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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | // Geometric Tools, LLC
// Copyright (c) 1998-2014
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
//
// File Version: 5.0.1 (2010/10/01)
#ifndef WM5APPRPOLYFIT3_H
#define WM5APPRPOLYFIT3_H
#include "Wm5MathematicsLIB.h"
namespace Wm5
{
// The samples are (x[i],y[i],w[i]) for 0 <= i < S. Think of w as a function
// of x and y, say w = f(x,y). The function fits the samples with a
// polynomial of degree d0 in x and degree d1 in y, say
// w = sum_{i=0}^{d0} sum_{j=0}^{d1} c[i][j]*x^i*y^j
// The method is a least-squares fitting algorithm. The returned array
// stores the c[i][j] values according to returned[i+(d0+1)*j] = c[i][j]
// for a total of (d0+1)*(d1+1) coefficients. The caller is responsible for
// deleting the input arrays if they were dynamically allocated. The caller
// is also responsible for deleting the returned array.
//
// WARNING. The fitting algorithm for polynomial terms
// (1,x,x^2,...,x^d0), (1,y,y^2,...,y^d1)
// is known to be nonrobust for large degrees and for large magnitude data.
// One alternative is to use orthogonal polynomials
// (f[0](x),...,f[d0](x)), (g[0](y),...,g[d1](y))
// and apply the least-squares algorithm to these. Another alternative is to
// transform
// (x',y',w') = ((x-xcen)/rng, (y-ycen)/rng, w/rng)
// where xmin = min(x[i]), xmax = max(x[i]), xcen = (xmin+xmax)/2,
// ymin = min(y[i]), ymax = max(y[i]), ycen = (ymin+ymax)/2, and
// rng = max(xmax-xmin,ymax-ymin). Fit the (x',y',w') points,
// w' = sum_{i=0}^{d0} sum_{j=0}^{d1} c'[i][j]*(x')^i*(y')^j
// The original polynomial is evaluated as
// w = rng * sum_{i=0}^{d0} sum_{j=0}^{d1} c'[i][j] *
// ((x-xcen)/rng)^i * ((y-ycen)/rng)^j
template <typename Real> WM5_MATHEMATICS_ITEM
Real* PolyFit3 (int numSamples, const Real* xSamples, const Real* ySamples,
const Real* wSamples, int xDegree, int yDegree);
}
#endif
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