<ReturnT>(0), _xo(0.0), _yo(0.0) {}
};
 
template <typename ReturnT>
class GaussianFunction1 : public Function1<ReturnT> {
public:
    explicit GaussianFunction1(double sigma)  
            : Function1<ReturnT>(1), _multFac(1.0 / std::sqrt(lsst::geom::TWOPI)) {
        this->_params[0] = sigma;
    }
    GaussianFunction1(GaussianFunction1 const&) = default;
    GaussianFunction1(GaussianFunction1&&) = default;
    GaussianFunction1& operator=(GaussianFunction1 const&) = default;
    GaussianFunction1& operator=(GaussianFunction1&&) = default;
    ~GaussianFunction1() noexcept override = default;
 
    std::shared_ptr<Function1<ReturnT>> clone() const override {
        return std::shared_ptr<Function1<ReturnT>>(new GaussianFunction1(this->_params[0]));
    }
 
    ReturnT operator()(double x) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        return static_cast<ReturnT>((_multFac / this->_params[0]) *
                                    std::exp(-(x * x) / (2.0 * this->_params[0] * this->_params[0])));
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "GaussianFunction1 [" << _multFac << "]: ";
        os << Function1<ReturnT>::toString(prefix);
        return os.str();
    }
 
private:
    const double _multFac;  
 
protected:
    /* Default constructor: intended only for serialization */
    explicit GaussianFunction1() : Function1<ReturnT>(1), _multFac(1.0 / std::sqrt(lsst::geom::TWOPI)) {}
};
 
template <typename ReturnT>
class GaussianFunction2 : public Function2<ReturnT> {
public:
    explicit GaussianFunction2(double sigma1,       
                               double sigma2,       
                               double angle = 0.0)  
            : Function2<ReturnT>(3), _multFac(1.0 / (lsst::geom::TWOPI)) {
        this->_params[0] = sigma1;
        this->_params[1] = sigma2;
        this->_params[2] = angle;
        _updateCache();
    }
 
    GaussianFunction2(GaussianFunction2 const&) = default;
    GaussianFunction2(GaussianFunction2&&) = default;
    GaussianFunction2& operator=(GaussianFunction2 const&) = default;
    GaussianFunction2& operator=(GaussianFunction2&&) = default;
    ~GaussianFunction2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(
                new GaussianFunction2(this->_params[0], this->_params[1], this->_params[2]));
    }
 
    ReturnT operator()(double x, double y) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        if (_angle != this->_params[2]) {
            _updateCache();
        }
        double pos1 = (_cosAngle * x) + (_sinAngle * y);
        double pos2 = (-_sinAngle * x) + (_cosAngle * y);
        return static_cast<ReturnT>((_multFac / (this->_params[0] * this->_params[1])) *
                                    std::exp(-((pos1 * pos1) / (2.0 * this->_params[0] * this->_params[0])) -
                                             ((pos2 * pos2) / (2.0 * this->_params[1] * this->_params[1]))));
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "GaussianFunction2: ";
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
    bool isPersistable() const noexcept override { return true; }
 
protected:
    std::string getPersistenceName() const override;
 
    void write(afw::table::io::OutputArchiveHandle& handle) const override;
 
private:
    void _updateCache() const noexcept {
        _angle = this->_params[2];
        _sinAngle = std::sin(_angle);
        _cosAngle = std::cos(_angle);
    }
    const double _multFac;     
    mutable double _angle;     
    mutable double _sinAngle;  
    mutable double _cosAngle;  
 
protected:
    /* Default constructor: intended only for serialization */
    explicit GaussianFunction2()
            : Function2<ReturnT>(3),
              _multFac(1.0 / (lsst::geom::TWOPI)),
              _angle(0.0),
              _sinAngle(0.0),
              _cosAngle(1.0) {}
};
 
template <typename ReturnT>
class DoubleGaussianFunction2 : public Function2<ReturnT> {
public:
    explicit DoubleGaussianFunction2(
            double sigma1,      
            double sigma2 = 0,  
            double ampl2 = 0)   
            : Function2<ReturnT>(3), _multFac(1.0 / (lsst::geom::TWOPI)) {
        this->_params[0] = sigma1;
        this->_params[1] = sigma2;
        this->_params[2] = ampl2;
    }
 
