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564 lines
18 KiB
564 lines
18 KiB
3 months ago
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// David Eberly, Geometric Tools, Redmond WA 98052
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// Copyright (c) 1998-2021
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// Distributed under the Boost Software License, Version 1.0.
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// https://www.boost.org/LICENSE_1_0.txt
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// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
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// Version: 4.0.2019.08.13
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#pragma once
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#include <Mathematics/Logger.h>
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#include <Mathematics/Math.h>
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#include <Mathematics/Array2.h>
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#include <algorithm>
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#include <array>
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#include <cstring>
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// The interpolator is for uniformly spaced (x,y)-values. The input samples
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// F must be stored in row-major order to represent f(x,y); that is,
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// F[c + xBound*r] corresponds to f(x,y), where c is the index corresponding
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// to x and r is the index corresponding to y.
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namespace gte
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{
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template <typename Real>
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class IntpAkimaUniform2
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{
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public:
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// Construction and destruction.
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IntpAkimaUniform2(int xBound, int yBound, Real xMin, Real xSpacing,
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Real yMin, Real ySpacing, Real const* F)
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:
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mXBound(xBound),
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mYBound(yBound),
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mQuantity(xBound* yBound),
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mXMin(xMin),
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mXSpacing(xSpacing),
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mYMin(yMin),
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mYSpacing(ySpacing),
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mF(F),
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mPoly(xBound - 1, mYBound - 1)
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{
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// At least a 3x3 block of data points is needed to construct the
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// estimates of the boundary derivatives.
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LogAssert(mXBound >= 3 && mYBound >= 3 && mF != nullptr, "Invalid input.");
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LogAssert(mXSpacing > (Real)0 && mYSpacing > (Real)0, "Invalid input.");
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mXMax = mXMin + mXSpacing * static_cast<Real>(mXBound - 1);
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mYMax = mYMin + mYSpacing * static_cast<Real>(mYBound - 1);
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// Create a 2D wrapper for the 1D samples.
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Array2<Real> Fmap(mXBound, mYBound, const_cast<Real*>(mF));
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// Construct first-order derivatives.
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Array2<Real> FX(mXBound, mYBound), FY(mXBound, mYBound);
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GetFX(Fmap, FX);
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GetFY(Fmap, FY);
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// Construct second-order derivatives.
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Array2<Real> FXY(mXBound, mYBound);
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GetFXY(Fmap, FXY);
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// Construct polynomials.
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GetPolynomials(Fmap, FX, FY, FXY);
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}
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~IntpAkimaUniform2() = default;
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// Member access.
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inline int GetXBound() const
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{
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return mXBound;
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}
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inline int GetYBound() const
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{
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return mYBound;
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}
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inline int GetQuantity() const
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{
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return mQuantity;
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}
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inline Real const* GetF() const
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{
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return mF;
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}
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inline Real GetXMin() const
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{
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return mXMin;
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}
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inline Real GetXMax() const
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{
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return mXMax;
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}
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inline Real GetXSpacing() const
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{
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return mXSpacing;
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}
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inline Real GetYMin() const
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{
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return mYMin;
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}
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inline Real GetYMax() const
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{
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return mYMax;
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}
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inline Real GetYSpacing() const
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{
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return mYSpacing;
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}
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// Evaluate the function and its derivatives. The functions clamp the
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// inputs to xmin <= x <= xmax and ymin <= y <= ymax. The first
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// operator is for function evaluation. The second operator is for
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// function or derivative evaluations. The xOrder argument is the
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// order of the x-derivative and the yOrder argument is the order of
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// the y-derivative. Both orders are zero to get the function value
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// itself.
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Real operator()(Real x, Real y) const
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{
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x = std::min(std::max(x, mXMin), mXMax);
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y = std::min(std::max(y, mYMin), mYMax);
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int ix, iy;
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Real dx, dy;
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XLookup(x, ix, dx);
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YLookup(y, iy, dy);
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return mPoly[iy][ix](dx, dy);
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}
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Real operator()(int xOrder, int yOrder, Real x, Real y) const
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{
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x = std::min(std::max(x, mXMin), mXMax);
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y = std::min(std::max(y, mYMin), mYMax);
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int ix, iy;
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Real dx, dy;
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XLookup(x, ix, dx);
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YLookup(y, iy, dy);
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return mPoly[iy][ix](xOrder, yOrder, dx, dy);
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}
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private:
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class Polynomial
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{
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public:
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Polynomial()
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{
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for (size_t i = 0; i < 4; ++i)
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{
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mCoeff[i].fill((Real)0);
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}
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}
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// P(x,y) = (1,x,x^2,x^3)*A*(1,y,y^2,y^3). The matrix term A[ix][iy]
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// corresponds to the polynomial term x^{ix} y^{iy}.
