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TTInterpolate.h
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TTInterpolate.h
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/** @file TTInterpolate.h
*
* @ingroup foundationLibrary
*
* @brief Interpolation Utilities
* @details Defines several functions for <a href="http://en.wikipedia.org/wiki/Interpolation">interpolating</a> between discrete data points such as those found in an array or matrix. These methods are commonly used in digital audio whenever we alter the rate at which a signal is read.
* These functions require known discrete values to be passed by reference along with a double between 0 and 1 representing the fractional location desired. They return the interpolated value.
*
* @author Timothy Place
*
* @copyright Copyright © 2012, Timothy Place @n
* This code is licensed under the terms of the "New BSD License" @n
* http://creativecommons.org/licenses/BSD/
*/
#ifndef __TT_INTERPOLATE_H__
#define __TT_INTERPOLATE_H__
#include "TTBase.h"
/** Isolate the fractional part from a double.
Essentially wraps the <a href="http://www.cplusplus.com/reference/clibrary/cmath/modf/">modf()</a> function, but protects Jamoma in case it ever prooves uneven in implementation. NW: FAILS TO BUILD but Xcode can't find error.
@param aa Double whose fractional part you would like.
@return The fractional portion of aa.
double TTSplitFractional(double& aa)
{
double discard;
return modf(aa,&discard);
}
*/
/** Linear interpolation.
@param x0 Sample value at prior integer index
@param x1 Sample value at next integer index
@param delta Linear interpolation between x0 (delta=0) and x1 (delta=1)
@return The interpolated value.
@seealso TTInterpolateCosine
@seealso TTInterpolateCubic
@seealso TTInterpolateSpline
@seealso TTInterpolateHermite
*/
template<class T>
T TTInterpolateLinear(const T& x0, const T& x1, const double& delta)
{
return x0 + delta * (x1-x0);
}
/** Cosine interpolation
@param x Sample value at prior integer index
@param y Sample value at next integer index
@param a Fractional location between x (0) and y (1)
@return The interpolated value.
@seealso TTInterpolateLinear
@seealso TTInterpolateHermite
@seealso TTInterpolateCubic
@seealso TTInterpolateSpline
*/
template<class T>
T TTInterpolateCosine(const T& x, const T& y, const double& a)
{
T a2 = 0.5 * (1.0 - cos(a * kTTPi));
return x + a2 * (y-x);
}
/** Cubic interpolation
@details This interpolation algorithms calculate the coefficients a, b, c, d
of the 3rd order polynomial
f(delta) = a*aDelta^3 + b*aDelta^2 + c*aDelta + d
= ( (a*aDelta + b )*aDelta + c)*aDelta + d)
so that the function fulfill the following four conditions:
-# f(0) = x1 @n
-# f(1) = x2 @n
-# f'(0) = (x2-x0)/2 @n
-# f'(1) = (x3-x1)/2
The two last conditions use a symetric estimate of the difference at the end points
of the region to interpolate over: 0 ≤ aDelta ≤ 1
These asumptions ensure that the resulting interpolated function, when moving over several
subsequent sections, is:
-# Continuous (no sudden jump)
-# Has a continuous derivative (no break pints with hard edges)
However, the 2nd order derivate will generally be discontinuous on the points connecting two sections.
@param x0 Sample value at integer index prior to x0
@param x1 Sample value at prior integer index
@param x2 Sample value at next integer index
@param x3 Sample value at integer index after x2
@param aDelta Fractional location where we want to do the interpolation. @n
aDelta = 0 => interpolatedeValue = x1 @n
aDelta = 1 => interpolatedeValue = x2
@return The interpolated value.
@seealso TTInterpolateLinear
@seealso TTInterpolateCosine
@seealso TTInterpolateHermite
@seealso TTInterpolateSpline
*/
template<class T>
T TTInterpolateCubic(const T& x0, const T& x1, const T& x2, const T& x3, const double& aDelta)
{
T a = (-x0 + 3.*x1 - 3*x2 + x3)*0.5;
T b = x0 - 2.5*x1 + 2.*x2 - 0.5*x3;
T c = (x2-x0)*0.5;
//T a = x3 - x2 - x0 + x1;
//T b = x0 - x1 - f0;
//T c = x2 - x0;
// T d = x1;
return ( (a*aDelta + b)*aDelta + c)*aDelta + x1;
}
/** Spline interpolation based on the Breeuwsma catmull-rom spline
@param w Sample value at integer index prior to x
@param x Sample value at prior integer index
@param y Sample value at next integer index
@param z Sample value at integer index after y
@param a Fractional location between x (0) and y (1)
@return The interpolated value.
@seealso TTInterpolateLinear
@seealso TTInterpolateCosine
@seealso TTInterpolateCubic
@seealso TTInterpolateHermite
*/
template<class T>
T TTInterpolateSpline(const T& w, const T& x, const T& y, const T& z, const double& a)
{
T a2 = a*a;
T f0 = -0.5*w + 1.5*x - 1.5*y + 0.5*z;
T f1 = w - 2.5*x + 2.0*y - 0.5*z;
T f2 = -0.5*w + 0.5*y;
return f0*a*a2 + f1*a2 + f2*a + x;
}
/** Hermite interpolation
@param w Sample value at integer index prior to x
@param x Sample value at prior integer index
@param y Sample value at next integer index
@param z Sample value at integer index after y
@param a Fractional location between x (0) and y (1)
@return The interpolated value.
@seealso TTInterpolateLinear
@seealso TTInterpolateCosine
@seealso TTInterpolateCubic
@seealso TTInterpolateSpline
*/
template<class T>
T TTInterpolateHermite(const T& w, const T& x, const T& y, const T& z, const double& a, const double& bias, const double& tension)
{
T aa = a*a;
T aaa = a*aa;
T bp = 1+bias;
T bm = 1-bias;
T mt = (1-tension)*0.5;
T m0 = ((x-w)*bp + (y-x)*bm) * mt;
T m1 = ((y-x)*bp + (z-y)*bm) * mt;
T a0 = 2*aaa - 3*aa + 1;
T a1 = aaa - 2*aa + a;
T a2 = aaa - aa;
T a3 = -2*aaa + 3*aa;
return a0*x + a1*m0 + a2*m1 + a3*y;
}
#endif // __TT_INTERPOLATE_H__