    DoubleGaussianFunction2(DoubleGaussianFunction2 const&) = default;
    DoubleGaussianFunction2(DoubleGaussianFunction2&&) = default;
    DoubleGaussianFunction2& operator=(DoubleGaussianFunction2 const&) = default;
    DoubleGaussianFunction2& operator=(DoubleGaussianFunction2&&) = default;
    ~DoubleGaussianFunction2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(
                new DoubleGaussianFunction2(this->_params[0], this->_params[1], this->_params[2]));
    }
 
    ReturnT operator()(double x, double y) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        double radSq = (x * x) + (y * y);
        double sigma1Sq = this->_params[0] * this->_params[0];
        double sigma2Sq = this->_params[1] * this->_params[1];
        double b = this->_params[2];
        return static_cast<ReturnT>(
                (_multFac / (sigma1Sq + (b * sigma2Sq))) *
                (std::exp(-radSq / (2.0 * sigma1Sq)) + (b * std::exp(-radSq / (2.0 * sigma2Sq)))));
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "DoubleGaussianFunction2 [" << _multFac << "]: ";
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
    bool isPersistable() const noexcept override { return true; }
 
protected:
    std::string getPersistenceName() const override;
 
    void write(afw::table::io::OutputArchiveHandle& handle) const override;
 
private:
    const double _multFac;  
 
protected:
    /* Default constructor: intended only for serialization */
    explicit DoubleGaussianFunction2() : Function2<ReturnT>(3), _multFac(1.0 / (lsst::geom::TWOPI)) {}
};
 
template <typename ReturnT>
class PolynomialFunction1 : public Function1<ReturnT> {
public:
    explicit PolynomialFunction1(unsigned int order)  
            : Function1<ReturnT>(order + 1) {}
 
    explicit PolynomialFunction1(std::vector<double> params)  
            : Function1<ReturnT>(params) {
        if (params.size() < 1) {
            throw LSST_EXCEPT(lsst::pex::exceptions::InvalidParameterError,
                              "PolynomialFunction1 called with empty vector");
        }
    }
 
    PolynomialFunction1(PolynomialFunction1 const&) = default;
    PolynomialFunction1(PolynomialFunction1&&) = default;
    PolynomialFunction1& operator=(PolynomialFunction1 const&) = default;
    PolynomialFunction1& operator=(PolynomialFunction1&&) = default;
    ~PolynomialFunction1() noexcept override = default;
 
    std::shared_ptr<Function1<ReturnT>> clone() const override {
        return std::shared_ptr<Function1<ReturnT>>(new PolynomialFunction1(this->_params));
    }
 
    bool isLinearCombination() const noexcept override { return true; };
 
    ReturnT operator()(double x) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        int const order = static_cast<int>(this->_params.size()) - 1;
        double retVal = this->_params[order];
        for (int ii = order - 1; ii >= 0; --ii) {
            retVal = (retVal * x) + this->_params[ii];
        }
        return static_cast<ReturnT>(retVal);
    }
 
    unsigned int getOrder() const noexcept { return this->getNParameters() - 1; };
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "PolynomialFunction1 []: ";
        os << Function1<ReturnT>::toString(prefix);
        return os.str();
    }
 
protected:
    /* Default constructor: intended only for serialization */
    explicit PolynomialFunction1() : Function1<ReturnT>(1) {}
};
 
template <typename ReturnT>
class PolynomialFunction2 : public BasePolynomialFunction2<ReturnT> {
public:
    explicit PolynomialFunction2(unsigned int order)  
            : BasePolynomialFunction2<ReturnT>(order), _oldY(0), _xCoeffs(this->_order + 1) {}
 
    explicit PolynomialFunction2(
            std::vector<double> params)  
            : BasePolynomialFunction2<ReturnT>(params), _oldY(0), _xCoeffs(this->_order + 1) {}
 
    PolynomialFunction2(PolynomialFunction2 const&) = default;
    PolynomialFunction2(PolynomialFunction2&&) = default;
    PolynomialFunction2& operator=(PolynomialFunction2 const&) = default;
    PolynomialFunction2& operator=(PolynomialFunction2&&) = default;
    ~PolynomialFunction2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(new PolynomialFunction2(this->_params));
    }
 
    ReturnT operator()(double x, double y) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        /* Solve as follows:
        - f(x,y) = Cx0 + Cx1 x + Cx2 x^2 + Cx3 x^3 + ...
        where:
          Cx0 = P0 + P2 y + P5 y^2 + P9 y^3 + ...
          Cx1 = P1 + P4 y + P8 y2 + ...
          Cx2 = P3 + P7 y + ...
          Cx3 = P6 + ...
          ...
 
