spline.h
// ---------------------------------------------------------------------------
// This file is part of reSID, a MOS6581 SID emulator engine.
// Copyright (C) 2010 Dag Lem <resid@nimrod.no>
//
// 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 2 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 GNU General Public License
// along with this program; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
// ---------------------------------------------------------------------------
#ifndef RESID_SPLINE_H
#define RESID_SPLINE_H
namespace reSID
{
// Our objective is to construct a smooth interpolating single-valued function
// y = f(x).
//
// Catmull-Rom splines are widely used for interpolation, however these are
// parametric curves [x(t) y(t) ...] and can not be used to directly calculate
// y = f(x).
// For a discussion of Catmull-Rom splines see Catmull, E., and R. Rom,
// "A Class of Local Interpolating Splines", Computer Aided Geometric Design.
//
// Natural cubic splines are single-valued functions, and have been used in
// several applications e.g. to specify gamma curves for image display.
// These splines do not afford local control, and a set of linear equations
// including all interpolation points must be solved before any point on the
// curve can be calculated. The lack of local control makes the splines
// more difficult to handle than e.g. Catmull-Rom splines, and real-time
// interpolation of a stream of data points is not possible.
// For a discussion of natural cubic splines, see e.g. Kreyszig, E., "Advanced
// Engineering Mathematics".
//
// Our approach is to approximate the properties of Catmull-Rom splines for
// piecewice cubic polynomials f(x) = ax^3 + bx^2 + cx + d as follows:
// Each curve segment is specified by four interpolation points,
// p0, p1, p2, p3.
// The curve between p1 and p2 must interpolate both p1 and p2, and in addition
// f'(p1.x) = k1 = (p2.y - p0.y)/(p2.x - p0.x) and
// f'(p2.x) = k2 = (p3.y - p1.y)/(p3.x - p1.x).
//
// The constraints are expressed by the following system of linear equations
//
// [ 1 xi xi^2 xi^3 ] [ d ] [ yi ]
// [ 1 2*xi 3*xi^2 ] * [ c ] = [ ki ]
// [ 1 xj xj^2 xj^3 ] [ b ] [ yj ]
// [ 1 2*xj 3*xj^2 ] [ a ] [ kj ]
//
// Solving using Gaussian elimination and back substitution, setting
// dy = yj - yi, dx = xj - xi, we get
//
// a = ((ki + kj) - 2*dy/dx)/(dx*dx);
// b = ((kj - ki)/dx - 3*(xi + xj)*a)/2;
// c = ki - (3*xi*a + 2*b)*xi;
// d = yi - ((xi*a + b)*xi + c)*xi;
//
// Having calculated the coefficients of the cubic polynomial we have the
// choice of evaluation by brute force
//
// for (x = x1; x <= x2; x += res) {
// y = ((a*x + b)*x + c)*x + d;
// plot(x, y);
// }
//
// or by forward differencing
//
// y = ((a*x1 + b)*x1 + c)*x1 + d;
// dy = (3*a*(x1 + res) + 2*b)*x1*res + ((a*res + b)*res + c)*res;
// d2y = (6*a*(x1 + res) + 2*b)*res*res;
// d3y = 6*a*res*res*res;
//
// for (x = x1; x <= x2; x += res) {
// plot(x, y);
// y += dy; dy += d2y; d2y += d3y;
// }
//
// See Foley, Van Dam, Feiner, Hughes, "Computer Graphics, Principles and
// Practice" for a discussion of forward differencing.
//
// If we have a set of interpolation points p0, ..., pn, we may specify
// curve segments between p0 and p1, and between pn-1 and pn by using the
// following constraints:
// f''(p0.x) = 0 and
// f''(pn.x) = 0.
//
// Substituting the results for a and b in
//
// 2*b + 6*a*xi = 0
//
// we get
//
// ki = (3*dy/dx - kj)/2;
//
// or by substituting the results for a and b in
//
// 2*b + 6*a*xj = 0
//
// we get
//
// kj = (3*dy/dx - ki)/2;
//
// Finally, if we have only two interpolation points, the cubic polynomial
// will degenerate to a straight line if we set
//
// ki = kj = dy/dx;
//
#if SPLINE_BRUTE_FORCE
#define interpolate_segment interpolate_brute_force
#else
#define interpolate_segment interpolate_forward_difference
#endif
// ----------------------------------------------------------------------------
// Calculation of coefficients.
