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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 50 51 52 53 54 55 56 57 58 | // 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 WM5APPRPARABOLOIDFIT3_H
#define WM5APPRPARABOLOIDFIT3_H
#include "Wm5MathematicsLIB.h"
#include "Wm5Vector3.h"
namespace Wm5
{
// Least-squares fit of a paraboloid to a set of point. The paraboloid is
// of the form z = c0*x^2+c1*x*y+c2*y^2+c3*x+c4*y+c5. A successful fit is
// indicated by return value of 'true'.
//
// Given a set of samples (x_i,y_i,z_i) for 0 <= i < N, and assuming
// that the true values lie on a paraboloid
// z = p0*x*x + p1*x*y + p2*y*y + p3*x + p4*y + p5 = Dot(P,Q(x,y))
// where P = (p0,p1,p2,p3,p4,p5) and Q(x,y) = (x*x,x*y,y*y,x,y,1),
// select P to minimize the sum of squared errors
// E(P) = sum_{i=0}^{N-1} [Dot(P,Q_i)-z_i]^2
// where Q_i = Q(x_i,y_i).
//
// The minimum occurs when the gradient of E is the zero vector,
// grad(E) = 2 sum_{i=0}^{N-1} [Dot(P,Q_i)-z_i] Q_i = 0
// Some algebra converts this to a system of 6 equations in 6 unknowns:
// [(sum_{i=0}^{N-1} Q_i Q_i^t] P = sum_{i=0}^{N-1} z_i Q_i
// The product Q_i Q_i^t is a product of the 6x1 matrix Q_i with the
// 1x6 matrix Q_i^t, the result being a 6x6 matrix.
//
// Define the 6x6 symmetric matrix A = sum_{i=0}^{N-1} Q_i Q_i^t and the 6x1
// vector B = sum_{i=0}^{N-1} z_i Q_i. The choice for P is the solution to
// the linear system of equations A*P = B. The entries of A and B indicate
// summations over the appropriate product of variables. For example,
// s(x^3 y) = sum_{i=0}^{N-1} x_i^3 y_i.
//
// +- -++ + +- -+
// | s(x^4) s(x^3 y) s(x^2 y^2) s(x^3) s(x^2 y) s(x^2) ||p0| |s(z x^2)|
// | s(x^2 y^2) s(x y^3) s(x^2 y) s(x y^2) s(x y) ||p1| |s(z x y)|
// | s(y^4) s(x y^2) s(y^3) s(y^2) ||p2| = |s(z y^2)|
// | s(x^2) s(x y) s(x) ||p3| |s(z x) |
// | s(y^2) s(y) ||p4| |s(z y) |
// | s(1) ||p5| |s(z) |
// +- -++ + +- -+
template <typename Real> WM5_MATHEMATICS_ITEM
bool ParaboloidFit3 (int numPoints, const Vector3<Real>* points,
Real coeff[6]);
}
#endif
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