        Compute Cx0, Cx1...Cxn by solving 1-d polynomials in y in the usual way.
        These values are cached and only recomputed for new values of Y or if the parameters change.
 
        Then compute f(x,y) by solving the 1-d polynomial in x in the usual way.
        */
        const int maxXCoeffInd = this->_order;
 
        if ((y != _oldY) || !this->_isCacheValid) {
            // update _xCoeffs cache
            // note: paramInd is decremented in both of the following loops
            int paramInd = static_cast<int>(this->_params.size()) - 1;
 
            // initialize _xCoeffs to coeffs for pure y^n; e.g. for 3rd order:
            // _xCoeffs[0] = _params[9], _xCoeffs[1] = _params[8], ... _xCoeffs[3] = _params[6]
            for (int xCoeffInd = 0; xCoeffInd <= maxXCoeffInd; ++xCoeffInd, --paramInd) {
                _xCoeffs[xCoeffInd] = this->_params[paramInd];
            }
 
            // finish computing _xCoeffs
            for (int xCoeffInd = 0, endXCoeffInd = maxXCoeffInd; paramInd >= 0; --paramInd) {
                _xCoeffs[xCoeffInd] = (_xCoeffs[xCoeffInd] * y) + this->_params[paramInd];
                ++xCoeffInd;
                if (xCoeffInd >= endXCoeffInd) {
                    xCoeffInd = 0;
                    --endXCoeffInd;
                }
            }
 
            _oldY = y;
            this->_isCacheValid = true;
        }
 
        // use _xCoeffs to compute result
        double retVal = _xCoeffs[maxXCoeffInd];
        for (int xCoeffInd = maxXCoeffInd - 1; xCoeffInd >= 0; --xCoeffInd) {
            retVal = (retVal * x) + _xCoeffs[xCoeffInd];
        }
        return static_cast<ReturnT>(retVal);
    }
 
    std::vector<double> getDFuncDParameters(double x, double y) const override;
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "PolynomialFunction2 [" << this->_order << "]: ";
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
    bool isPersistable() const noexcept override { return true; }
 
protected:
    std::string getPersistenceName() const override;
 
    void write(afw::table::io::OutputArchiveHandle& handle) const override;
 
private:
    mutable double _oldY;                  
    mutable std::vector<double> _xCoeffs;  
 
protected:
    /* Default constructor: intended only for serialization */
    explicit PolynomialFunction2() : BasePolynomialFunction2<ReturnT>(), _oldY(0), _xCoeffs(0) {}
};
 
template <typename ReturnT>
class Chebyshev1Function1 : public Function1<ReturnT> {
public:
    explicit Chebyshev1Function1(unsigned int order,  
                                 double minX = -1,    
                                 double maxX = 1)     
            : Function1<ReturnT>(order + 1) {
        _initialize(minX, maxX);
    }
 
    explicit Chebyshev1Function1(std::vector<double> params,  
                                 double minX = -1,            
                                 double maxX = 1)             
            : Function1<ReturnT>(params) {
        if (params.size() < 1) {
            throw LSST_EXCEPT(lsst::pex::exceptions::InvalidParameterError,
                              "Chebyshev1Function1 called with empty vector");
        }
        _initialize(minX, maxX);
    }
 
    Chebyshev1Function1(Chebyshev1Function1 const&) = default;
    Chebyshev1Function1(Chebyshev1Function1&&) = default;
    Chebyshev1Function1& operator=(Chebyshev1Function1 const&) = default;
    Chebyshev1Function1& operator=(Chebyshev1Function1&&) = default;
    ~Chebyshev1Function1() noexcept override = default;
 
    std::shared_ptr<Function1<ReturnT>> clone() const override {
        return std::shared_ptr<Function1<ReturnT>>(new Chebyshev1Function1(this->_params, _minX, _maxX));
    }
 
    double getMinX() const noexcept { return _minX; };
 
    double getMaxX() const noexcept { return _maxX; };
 
    unsigned int getOrder() const noexcept { return this->getNParameters() - 1; };
 
    bool isLinearCombination() const noexcept override { return true; };
 
