al_Interpolation.hpp

#ifndef INCLUDE_AL_MATH_INTERPOLATION_HPP
#define INCLUDE_AL_MATH_INTERPOLATION_HPP
 
/*  Allocore --
  Multimedia / virtual environment application class library
 
  Copyright (C) 2009. AlloSphere Research Group, Media Arts & Technology, UCSB.
  Copyright (C) 2012. The Regents of the University of California.
  All rights reserved.
 
  Redistribution and use in source and binary forms, with or without
  modification, are permitted provided that the following conditions are met:
 
    Redistributions of source code must retain the above copyright notice,
    this list of conditions and the following disclaimer.
 
    Redistributions in binary form must reproduce the above copyright
    notice, this list of conditions and the following disclaimer in the
    documentation and/or other materials provided with the distribution.
 
    Neither the name of the University of California nor the names of its
    contributors may be used to endorse or promote products derived from
    this software without specific prior written permission.
 
  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
  AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
  LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
  POSSIBILITY OF SUCH DAMAGE.
 
 
  File description:
  A collection of generic interpolation functions
 
  File author(s):
  Lance Putnam, 2006, putnam.lance@gmail.com
*/
 
namespace al {
 
namespace ipl {
 
 
template <class Tf, class Tv>
Tv bezier(Tf frac, const Tv& x2, const Tv& x1, const Tv& x0);
 
 
template <class Tf, class Tv>
Tv bezier(Tf frac, const Tv& x3, const Tv& x2, const Tv& x1, const Tv& x0);
 
 
template <class Tf, class Tv>
Tv casteljau(const Tf& frac, const Tv& a, const Tv& b, const Tv& c,
             const Tv& d);
 
template <class Tp, class Tv>
Tv hermite(Tp f, const Tv& w, const Tv& x, const Tv& y, const Tv& z, Tp tension,
           Tp bias);
 
 
template <class T>
void lagrange(T* h, T delay, int order);
 
template <class T>
void lagrange1(T* h, T delay);
 
template <class T>
void lagrange2(T* h, T delay);
 
template <class T>
void lagrange3(T* h, T delay);
 
// Various functions to perform Waring-Lagrange interpolation.
//    These are much faster than using a general purpose FIR filter since
//    the coefs are computed directly and nested multiplication is used
//    rather than directly evaluating the polynomial (FIR).
 
 
template <class Tf, class Tv>
void cardinalSpline(Tv* w, const Tv* x, const Tf& f, double b);
 
 
template <class Tf, class Tv>
Tv cubic(Tf frac, const Tv& w, const Tv& x, const Tv& y, const Tv& z);
 
 
template <class Tf, class Tv>
Tv cubic2(Tf frac, const Tv& w, const Tv& x, const Tv& y, const Tv& z);
 
template <class Tf, class Tv>
Tv linear(Tf frac, const Tv& x, const Tv& y);
 
template <class Tf, class Tv>
Tv linear(Tf frac, const Tv& x, const Tv& y, const Tv& z);
 
template <class Tf, class Tv>
Tv linearCyclic(Tf frac, const Tv& x, const Tv& y, const Tv& z);
 
template <class Tf, class Tv>
void linear(Tv* dst, const Tv* xs, const Tv* xp1s, int len, const Tf& frac);
 
template <class Tf, class Tv>
Tv nearest(Tf frac, const Tv& x, const Tv& y);
 
template <class Tf, class Tv>
inline Tv bilinear(const Tf& fracX, const Tf& fracY, const Tv& xy, const Tv& Xy,
                   const Tv& xY, const Tv& XY) {
  return linear(fracY, linear(fracX, xy, Xy), linear(fracX, xY, XY));
}
 
template <class Tf2, class Tv>
inline Tv bilinear(const Tf2& f, const Tv& xy, const Tv& Xy, const Tv& xY,
                   const Tv& XY) {
  return bilinear(f[0], f[1], xy, Xy, xY, XY);
}
 
template <class Tf, class Tv>
inline Tv trilinear(const Tf& fracX, const Tf& fracY, const Tf& fracZ,
                    const Tv& xyz, const Tv& Xyz, const Tv& xYz, const Tv& XYz,
                    const Tv& xyZ, const Tv& XyZ, const Tv& xYZ,
                    const Tv& XYZ) {
  return linear(fracZ, bilinear(fracX, fracY, xyz, Xyz, xYz, XYz),
                bilinear(fracX, fracY, xyZ, XyZ, xYZ, XYZ));
}
 
 
template <class Tf3, class Tv>
inline Tv trilinear(const Tf3& f, const Tv& xyz, const Tv& Xyz, const Tv& xYz,
                    const Tv& XYz, const Tv& xyZ, const Tv& XyZ, const Tv& xYZ,
                    const Tv& XYZ) {
  return trilinear(f[0], f[1], f[2], xyz, Xyz, xYz, XYz, xyZ, XyZ, xYZ, XYZ);
}
 