// ----------------------------------------------------------------------------
inline
void cubic_coefficients(double x1, double y1, double x2, double y2,
double k1, double k2,
double& a, double& b, double& c, double& d)
{
double dx = x2 - x1, dy = y2 - y1;
a = ((k1 + k2) - 2*dy/dx)/(dx*dx);
b = ((k2 - k1)/dx - 3*(x1 + x2)*a)/2;
c = k1 - (3*x1*a + 2*b)*x1;
d = y1 - ((x1*a + b)*x1 + c)*x1;
}
// ----------------------------------------------------------------------------
// Evaluation of cubic polynomial by brute force.
// ----------------------------------------------------------------------------
template<class PointPlotter>
inline
void interpolate_brute_force(double x1, double y1, double x2, double y2,
double k1, double k2,
PointPlotter plot, double res)
{
double a, b, c, d;
cubic_coefficients(x1, y1, x2, y2, k1, k2, a, b, c, d);
// Calculate each point.
for (double x = x1; x <= x2; x += res) {
double y = ((a*x + b)*x + c)*x + d;
plot(x, y);
}
}
// ----------------------------------------------------------------------------
// Evaluation of cubic polynomial by forward differencing.
// ----------------------------------------------------------------------------
template<class PointPlotter>
inline
void interpolate_forward_difference(double x1, double y1, double x2, double y2,
double k1, double k2,
PointPlotter plot, double res)
{
double a, b, c, d;
cubic_coefficients(x1, y1, x2, y2, k1, k2, a, b, c, d);
double y = ((a*x1 + b)*x1 + c)*x1 + d;
double dy = (3*a*(x1 + res) + 2*b)*x1*res + ((a*res + b)*res + c)*res;
double d2y = (6*a*(x1 + res) + 2*b)*res*res;
double d3y = 6*a*res*res*res;
// Calculate each point.
for (double x = x1; x <= x2; x += res) {
plot(x, y);
y += dy; dy += d2y; d2y += d3y;
}
}
template<class PointIter>
inline
double x(PointIter p)
{
return (*p)[0];
}
template<class PointIter>
inline
double y(PointIter p)
{
return (*p)[1];
}
// ----------------------------------------------------------------------------
// Evaluation of complete interpolating function.
// Note that since each curve segment is controlled by four points, the
// end points will not be interpolated. If extra control points are not
// desirable, the end points can simply be repeated to ensure interpolation.
// Note also that points of non-differentiability and discontinuity can be
// introduced by repeating points.
// ----------------------------------------------------------------------------
template<class PointIter, class PointPlotter>
inline
void interpolate(PointIter p0, PointIter pn, PointPlotter plot, double res)
{
double k1, k2;
// Set up points for first curve segment.
PointIter p1 = p0; ++p1;
PointIter p2 = p1; ++p2;
PointIter p3 = p2; ++p3;
// Draw each curve segment.
for (; p2 != pn; ++p0, ++p1, ++p2, ++p3) {
// p1 and p2 equal; single point.
if (x(p1) == x(p2)) {
continue;
}
// Both end points repeated; straight line.
if (x(p0) == x(p1) && x(p2) == x(p3)) {
k1 = k2 = (y(p2) - y(p1))/(x(p2) - x(p1));
}
// p0 and p1 equal; use f''(x1) = 0.
else if (x(p0) == x(p1)) {
k2 = (y(p3) - y(p1))/(x(p3) - x(p1));
k1 = (3*(y(p2) - y(p1))/(x(p2) - x(p1)) - k2)/2;
}
// p2 and p3 equal; use f''(x2) = 0.
else if (x(p2) == x(p3)) {
k1 = (y(p2) - y(p0))/(x(p2) - x(p0));
k2 = (3*(y(p2) - y(p1))/(x(p2) - x(p1)) - k1)/2;
}
// Normal curve.
else {
k1 = (y(p2) - y(p0))/(x(p2) - x(p0));
k2 = (y(p3) - y(p1))/(x(p3) - x(p1));
}
interpolate_segment(x(p1), y(p1), x(p2), y(p2), k1, k2, plot, res);
}
}
// ----------------------------------------------------------------------------
// Class for plotting integers into an array.
// ----------------------------------------------------------------------------
template<class F>
class PointPlotter
{
protected:
F* f;
public:
PointPlotter(F* arr) : f(arr)
{
}
void operator ()(double x, double y)
{
// Clamp negative values to zero.
if (y < 0) {
y = 0;
}
f[int(x)] = F(y + 0.5);
}
};
} // namespace reSID
#endif // not RESID_SPLINE_H
reSID
Definition: dac.cc:45
reSID::PointPlotter
Definition: spline.h:253