    ReturnT operator()(double x) const override {
        double xPrime = (x + _offset) * _scale;
 
        // Clenshaw function for solving the Chebyshev polynomial
        // Non-recursive version from Kresimir Cosic
        int const order = _order;
        if (order == 0) {
            return this->_params[0];
        } else if (order == 1) {
            return this->_params[0] + (this->_params[1] * xPrime);
        }
        double cshPrev = this->_params[order];
        double csh = (2 * xPrime * this->_params[order]) + this->_params[order - 1];
        for (int i = order - 2; i > 0; --i) {
            double cshNext = (2 * xPrime * csh) + this->_params[i] - cshPrev;
            cshPrev = csh;
            csh = cshNext;
        }
        return (xPrime * csh) + this->_params[0] - cshPrev;
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "Chebyshev1Function1 [" << _minX << ", " << _maxX << "]: ";
        os << Function1<ReturnT>::toString(prefix);
        return os.str();
    }
 
private:
    double _minX;         
    double _maxX;         
    double _scale;        
    double _offset;       
    unsigned int _order;  
 
    void _initialize(double minX, double maxX) {
        _minX = minX;
        _maxX = maxX;
        _scale = 2.0 / (_maxX - _minX);
        _offset = -(_minX + _maxX) * 0.5;
        _order = this->getNParameters() - 1;
    }
 
protected:
    /* Default constructor: intended only for serialization */
    explicit Chebyshev1Function1()
            : Function1<ReturnT>(1), _minX(0.0), _maxX(0.0), _scale(1.0), _offset(0.0), _order(0) {}
};
 
template <typename ReturnT>
class Chebyshev1Function2 : public BasePolynomialFunction2<ReturnT> {
public:
    explicit Chebyshev1Function2(unsigned int order,  
                                 lsst::geom::Box2D const& xyRange = lsst::geom::Box2D(
                                         lsst::geom::Point2D(-1.0, -1.0),
                                         lsst::geom::Point2D(1.0, 1.0)))  
            : BasePolynomialFunction2<ReturnT>(order),
              _oldYPrime(0),
              _yCheby(this->_order + 1),
              _xCoeffs(this->_order + 1) {
        _initialize(xyRange);
    }
 
    explicit Chebyshev1Function2(std::vector<double> params,  
                                 lsst::geom::Box2D const& xyRange = lsst::geom::Box2D(
                                         lsst::geom::Point2D(-1.0, -1.0),
                                         lsst::geom::Point2D(1.0, 1.0)))  
            : BasePolynomialFunction2<ReturnT>(params),
              _oldYPrime(0),
              _yCheby(this->_order + 1),
              _xCoeffs(this->_order + 1) {
        _initialize(xyRange);
    }
 
    Chebyshev1Function2(Chebyshev1Function2 const&) = default;
    Chebyshev1Function2(Chebyshev1Function2&&) = default;
    Chebyshev1Function2& operator=(Chebyshev1Function2 const&) = default;
    Chebyshev1Function2& operator=(Chebyshev1Function2&&) = default;
    ~Chebyshev1Function2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(
                new Chebyshev1Function2(this->_params, this->getXYRange()));
    }
 
    lsst::geom::Box2D getXYRange() const noexcept {
        return lsst::geom::Box2D(lsst::geom::Point2D(_minX, _minY), lsst::geom::Point2D(_maxX, _maxY));
    };
 
    virtual Chebyshev1Function2 truncate(int truncOrder  
                                         ) const {
        if (truncOrder > this->_order) {
            std::ostringstream os;
            os << "truncated order=" << truncOrder << " must be <= original order=" << this->_order;
            throw LSST_EXCEPT(lsst::pex::exceptions::InvalidParameterError, os.str());
        }
        int truncNParams = this->nParametersFromOrder(truncOrder);
        std::vector<double> truncParams(this->_params.begin(), this->_params.begin() + truncNParams);
        return Chebyshev1Function2(truncParams, this->getXYRange());
    }
 
    ReturnT operator()(double x, double y) const override {
        /* Solve as follows:
        - f(x,y) = Cy0 T0(y') + Cy1 T1(y') + Cy2 T2(y') + Cy3 T3(y') + ...
        where:
          Cy0 = P0 T0(x') + P1 T1(x') + P3 T2(x') + P6 T3(x') + ...
          Cy1 = P2 T0(x') + P4 T1(x') + P7 T2(x') + ...
          Cy2 = P5 T0(x') + P8 T1(x') + ...
          Cy3 = P9 T0(x') + ...
          ...
 
