o3de/Gems/Atom/Asset/ImageProcessingAtom/Code/Source/Converters/FIR-Windows.h

/*
 * Copyright (c) Contributors to the Open 3D Engine Project.
 * For complete copyright and license terms please see the LICENSE at the root of this distribution.
 *
 * SPDX-License-Identifier: Apache-2.0 OR MIT
 *
 */



// Description : A collection of window functions with Finite Impulse Response  FIR
//               and some helper window functions with Infinite Impulse Response  IIR

#pragma once
#include <math.h>
#include <ImageProcessing_Traits_Platform.h>
#include <AzCore/Math/MathUtils.h>

/* ####################################################################################################################
 */
#ifndef M_PI
#define M_PI     3.14159265358979323846 /* pi */
#endif

//the original file was from cry's RC.exe with all the filters. We decided we would only use few of them for users
namespace ImageProcessingAtom
{
    template<class F>
    F cube(const F& op) { return op * op * op; }

    template<class F>
    F square(F fOp) { return(fOp * fOp); }

    enum EWindowFunction
    {
        eWindowFunction_Combiner        =  0,

        /*--------------- unit-area filters for unit-spaced samples ----------------*/
        eWindowFunction_Point           =  1,

        eWindowFunction_Box             =  2, // box, pulse, Fourier window,  1st order (constant) b-spline
        eWindowFunction_Triangle        =  3, // triangle, Bartlett window,  2nd order (linear) b-spline
        eWindowFunction_Linear          =  eWindowFunction_Triangle,
        eWindowFunction_Bartlett        =  eWindowFunction_Triangle,

        eWindowFunction_Quadric         =  4, // 3rd order (quadratic) b-spline
        eWindowFunction_Bilinear        =  eWindowFunction_Quadric,
        eWindowFunction_Welch           =  eWindowFunction_Quadric,
        eWindowFunction_Cubic           =  5, // 4th order (cubic) b-spline
        eWindowFunction_Hermite         =  6, // 4th order (cubic hermite) b-spline
        eWindowFunction_Catrom          =  7, // Catmull-Rom spline, Overhauser spline

        eWindowFunction_Sine            =  8, // IIR
        eWindowFunction_Sinc            =  9, // Sinc, perfect lowpass filter (infinite)
        eWindowFunction_Bessel          = 10, // Bessel (for circularly symm. 2-d filt, inf)
        eWindowFunction_Lanczos         = 11, // Lanczos filtering, windowed Sinc

        /*------------------ filters for non-unit spaced samples -------------------*/
        eWindowFunction_Gaussian        = 12, // Gaussian (infinite)
        eWindowFunction_Normal          = 13, // Normal distribution (infinite)

        /*------------------------- parameterized filters --------------------------*/
        eWindowFunction_Mitchell        = 14, // Mitchell & Netravali's two-param cubic

        /*--------------------------- window functions -----------------------------*/
        eWindowFunction_Hann            = 15, // Hanning window
        eWindowFunction_BartlettHann    = 16,
        eWindowFunction_Hamming         = 17, // Hamming window
        eWindowFunction_Blackman        = 18, // Blackman window
        eWindowFunction_BlackmanHarris  = 19,
        eWindowFunction_BlackmanNuttall = 20,
        eWindowFunction_Flattop         = 21,

        /*------------------------- parameterized windows --------------------------*/
        eWindowFunction_Kaiser          = 22, // parameterized Kaiser window

        /*---------------------------- custom windows ------------------------------*/
        eWindowFunction_SigmaSix        = 23, // two Normal distributions
        eWindowFunction_KaiserSinc      = 24, // Kaiser and Sinc

        eWindowFunction_Num             = eWindowFunction_KaiserSinc + 1,
    };

    enum EWindowEvaluation
    {
        eWindowEvaluation_Sum,
        eWindowEvaluation_Max,
        eWindowEvaluation_Min,
    };

    /* ####################################################################################################################
     */
    template<typename T = float>
    class IWindowFunction
    {
    public:
        virtual ~IWindowFunction() {}

        virtual const char* getName() const = 0;
        virtual T getLength() const = 0;

        virtual bool isCardinal() const = 0;
        virtual bool isInfinite() const = 0;
        virtual bool isUnitSpaced() const = 0;
        virtual bool isCentered() const = 0;

    public:
        virtual T operator () (T pos) const = 0;
    };

