Wm4ApprPolyFit3.h

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// Wild Magic Source Code
// David Eberly
// http://www.geometrictools.com
// Copyright (c) 1998-2007
//
// This library is free software; you can redistribute it and/or modify it
// under the terms of the GNU Lesser General Public License as published by
// the Free Software Foundation; either version 2.1 of the License, or (at
// your option) any later version.  The license is available for reading at
// either of the locations:
//     http://www.gnu.org/copyleft/lgpl.html
//     http://www.geometrictools.com/License/WildMagicLicense.pdf
// The license applies to versions 0 through 4 of Wild Magic.
//
// Version: 4.0.0 (2006/06/28)

#ifndef WM4APPRPOLYFIT3_H
#define WM4APPRPOLYFIT3_H

#include "Wm4FoundationLIB.h"
#include "Wm4System.h"

namespace Wm4
{

// 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 <class Real> WM4_FOUNDATION_ITEM
Real* PolyFit3 (int iSamples, const Real* afX, const Real* afY,
    const Real* afW, int iXDegree, int iYDegree);

}

#endif