        First compute Tn(x') for each n
        Then use that to compute Cy0, Cy1, ...Cyn
        Then solve the y Chebyshev polynomial using the Clenshaw algorithm
        */
        double const xPrime = (x + _offsetX) * _scaleX;
        double const yPrime = (y + _offsetY) * _scaleY;
 
        const int nParams = static_cast<int>(this->_params.size());
        const int order = this->_order;
 
        if (order == 0) {
            return this->_params[0];  // No caching required
        }
 
        if ((yPrime != _oldYPrime) || !this->_isCacheValid) {
            // update cached _yCheby and _xCoeffs
            _yCheby[0] = 1.0;
            _yCheby[1] = yPrime;
            for (int chebyInd = 2; chebyInd <= order; chebyInd++) {
                _yCheby[chebyInd] = (2 * yPrime * _yCheby[chebyInd - 1]) - _yCheby[chebyInd - 2];
            }
 
            for (int coeffInd = 0; coeffInd <= order; coeffInd++) {
                _xCoeffs[coeffInd] = 0;
            }
            for (int coeffInd = 0, endCoeffInd = 0, paramInd = 0; paramInd < nParams; paramInd++) {
                _xCoeffs[coeffInd] += this->_params[paramInd] * _yCheby[endCoeffInd];
                --coeffInd;
                ++endCoeffInd;
                if (coeffInd < 0) {
                    coeffInd = endCoeffInd;
                    endCoeffInd = 0;
                }
            }
 
            _oldYPrime = yPrime;
            this->_isCacheValid = true;
        }
 
        // Clenshaw function for solving the Chebyshev polynomial
        // Non-recursive version from Kresimir Cosic
        if (order == 1) {
            return _xCoeffs[0] + (_xCoeffs[1] * xPrime);
        }
        double cshPrev = _xCoeffs[order];
        double csh = (2 * xPrime * _xCoeffs[order]) + _xCoeffs[order - 1];
        for (int i = order - 2; i > 0; --i) {
            double cshNext = (2 * xPrime * csh) + _xCoeffs[i] - cshPrev;
            cshPrev = csh;
            csh = cshNext;
        }
        return (xPrime * csh) + _xCoeffs[0] - cshPrev;
    }
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "Chebyshev1Function2 [";
        os << this->_order << ", " << this->getXYRange() << "]";
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
    bool isPersistable() const noexcept override { return true; }
 
protected:
    std::string getPersistenceName() const override;
 
    void write(afw::table::io::OutputArchiveHandle& handle) const override;
 
private:
    mutable double _oldYPrime;
    mutable std::vector<double> _yCheby;   
    mutable std::vector<double> _xCoeffs;  
    double _minX;                          
    double _minY;                          
    double _maxX;                          
    double _maxY;                          
    double _scaleX;                        
    double _scaleY;                        
    double _offsetX;                       
    double _offsetY;                       
 
    void _initialize(lsst::geom::Box2D const& xyRange) {
        _minX = xyRange.getMinX();
        _minY = xyRange.getMinY();
        _maxX = xyRange.getMaxX();
        _maxY = xyRange.getMaxY();
        _scaleX = 2.0 / (_maxX - _minX);
        _scaleY = 2.0 / (_maxY - _minY);
        _offsetX = -(_minX + _maxX) * 0.5;
        _offsetY = -(_minY + _maxY) * 0.5;
    }
 
protected:
    /* Default constructor: intended only for serialization */
    explicit Chebyshev1Function2()
            : BasePolynomialFunction2<ReturnT>(),
              _oldYPrime(0),
              _yCheby(0),
              _xCoeffs(0),
              _minX(0.0),
              _minY(0.0),
              _maxX(0.0),
              _maxY(0.0),
              _scaleX(1.0),
              _scaleY(1.0),
              _offsetX(0.0),
              _offsetY(0.0) {}
};
 
template <typename ReturnT>
class LanczosFunction1 : public Function1<ReturnT> {
public:
    explicit LanczosFunction1(unsigned int n,        
                              double xOffset = 0.0)  
            : Function1<ReturnT>(1), _invN(1.0 / static_cast<double>(n)) {
        this->_params[0] = xOffset;
    }
 