//------------------------------------------------------------------------------
// Implementation
//------------------------------------------------------------------------------
 
template <class Tf, class Tv>
inline Tv bezier(Tf d, const Tv& x2, const Tv& x1, const Tv& x0) {
  Tf d2 = d * d;
  Tf dm1 = Tf(1) - d;
  Tf dm12 = dm1 * dm1;
  return dm12 * x2 + Tf(2) * dm1 * d * x1 + d2 * x0;
 
  //  x2 (1-d)(1-d) + 2 x1 (1-d) d + x0 d d
  //  x2 - d 2 x2 + d d x2 + d 2 x1 - d d 2 x1 + d d x0
  //  x2 - d (2 x2 + d x2 + 2 x1 - d 2 x1 + d x0)
  //  x2 - d (2 (x2 + x1) + d (x2 - 2 x1 + x0))
 
  //  float c2 = x2 - 2.f * x1 + x0;
  //  float c1 = 2.f * (x2 + x1);
  //  return x2 - (d * c2 + c1) * d;
}
 
template <class Tf, class Tv>
inline Tv bezier(Tf d, const Tv& x3, const Tv& x2, const Tv& x1, const Tv& x0) {
  Tv c1 = Tf(3) * (x2 - x3);
  Tv c2 = Tf(3) * (x1 - x2) - c1;
  Tv c3 = x0 - x3 - c1 - c2;
  return ((c3 * d + c2) * d + c1) * d + x3;
}
 
template <class Tf, class Tv>
Tv casteljau(const Tf& f, const Tv& a, const Tv& b, const Tv& c, const Tv& d) {
  Tv ab = linear(f, a, b);
  Tv bc = linear(f, b, c);
  Tv cd = linear(f, c, d);
  return linear(f, linear(f, ab, bc), linear(f, bc, cd));
}
 
template <class Tf, class Tv>
inline void cardinalSpline(Tv* w, const Tv* x, const Tf& f, double b) {
  // c = (1-b);    // tension
  b *= (x[2] - x[1]);  // make domain [t[1], t[2]]
 
  // evaluate the Hermite basis functions
  Tf h00 = (1 + 2 * f) * (f - 1) * (f - 1);
  Tf h10 = f * (f - 1) * (f - 1);
  Tf h01 = f * f * (3 - 2 * f);
  Tf h11 = f * f * (f - 1);
 
  /*
  The general cubic Hermite spline is
    p(f) = h00 * p0 + h10 * m0 + h01 * p1 + h11 * m1
 
  For a cardinal spline, the tangent at point k is
    m[k] = (1-c) * (p[k+1] - p[k-1]) / (x[k+1] - x[k-1])
 
  To get the weights for each point, we plug the tangents into the general
  spline equation and factor out the p[k].
  */
 
  w[0] = (-b * h10 / (x[2] - x[0]));
  w[1] = (h00 - b * h11 / (x[3] - x[1]));
  w[2] = (h01 + b * h10 / (x[2] - x[0]));
  w[3] = (+b * h11 / (x[3] - x[1]));
}
 
template <class Tf, class Tv>
inline Tv cubic(Tf f, const Tv& w, const Tv& x, const Tv& y, const Tv& z) {
  //  Tv c3 = (x - y)*(Tf)1.5 + (z - w)*(Tf)0.5;
  //  Tv c2 = w - x*(Tf)2.5 + y*(Tf)2. - z*(Tf)0.5;
  //  Tv c1 = (y - w)*(Tf)0.5;
  //  return ((c3 * f + c2) * f + c1) * f + x;
 
  // -w + 3x - 3y + z
  // 2w - 5x + 4y - z
  // c2 = w - 2x + y - c3
 
  //  Tv c3 = (x - y)*(Tf)3 + z - w;
  //  Tv c2 = w - x*(Tf)2 + y - c3;
  //  Tv c1 = y - w;
  //  return (((c3 * f + c2) * f + c1)) * f * (Tf)0.5 + x;
 
  //  Tv c3 = (x - y)*(Tf)1.5 + (z - w)*(Tf)0.5;
  //  Tv c2 = (y + w)*(Tf)0.5 - x - c3;
  //  Tv c1 = (y - w)*(Tf)0.5;
  //  return ((c3 * f + c2) * f + c1) * f + x;
 
  Tv c1 = (y - w) * Tf(0.5);
  Tv c3 = (x - y) * Tf(1.5) + (z - w) * Tf(0.5);
  Tv c2 = c1 + w - x - c3;
  return ((c3 * f + c2) * f + c1) * f + x;
}
 
template <class T>
void cubic(T* dst, const T* xm1s, const T* xs, const T* xp1s, const T* xp2s,
           int len, T f) {
  for (int i = 0; i < len; ++i)
    dst[i] = cubic(f, xm1s[i], xs[i], xp1s[i], xp2s[i]);
}
 
// From http://astronomy.swin.edu.au/~pbourke/other/interpolation/ (Paul Bourke)
template <class Tf, class Tv>
inline Tv cubic2(Tf f, const Tv& w, const Tv& x, const Tv& y, const Tv& z) {
  Tv c3 = z - y - w + x;
  Tv c2 = w - x - c3;
  Tv c1 = y - w;
  return ((c3 * f + c2) * f + c1) * f + x;
}
 