    /* ####################################################################################################################
     * box, pulse, Fourier window,
     * box function also know as rectangle function
     * 1st order (constant) b-spline
     */
    template<typename T = double>
    class BoxWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Box-window";
        }
        virtual T getLength() const
        {
            return 0.5;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos <  0.0)
            {
                pos = -pos;
            }
            if (pos <= 0.5)
            {
                return 1.0;
            }
            return 0.0;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * triangle, Bartlett window,
     * triangle function also known as lambda function
     * 2nd order (linear) b-spline
     */
    template<typename T = double>
    class TriangleWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Triangle-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 1.0)
            {
                return 1.0 - pos;
            }
            return 0.0;
        }
    };

    /* ####################################################################################################################
     * 3rd order (quadratic) b-spline
     */
    template<typename T = double>
    class QuadricWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Quadric-window";
        }
        virtual T getLength() const
        {
            return 1.5;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 0.5)
            {
                return 0.75 - square(pos);
            }
            if (pos < 1.5)
            {
                return 0.50 * square(pos - 1.5);
            }
            return 0.0;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * 4th order (cubic) b-spline
     */
    template<typename T = double>
    class CubicWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Cubic-window";
        }
        virtual T getLength() const
        {
            return 2.0;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 1.0)
            {
                return 0.5 * cube(pos) - square(pos) + 2.0 / 3.0;
            }
            if (pos < 2.0)
            {
                return cube(2.0 - pos) / 6.0;
            }
            return 0.0;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Hermite filter
     * f(x) = 2|x|^3 - 3|x|^2 + 1, -1 <= x <= 1
     */
    template<typename T = double>
    class HermiteWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Hermite-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 1.0)
            {
                return 2.0 * cube(pos) - 3.0 * square(pos) + 1.0;
            }
            return 0.0;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Catmull-Rom spline, Overhauser spline
     */
    template<typename T = double>
    class CatromWindowFunction
        : public IWindowFunction<T>
    {
    public:
        CatromWindowFunction() { }

    public:
        virtual const char* getName() const
        {
            return "Catrom-window";
        }
        virtual T getLength() const
        {
            return 2.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 1.0)
            {
                return 1.5 * cube(pos) - 2.5 * square(pos)             + 1.0;
            }
            if (pos < 2.0)
            {
                return -0.5 * cube(pos) + 2.5 * square(pos) - 4.0 * pos + 2.0;
            }
            return 0.0;
        }
    };

    /* ####################################################################################################################
     */
    template<typename T = double>
    class SineWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Sine-window";
        }
        virtual T getLength() const
        {
            return 0.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return sin(pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * sinc, perfect lowpass filter (infinite)
     *
     * Note: Some people say sinc(x) is sin(x)/x.  Others say it's
     * sin(PI*x)/(PI*x), a horizontal compression of the former which is
     * zero at integer values.  We use the latter, whose Fourier transform
     * is a canonical rectangle function (edges at -1/2, +1/2, height 1).
     */
    template<typename T = double>
    class SincWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Sinc-window";
        }
        virtual T getLength() const
        {
            return 4.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos == 0.0)
            {
                return 1.0;
            }

            return sin(M_PI * pos) / (M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Bessel (for circularly symm. 2-d filt, infinite)
     * See Pratt "Digital Image Processing" p. 97 for Bessel functions
     */
    template<typename T = double>
    class BesselWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Bessel-window";
        }
        virtual T getLength() const
        {
            return 3.2383;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos == 0.0)
            {
                return M_PI / 4.0;
            }
            return AZ_TRAIT_IMAGEPROCESSING_BESSEL_FUNCTION_FIRST_ORDER(M_PI * pos) / (2.0 * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Lanczos filter
     */
    template<typename T = double>
    class LanczosWindowFunction
        : public SincWindowFunction<T>
    {
    public:
        LanczosWindowFunction(T tap = 3.0)
        {
            this->tap = AZ::GetMax(3.0, tap);
        }

    protected:
        T tap;

