// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2021
// Distributed under the Boost Software License, Version 1.0.
// https://www.boost.org/LICENSE_1_0.txt
// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
// Version: 4.0.2019.12.05
#pragma once
#include <Mathematics/ApprQuery.h>
#include <Mathematics/Array2.h>
#include <Mathematics/GMatrix.h>
#include <array>
// The samples are (x[i],y[i],z[i],w[i]) for 0 <= i < S. Think of w as a
// function of x, y, and z, say w = f(x,y,z). The function fits the samples
// with a polynomial of degree d0 in x, degree d1 in y, and degree d2 in z,
// say
// w = sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c[i][j][k]*x^i*y^j*z^k
// The method is a least-squares fitting algorithm. The mParameters stores
// c[i][j][k] = mParameters[i+(d0+1)*(j+(d1+1)*k)] for a total of
// (d0+1)*(d1+1)*(d2+1) coefficients. The observation type is
// std::array<Real,4>, which represents a tuple (x,y,z,w).
//
// WARNING. The fitting algorithm for polynomial terms
// (1,x,x^2,...,x^d0), (1,y,y^2,...,y^d1), (1,z,z^2,...,z^d2)
// 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)), (h[0](z),...,h[d2](z))
// and apply the least-squares algorithm to these. Another alternative is to
// transform
// (x',y',z',w') = ((x-xcen)/rng, (y-ycen)/rng, (z-zcen)/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, zmin = min(z[i]),
// zmax = max(z[i]), zcen = (zmin+zmax)/2, and
// rng = max(xmax-xmin,ymax-ymin,zmax-zmin). Fit the (x',y',z',w') points,
// w' = sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c'[i][j][k] *
// (x')^i*(y')^j*(z')^k
// The original polynomial is evaluated as
// w = rng * sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c'[i][j][k] *
// ((x-xcen)/rng)^i * ((y-ycen)/rng)^j * ((z-zcen)/rng)^k
namespace gte
{
template <typename Real>
class ApprPolynomial4 : public ApprQuery<Real, std::array<Real, 4>>
{
public:
// Initialize the model parameters to zero.
ApprPolynomial4(int xDegree, int yDegree, int zDegree)
:
mXDegree(xDegree),
mYDegree(yDegree),
mZDegree(zDegree),
mXDegreeP1(xDegree + 1),
mYDegreeP1(yDegree + 1),
mZDegreeP1(zDegree + 1),
mSize(mXDegreeP1* mYDegreeP1* mZDegreeP1),
mParameters(mSize, (Real)0),
mYZCoefficient(mYDegreeP1 * mZDegreeP1, (Real)0),
mZCoefficient(mZDegreeP1, (Real)0)
{
mXDomain[0] = std::numeric_limits<Real>::max();
mXDomain[1] = -mXDomain[0];
mYDomain[0] = std::numeric_limits<Real>::max();
mYDomain[1] = -mYDomain[0];
mZDomain[0] = std::numeric_limits<Real>::max();
mZDomain[1] = -mZDomain[0];
}
// Basic fitting algorithm. See ApprQuery.h for the various Fit(...)
// functions that you can call.
virtual bool FitIndexed(
size_t numObservations, std::array<Real, 4> const* observations,
size_t numIndices, int const* indices) override
{
if (this->ValidIndices(numObservations, observations, numIndices, indices))
{
int s, i0, j0, k0, n0, i1, j1, k1, n1;
// Compute the powers of x, y, and z.
int numSamples = static_cast<int>(numIndices);
int twoXDegree = 2 * mXDegree;
int twoYDegree = 2 * mYDegree;
int twoZDegree = 2 * mZDegree;
Array2<Real> xPower(twoXDegree + 1, numSamples);
Array2<Real> yPower(twoYDegree + 1, numSamples);
Array2<Real> zPower(twoZDegree + 1, numSamples);
for (s = 0; s < numSamples; ++s)
{
Real x = observations[indices[s]][0];
Real y = observations[indices[s]][1];
Real z = observations[indices[s]][2];
mXDomain[0] = std::min(x, mXDomain[0]);
mXDomain[1] = std::max(x, mXDomain[1]);
mYDomain[0] = std::min(y, mYDomain[0]);
mYDomain[1] = std::max(y, mYDomain[1]);
mZDomain[0] = std::min(z, mZDomain[0]);
mZDomain[1] = std::max(z, mZDomain[1]);
xPower[s][0] = (Real)1;
for (i0 = 1; i0 <= twoXDegree; ++i0)
{
xPower[s][i0] = x * xPower[s][i0 - 1];
}
yPower[s][0] = (Real)1;
for (j0 = 1; j0 <= twoYDegree; ++j0)
{
yPower[s][j0] = y * yPower[s][j0 - 1];
}
zPower[s][0] = (Real)1;
for (k0 = 1; k0 <= twoZDegree; ++k0)
{
zPower[s][k0] = z * zPower[s][k0 - 1];
}
}
// Matrix A is the Vandermonde matrix and vector B is the
// right-hand side of the linear system A*X = B.