// From http://astronomy.swin.edu.au/~pbourke/other/interpolation/ (Paul Bourke)
/*
   Tension: 1 is high, 0 normal, -1 is low
   Bias: 0 is even,
         positive is towards first segment,
         negative towards the other
*/
template <class Tp, class Tv>
inline Tv hermite(Tp f, const Tv& w, const Tv& x, const Tv& y, const Tv& z,
                  Tp tension, Tp bias) {
  tension = (Tp(1) - tension) * Tp(0.5);
 
  // compute endpoint tangents
  // Tv m0 = ((x-w)*(1+bias) + (y-x)*(1-bias))*tension;
  // Tv m1 = ((y-x)*(1+bias) + (z-y)*(1-bias))*tension;
  Tv m0 = ((x * Tv(2) - w - y) * bias + y - w) * tension;
  Tv m1 = ((y * Tv(2) - x - z) * bias + z - x) * tension;
 
  //  x - w + x b - w b + y - x - y b + x b
  //  -w + 2x b - w b + y - y b
  //  b(2x - w - y) + y - w
  //
  //  y - x + y b - x b + z - y - z b + y b
  //  -x + 2y b - x b + z - z b
  //  b(2y - x - z) + z - x
 
  Tp f2 = f * f;
  Tp f3 = f2 * f;
 
  // compute hermite basis functions
  Tp a3 = Tp(-2) * f3 + Tp(3) * f2;
  Tp a0 = Tp(1) - a3;
  Tp a2 = f3 - f2;
  Tp a1 = f3 - Tp(2) * f2 + f;
 
  return x * a0 + m0 * a1 + m1 * a2 + y * a3;
}
 
template <class T>
void lagrange(T* a, T delay, int order) {
  for (int i = 0; i <= order; ++i) {
    T coef = T(1);
    T i_f = T(i);
    for (int j = 0; j <= order; ++j) {
      if (j != i) {
        T j_f = (T)j;
        coef *= (delay - j_f) / (i_f - j_f);
      }
    }
    *a++ = coef;
  }
}
 
template <class T>
inline void lagrange1(T* h, T d) {
  h[0] = T(1) - d;
  h[1] = d;
}
 
template <class T>
inline void lagrange2(T* h, T d) {
  h[0] = (d - T(1)) * (d - T(2)) * T(0.5);
  h[1] = -d * (d - T(2));
  h[2] = d * (d - T(1)) * T(0.5);
}
 
template <class T>
inline void lagrange3(T* h, T d) {
  T d1 = d - T(1);
  T d2 = d - T(2);
  T d3 = d - T(3);
  h[0] = -d1 * d2 * d3 * T(1. / 6.);
  h[1] = d * d2 * d3 * T(0.5);
  h[2] = -d * d1 * d3 * T(0.5);
  h[3] = d * d1 * d2 * T(1. / 6.);
}
 
/*
x1 (1 - d) + x0 d
x1 - x1 d + x0 d
x1 + (x0 - x1) d
 
x2 (d - 1) (d - 2) /2 - x1 d (d - 2) + x0 d (d - 1) /2
d d /2 x2 - d 3/2 x2 + x2 - d d x1 + d 2 x1 + d d /2 x0 - d /2 x0
d d /2 x2 - d d x1 + d d /2 x0 - d 3/2 x2 + d 2 x1 - d /2 x0 + x2
d (d (/2 x2 - x1 + /2 x0) - 3/2 x2 + 2 x1 - /2 x0) + x2
*/
 
template <class Tf, class Tv>
inline Tv linear(Tf f, const Tv& x, const Tv& y) {
  return (y - x) * f + x;
}
 
template <class Tf, class Tv>
inline Tv linear(Tf frac, const Tv& x, const Tv& y, const Tv& z) {
  frac *= Tf(2);
  if (frac < Tf(1)) return ipl::linear(frac, x, y);
  return ipl::linear(frac - Tf(1), y, z);
}
 
template <class Tf, class Tv>
void linear(Tv* dst, const Tv* xs, const Tv* xp1s, int len, const Tf& f) {
  for (int i = 0; i < len; ++i) dst[i] = linear(f, xs[i], xp1s[i]);
}
 
template <class Tf, class Tv>
inline Tv linearCyclic(Tf frac, const Tv& x, const Tv& y, const Tv& z) {
  frac *= Tf(3);
  if (frac <= Tf(1))
    return ipl::linear(frac, x, y);
  else if (frac >= Tf(2))
    return ipl::linear(frac - Tf(2), z, x);
  return ipl::linear(frac - Tf(1), y, z);
}
 
template <class Tf, class Tv>
inline Tv nearest(Tf f, const Tv& x, const Tv& y) {
  return (f < Tf(0.5)) ? x : y;
}
 
}  // namespace ipl
}  // namespace al
 
#endif