GteApprPolynomial3.h

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// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2017
// 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: 3.0.0 (2016/06/19)

#pragma once

#include <LowLevel/GteArray2.h>
#include <Mathematics/GteGMatrix.h>
#include <Mathematics/GteApprQuery.h>
#include <array>

// 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 mParameters stores
// c[i][j] = mParameters[i+(d0+1)*j] for a total of (d0+1)*(d1+1)
// coefficients.  The observation type is std::array<Real,3>, which represents
// a triple (x,y,w).
//
// 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

namespace gte
{

template <typename Real>
class ApprPolynomial3
    :
    public ApprQuery<Real, ApprPolynomial3<Real>, std::array<Real, 3>>
{
public:
    // Initialize the model parameters to zero.
    ApprPolynomial3(int xDegree, int yDegree);

    // The minimum number of observations required to fit the model.
    int GetMinimumRequired() const;

    // Estimate the model parameters for all observations specified by the
    // indices.  This function is called by the base-class Fit(...) functions.
    bool Fit(std::vector<std::array<Real, 3>> const& observations,
        std::vector<int> const& indices);

    // Compute the model error for the specified observation for the current
    // model parameters.  The returned value for observation (x0,y0,w0) is
    // |w(x0,y0) - w0|, where w(x,y) is the fitted polynomial.
    Real Error(std::array<Real, 3> const& observation) const;

    // Get the parameters of the model.
    std::vector<Real> const& GetParameters() const;

    // Evaluate the polynomial.  The domain intervals are provided so you can
    // interpolate ((x,y) in domain) or extrapolate ((x,y) not in domain).
    std::array<Real, 2> const& GetXDomain() const;
    std::array<Real, 2> const& GetYDomain() const;
    Real Evaluate(Real x, Real y) const;

private:
    int mXDegree, mYDegree, mXDegreeP1, mYDegreeP1, mSize;
    std::array<Real, 2> mXDomain, mYDomain;
    std::vector<Real> mParameters;

    // This array is used by Evaluate() to avoid reallocation of the 'vector'
    // for each call.  The member is mutable because, to the user, the call
    // to Evaluate does not modify the polynomial.
    mutable std::vector<Real> mYCoefficient;
};


template <typename Real>
ApprPolynomial3<Real>::ApprPolynomial3(int xDegree, int yDegree)
    :
    mXDegree(xDegree),
    mYDegree(yDegree),
    mXDegreeP1(xDegree + 1),
    mYDegreeP1(yDegree + 1),
    mSize(mXDegreeP1 * mYDegreeP1),
    mParameters(mSize),
    mYCoefficient(mYDegreeP1)
{
    mXDomain[0] = std::numeric_limits<Real>::max();
    mXDomain[1] = -mXDomain[0];
    mYDomain[0] = std::numeric_limits<Real>::max();
    mYDomain[1] = -mYDomain[0];
    std::fill(mParameters.begin(), mParameters.end(), (Real)0);
    std::fill(mYCoefficient.begin(), mYCoefficient.end(), (Real)0);
}

template <typename Real>
int ApprPolynomial3<Real>::GetMinimumRequired() const
{
    return mSize;
}

template <typename Real>
bool ApprPolynomial3<Real>::Fit(
    std::vector<std::array<Real, 3>> const& observations,
    std::vector<int> const& indices)
{
    if (indices.size() > 0)
    {
        int s, i0, j0, k0, i1, j1, k1;

        // Compute the powers of x and y.
        int numSamples = static_cast<int>(indices.size());
        int twoXDegree = 2 * mXDegree;
        int twoYDegree = 2 * mYDegree;
        Array2<Real> xPower(twoXDegree + 1, numSamples);
        Array2<Real> yPower(twoYDegree + 1, numSamples);
        for (s = 0; s < numSamples; ++s)
        {
            Real x = observations[indices[s]][0];
            Real y = observations[indices[s]][1];
            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]);

            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];
            }
        }

        // 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 (j0 = 0; j0 <= mYDegree; ++j0)
        {
            for (i0 = 0; i0 <= mXDegree; ++i0)
            {
                Real sum = (Real)0;
                k0 = i0 + mXDegreeP1 * j0;
                for (s = 0; s < numSamples; ++s)
                {
                    Real w = observations[indices[s]][2];
                    sum += w * xPower[s][i0] * yPower[s][j0];
                }

                B[k0] = sum;

                for (j1 = 0; j1 <= mYDegree; ++j1)
                {
                    for (i1 = 0; i1 <= mXDegree; ++i1)
                    {
                        sum = (Real)0;
                        k1 = i1 + mXDegreeP1 * j1;
                        for (s = 0; s < numSamples; ++s)
                        {
                            sum += xPower[s][i0 + i1] * yPower[s][j0 + j1];
                        }

                        A(k0, k1) = 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;
}

template <typename Real>
Real ApprPolynomial3<Real>::Error(std::array<Real, 3> const& observation)
const
{
    Real w = Evaluate(observation[0], observation[1]);
    Real error = std::abs(w - observation[2]);
    return error;
}

template <typename Real>
std::vector<Real> const& ApprPolynomial3<Real>::GetParameters() const
{
    return mParameters;
}

template <typename Real>
std::array<Real, 2> const& ApprPolynomial3<Real>::GetXDomain() const
{
    return mXDomain;
}

template <typename Real>
std::array<Real, 2> const& ApprPolynomial3<Real>::GetYDomain() const
{
    return mYDomain;
}

template <typename Real>
Real ApprPolynomial3<Real>::Evaluate(Real x, Real y) const
{
    int i0, i1;
    Real w;

    for (i1 = 0; i1 <= mYDegree; ++i1)
    {
        i0 = mXDegree;
        w = mParameters[i0 + mXDegreeP1 * i1];
        while (--i0 >= 0)
        {
            w = mParameters[i0 + mXDegreeP1 * i1] + w * x;
        }
        mYCoefficient[i1] = w;
    }

    i1 = mYDegree;
    w = mYCoefficient[i1];
    while (--i1 >= 0)
    {
        w = mYCoefficient[i1] + w * y;
    }

    return w;
}


}