    LanczosFunction1(LanczosFunction1 const&) = default;
    LanczosFunction1(LanczosFunction1&&) = default;
    LanczosFunction1& operator=(LanczosFunction1 const&) = default;
    LanczosFunction1& operator=(LanczosFunction1&&) = default;
    ~LanczosFunction1() noexcept override = default;
 
    std::shared_ptr<Function1<ReturnT>> clone() const override {
        return std::shared_ptr<Function1<ReturnT>>(new LanczosFunction1(this->getOrder(), this->_params[0]));
    }
 
    ReturnT operator()(double x) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        double xArg1 = (x - this->_params[0]) * lsst::geom::PI;
        double xArg2 = xArg1 * _invN;
        if (std::fabs(xArg1) > 1.0e-5) {
            return static_cast<ReturnT>(std::sin(xArg1) * std::sin(xArg2) / (xArg1 * xArg2));
        } else {
            return static_cast<ReturnT>(1);
        }
    }
 
    unsigned int getOrder() const noexcept { return static_cast<unsigned int>(0.5 + (1.0 / _invN)); };
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "LanczosFunction1 [" << this->getOrder() << "]: ";
        ;
        os << Function1<ReturnT>::toString(prefix);
        return os.str();
    }
 
private:
    double _invN;  // == 1/n
 
protected:
    /* Default constructor: intended only for serialization */
    explicit LanczosFunction1() : Function1<ReturnT>(1), _invN(1.0) {}
};
 
template <typename ReturnT>
class LanczosFunction2 : public Function2<ReturnT> {
public:
    explicit LanczosFunction2(unsigned int n,        
                              double xOffset = 0.0,  
                              double yOffset = 0.0)  
            : Function2<ReturnT>(2), _invN(1.0 / static_cast<double>(n)) {
        this->_params[0] = xOffset;
        this->_params[1] = yOffset;
    }
 
    LanczosFunction2(LanczosFunction2 const&) = default;
    LanczosFunction2(LanczosFunction2&&) = default;
    LanczosFunction2& operator=(LanczosFunction2 const&) = default;
    LanczosFunction2& operator=(LanczosFunction2&&) = default;
    ~LanczosFunction2() noexcept override = default;
 
    std::shared_ptr<Function2<ReturnT>> clone() const override {
        return std::shared_ptr<Function2<ReturnT>>(
                new LanczosFunction2(this->getOrder(), this->_params[0], this->_params[1]));
    }
 
    ReturnT operator()(double x, double y) const noexcept(IS_NOTHROW_INIT<ReturnT>) override {
        double xArg1 = (x - this->_params[0]) * lsst::geom::PI;
        double xArg2 = xArg1 * _invN;
        double xFunc = 1;
        if (std::fabs(xArg1) > 1.0e-5) {
            xFunc = std::sin(xArg1) * std::sin(xArg2) / (xArg1 * xArg2);
        }
        double yArg1 = (y - this->_params[1]) * lsst::geom::PI;
        double yArg2 = yArg1 * _invN;
        double yFunc = 1;
        if (std::fabs(yArg1) > 1.0e-5) {
            yFunc = std::sin(yArg1) * std::sin(yArg2) / (yArg1 * yArg2);
        }
        return static_cast<ReturnT>(xFunc * yFunc);
    }
 
    unsigned int getOrder() const noexcept { return static_cast<unsigned int>(0.5 + (1.0 / _invN)); };
 
    std::string toString(std::string const& prefix) const override {
        std::ostringstream os;
        os << "LanczosFunction2 [" << this->getOrder() << "]: ";
        ;
        os << Function2<ReturnT>::toString(prefix);
        return os.str();
    }
 
private:
    double _invN;  
 
protected:
    /* Default constructor: intended only for serialization */
    explicit LanczosFunction2() : Function2<ReturnT>(2), _invN(1.0) {}
};
}  // namespace math
}  // namespace afw
}  // namespace lsst
 
#endif  // #ifndef LSST_AFW_MATH_FUNCTIONLIBRARY_H