    public:
        virtual const char* getName() const
        {
            return "Lanczos-window";
        }
        virtual T getLength() const
        {
            return tap;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < tap)
            {
                return ((SincWindowFunction<T>) * this)(pos) * ((SincWindowFunction<T>) * this)(pos / tap);
            }
            return 0.0;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Gaussian filter (infinite)
     */
    template<typename T = double>
    class GaussianWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Gaussian-window";
        }
        virtual T getLength() const
        {
            return 1.25;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return exp(-2.0 * square(pos)) * sqrt(2.0 / M_PI);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * normal distribution (infinite)
     * Normal(x) = Gaussian(x/2)/2
     */
    template<typename T = double>
    class NormalWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Normal-window";
        }
        virtual T getLength() const
        {
            return 2.5;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return exp(-square(pos) / 2.0) / sqrt(2.0 * M_PI);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     */
    template<typename T = double>
    class SigmaSixWindowFunction
        : public IWindowFunction<T>
    {
    public:
        SigmaSixWindowFunction(T diameter = 1.0, T negative = 0.0)
        {
            // we aim for 6 * sigma = 99,99996% of all values
            const T sigma = 1.0 / 3.0;

            s2 = sigma * sigma * 2.0;
            d2 = s2 / (diameter * diameter);
            d = diameter;
            n = negative;
        }

    protected:
        T s2, d2, d, n;

    public:
        virtual const char* getName() const
        {
            return "SigmaSix-window";
        }
        virtual T getLength() const
        {
            return 1.44;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            T o = exp(-square(pos) / s2) - (1.0 - 0.9999996);
            T i = exp(-square(pos) / d2) - (1.0 - 0.9999996);
            return (pos >= d ? 0.0 : i) - o * n;
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Mitchell & Netravali's two-param cubic
     * see Mitchell & Netravali,
     * "Reconstruction Filters in Computer Graphics", SIGGRAPH 88
     */
    template<typename T = double>
    class MitchellWindowFunction
        : public IWindowFunction<T>
    {
    public:
        MitchellWindowFunction(T b = 1.0 / 3.0, T c = 1.0 / 3.0)
        {
            p0 = (6. -  2. * b) / 6.;
            p2 = (-18. + 12. * b +  6. * c) / 6.;
            p3 = (12. -  9. * b -  6. * c) / 6.;
            q0 = (8. * b + 24. * c) / 6.;
            q1 = (-12. * b - 48. * c) / 6.;
            q2 = (6. * b + 30. * c) / 6.;
            q3 = (-b -  6. * c) / 6.;
        }

    protected:
        T p0, p2, p3, q0, q1, q2, q3;

    public:
        virtual const char* getName() const
        {
            return "Mitchell-window";
        }
        virtual T getLength() const
        {
            return 2.0;
        }

        virtual bool isCardinal() const
        {
            return false;
        }
        virtual bool isInfinite() const
        {
            return false;
        }
        virtual bool isUnitSpaced() const
        {
            return false;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            if (pos < 0.0)
            {
                pos = -pos;
            }
            if (pos < 1.0)
            {
                return p3 * cube(pos) + p2 * square(pos)            + p0;
            }
            if (pos < 2.0)
            {
                return q3 * cube(pos) + q2 * square(pos) + q1 * pos + q0;
            }
            return 0.0;
        }
    };