GMatrix<Real> A(mSize, mSize);
GVector<Real> B(mSize);
for (k0 = 0; k0 <= mZDegree; ++k0)
{
for (j0 = 0; j0 <= mYDegree; ++j0)
{
for (i0 = 0; i0 <= mXDegree; ++i0)
{
Real sum = (Real)0;
n0 = i0 + mXDegreeP1 * (j0 + mYDegreeP1 * k0);
for (s = 0; s < numSamples; ++s)
{
Real w = observations[indices[s]][3];
sum += w * xPower[s][i0] * yPower[s][j0] * zPower[s][k0];
}
B[n0] = sum;
for (k1 = 0; k1 <= mZDegree; ++k1)
{
for (j1 = 0; j1 <= mYDegree; ++j1)
{
for (i1 = 0; i1 <= mXDegree; ++i1)
{
sum = (Real)0;
n1 = i1 + mXDegreeP1 * (j1 + mYDegreeP1 * k1);
for (s = 0; s < numSamples; ++s)
{
sum += xPower[s][i0 + i1] * yPower[s][j0 + j1] * zPower[s][k0 + k1];
}
A(n0, n1) = sum;
}
}
}
}
}
}
// Solve for the polynomial coefficients.
GVector<Real> coefficients = Inverse(A) * B;
bool hasNonzero = false;
for (int i = 0; i < mSize; ++i)
{
mParameters[i] = coefficients[i];
if (coefficients[i] != (Real)0)
{
hasNonzero = true;
}
}
return hasNonzero;
}
std::fill(mParameters.begin(), mParameters.end(), (Real)0);
return false;
}
// Get the parameters for the best fit.
std::vector<Real> const& GetParameters() const
{
return mParameters;
}
virtual size_t GetMinimumRequired() const override
{
return static_cast<size_t>(mSize);
}
// Compute the model error for the specified observation for the
// current model parameters. The returned value for observation
// (x0,y0,z0,w0) is |w(x0,y0,z0) - w0|, where w(x,y,z) is the fitted
// polynomial.
virtual Real Error(std::array<Real, 4> const& observation) const override
{
Real w = Evaluate(observation[0], observation[1], observation[2]);
Real error = std::fabs(w - observation[3]);
return error;
}
virtual void CopyParameters(ApprQuery<Real, std::array<Real, 4>> const* input) override
{
auto source = dynamic_cast<ApprPolynomial4 const*>(input);
if (source)
{
*this = *source;
}
}
// Evaluate the polynomial. The domain intervals are provided so you
// can interpolate ((x,y,z) in domain) or extrapolate ((x,y,z) not in
// domain).
std::array<Real, 2> const& GetXDomain() const
{
return mXDomain;
}
std::array<Real, 2> const& GetYDomain() const
{
return mYDomain;
}
std::array<Real, 2> const& GetZDomain() const
{
return mZDomain;
}
Real Evaluate(Real x, Real y, Real z) const
{
int i0, i1, i2;
Real w;
for (i2 = 0; i2 <= mZDegree; ++i2)
{
for (i1 = 0; i1 <= mYDegree; ++i1)
{
i0 = mXDegree;
w = mParameters[i0 + mXDegreeP1 * (i1 + mYDegreeP1 * i2)];
while (--i0 >= 0)
{
w = mParameters[i0 + mXDegreeP1 * (i1 + mYDegreeP1 * i2)] + w * x;
}
mYZCoefficient[i1 + mYDegree * i2] = w;
}
}
for (i2 = 0; i2 <= mZDegree; ++i2)
{
i1 = mYDegree;
w = mYZCoefficient[i1 + mYDegreeP1 * i2];
while (--i1 >= 0)
{
w = mParameters[i1 + mYDegreeP1 * i2] + w * y;
}
mZCoefficient[i2] = w;
}
i2 = mZDegree;
w = mZCoefficient[i2];
while (--i2 >= 0)
{
w = mZCoefficient[i2] + w * z;
}
return w;
}
private:
int mXDegree, mYDegree, mZDegree;
int mXDegreeP1, mYDegreeP1, mZDegreeP1, mSize;
std::array<Real, 2> mXDomain, mYDomain, mZDomain;
std::vector<Real> mParameters;
// These arrays are used by Evaluate() to avoid reallocation of the
// 'vector's for each call. The member is mutable because, to the
// user, the call to Evaluate does not modify the polynomial.
mutable std::vector<Real> mYZCoefficient;
mutable std::vector<Real> mZCoefficient;
};
}