    /* ####################################################################################################################
     * Hanning window (infinite)
     */
    template<typename T = double>
    class HannWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Hann-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.5 + 0.5 * cos(M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * BartlettHanning window (infinite)
     */
    template<typename T = double>
    class BartlettHannWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Bartlett-Hann-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.62 + 0.48 * (1.0 - pos) + 0.38 * cos(M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Hamming window (infinite)
     */
    template<typename T = double>
    class HammingWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Hamming-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.53836 + 0.46164 * cos(M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * Blackman window (infinite)
     */
    template<typename T = double>
    class BlackmanWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Blackman-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.42659
                   + 0.49656 * cos(M_PI * pos)
                   + 0.07685 * cos(2.0 * M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * BlackmanHarris window (infinite)
     */
    template<typename T = double>
    class BlackmanHarrisWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Blackman-Harris-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.35875
                   + 0.48829 * cos(M_PI * pos)
                   + 0.14128 * cos(2.0 * M_PI * pos)
                   + 0.01168 * cos(3.0 * M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * BlackmanNuttall window (infinite)
     */
    template<typename T = double>
    class BlackmanNuttallWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Blackman-Nuttall-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.3635819
                   + 0.4891775 * cos(M_PI * pos)
                   + 0.1365995 * cos(2.0 * M_PI * pos)
                   + 0.0106411 * cos(3.0 * M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * FlatTop window (infinite)
     */
    template<typename T = double>
    class FlatTopWindowFunction
        : public IWindowFunction<T>
    {
    public:
        virtual const char* getName() const
        {
            return "Flat-Top-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return 0.215578948
                   + 0.416631580 * cos(M_PI * pos)
                   + 0.277263158 * cos(2.0 * M_PI * pos)
                   + 0.083578947 * cos(3.0 * M_PI * pos)
                   + 0.006947368 * cos(4.0 * M_PI * pos);
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     * parameterized Kaiser window (infinite)
     * from Oppenheim & Schafer, Hamming
     */
    template<typename T = double>
    class KaiserWindowFunction
        : public IWindowFunction<T>
    {
    public:
        KaiserWindowFunction(T a = 6.5)
        {
            /* typically 4<a<9 */
            /* param a trades off main lobe width (sharpness) */
            /* for side lobe amplitude (ringing) */
            this->a   = a;
            this->i0a = 1. / bessel_i0(a);
        }

    protected:
        T a, i0a;

        /* modified zeroth order Bessel function of the first kind. */
        static T bessel_i0(T x)
        {
    #define EPSILON 1e-7
            const T y = square(x) / 4.0;

            T sum = 1.0;
            T t = y;

            for (int i = 2; t > EPSILON; i++)
            {
                sum += t;
                t *= y / square(i);
            }

            return sum;

    #undef EPSILON
        }

    public:
        virtual const char* getName() const
        {
            return "Kaiser-window";
        }
        virtual T getLength() const
        {
            return 1.0;
        }

        virtual bool isCardinal() const
        {
            return true;
        }
        virtual bool isInfinite() const
        {
            return true;
        }
        virtual bool isUnitSpaced() const
        {
            return true;
        }
        virtual bool isCentered() const
        {
            return true;
        }

    public:
        virtual T operator () (T pos) const
        {
            return i0a * bessel_i0(a * sqrt(1.0 - square(pos)));
        }
    };

    /* ####################################################################################################################
     */
    template<typename T = double>
    class CombinerWindowFunction
        : public IWindowFunction<T>
    {
    public:
        CombinerWindowFunction(IWindowFunction<T>* fu, IWindowFunction<T>* wi)
        {
            shaper     = fu;
            restrictor = wi;
        }

    protected:
        IWindowFunction<T>* shaper;
        IWindowFunction<T>* restrictor;

    public:
        virtual const char* getName() const
        {
            return "Combiner of two window-generators";
        }
        virtual T getLength() const
        {
            return shaper->getLength();
        }

        virtual bool isCardinal() const
        {
            return shaper->isCardinal() && restrictor->isCardinal();
        }
        virtual bool isInfinite() const
        {
            return shaper->isInfinite() && restrictor->isInfinite();
        }
        virtual bool isUnitSpaced() const
        {
            return shaper->isUnitSpaced() && restrictor->isUnitSpaced();
        }
        virtual bool isCentered() const
        {
            return shaper->isCentered() && restrictor->isCentered();
        }

    public:
        virtual T operator () (T pos) const
        {
            return (*shaper)(pos) * (*restrictor)(pos / shaper->getLength());
        }
    };

    /* --------------------------------------------------------------------------------------------------------------------
     */
    template<typename T = double>
    class PointWindowFunction
        : public BoxWindowFunction<T>
    {
    public:
        PointWindowFunction()
            : BoxWindowFunction<T>() { }

    public:
        virtual const char* getName() const
        {
            return "Point-window";
        }
        virtual T getLength() const
        {
            return 0.0;
        }
    };
} //end namespace ImageProcessingAtom