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Real& A(int ix, int iy)
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{
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return mCoeff[ix][iy];
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}
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Real operator()(Real x, Real y) const
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{
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std::array<Real, 4> B;
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for (int i = 0; i <= 3; ++i)
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{
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B[i] = mCoeff[i][0] + y * (mCoeff[i][1] + y * (mCoeff[i][2] + y * mCoeff[i][3]));
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}
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return B[0] + x * (B[1] + x * (B[2] + x * B[3]));
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}
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Real operator()(int xOrder, int yOrder, Real x, Real y) const
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{
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std::array<Real, 4> xPow;
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switch (xOrder)
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{
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case 0:
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xPow[0] = (Real)1;
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xPow[1] = x;
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xPow[2] = x * x;
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xPow[3] = x * x * x;
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break;
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case 1:
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xPow[0] = (Real)0;
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xPow[1] = (Real)1;
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xPow[2] = (Real)2 * x;
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xPow[3] = (Real)3 * x * x;
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break;
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case 2:
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xPow[0] = (Real)0;
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xPow[1] = (Real)0;
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xPow[2] = (Real)2;
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xPow[3] = (Real)6 * x;
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break;
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case 3:
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xPow[0] = (Real)0;
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xPow[1] = (Real)0;
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xPow[2] = (Real)0;
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xPow[3] = (Real)6;
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break;
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default:
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return (Real)0;
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}
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std::array<Real, 4> yPow;
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switch (yOrder)
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{
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case 0:
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yPow[0] = (Real)1;
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yPow[1] = y;
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yPow[2] = y * y;
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yPow[3] = y * y * y;
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break;
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case 1:
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yPow[0] = (Real)0;
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yPow[1] = (Real)1;
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yPow[2] = (Real)2 * y;
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yPow[3] = (Real)3 * y * y;
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break;
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case 2:
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yPow[0] = (Real)0;
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yPow[1] = (Real)0;
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yPow[2] = (Real)2;
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yPow[3] = (Real)6 * y;
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break;
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case 3:
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yPow[0] = (Real)0;
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yPow[1] = (Real)0;
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yPow[2] = (Real)0;
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yPow[3] = (Real)6;
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break;
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default:
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return (Real)0;
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}
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Real p = (Real)0;
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for (size_t iy = 0; iy <= 3; ++iy)
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{
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for (size_t ix = 0; ix <= 3; ++ix)
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{
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p += mCoeff[ix][iy] * xPow[ix] * yPow[iy];
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}
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}
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return p;
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}
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private:
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std::array<std::array<Real, 4>, 4> mCoeff;
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};
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// Support for construction.
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void GetFX(Array2<Real> const& F, Array2<Real>& FX)
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{
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Array2<Real> slope(mXBound + 3, mYBound);
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Real invDX = (Real)1 / mXSpacing;
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int ix, iy;
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for (iy = 0; iy < mYBound; ++iy)
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{
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for (ix = 0; ix < mXBound - 1; ++ix)
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{
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slope[iy][ix + 2] = (F[iy][ix + 1] - F[iy][ix]) * invDX;
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}
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slope[iy][1] = (Real)2 * slope[iy][2] - slope[iy][3];
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slope[iy][0] = (Real)2 * slope[iy][1] - slope[iy][2];
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slope[iy][mXBound + 1] = (Real)2 * slope[iy][mXBound] - slope[iy][mXBound - 1];
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slope[iy][mXBound + 2] = (Real)2 * slope[iy][mXBound + 1] - slope[iy][mXBound];
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}
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for (iy = 0; iy < mYBound; ++iy)
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{
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for (ix = 0; ix < mXBound; ++ix)
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{
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FX[iy][ix] = ComputeDerivative(slope[iy] + ix);
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}
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}
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}
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void GetFY(Array2<Real> const& F, Array2<Real>& FY)
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{
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Array2<Real> slope(mYBound + 3, mXBound);
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Real invDY = (Real)1 / mYSpacing;
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int ix, iy;
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for (ix = 0; ix < mXBound; ++ix)
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{
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for (iy = 0; iy < mYBound - 1; ++iy)
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{
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slope[ix][iy + 2] = (F[iy + 1][ix] - F[iy][ix]) * invDY;
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}
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slope[ix][1] = (Real)2 * slope[ix][2] - slope[ix][3];
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slope[ix][0] = (Real)2 * slope[ix][1] - slope[ix][2];
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slope[ix][mYBound + 1] = (Real)2 * slope[ix][mYBound] - slope[ix][mYBound - 1];
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slope[ix][mYBound + 2] = (Real)2 * slope[ix][mYBound + 1] - slope[ix][mYBound];
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}
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for (ix = 0; ix < mXBound; ++ix)
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{
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for (iy = 0; iy < mYBound; ++iy)
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{
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FY[iy][ix] = ComputeDerivative(slope[ix] + iy);
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}
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}
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}
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void GetFXY(Array2<Real> const& F, Array2<Real>& FXY)
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{
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int xBoundM1 = mXBound - 1;
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int yBoundM1 = mYBound - 1;
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int ix0 = xBoundM1, ix1 = ix0 - 1, ix2 = ix1 - 1;
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int iy0 = yBoundM1, iy1 = iy0 - 1, iy2 = iy1 - 1;
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int ix, iy;
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Real invDXDY = (Real)1 / (mXSpacing * mYSpacing);
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// corners
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FXY[0][0] = (Real)0.25 * invDXDY * (
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(Real)9 * F[0][0]
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- (Real)12 * F[0][1]
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+ (Real)3 * F[0][2]
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- (Real)12 * F[1][0]
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+ (Real)16 * F[1][1]
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- (Real)4 * F[1][2]
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+ (Real)3 * F[2][0]
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- (Real)4 * F[2][1]
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+ F[2][2]);
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FXY[0][xBoundM1] = (Real)0.25 * invDXDY * (
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(Real)9 * F[0][ix0]
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- (Real)12 * F[0][ix1]
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+ (Real)3 * F[0][ix2]
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- (Real)12 * F[1][ix0]
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+ (Real)16 * F[1][ix1]
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- (Real)4 * F[1][ix2]
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+ (Real)3 * F[2][ix0]
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- (Real)4 * F[2][ix1]
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+ F[2][ix2]);
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FXY[yBoundM1][0] = (Real)0.25 * invDXDY * (
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(Real)9 * F[iy0][0]
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- (Real)12 * F[iy0][1]
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+ (Real)3 * F[iy0][2]
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- (Real)12 * F[iy1][0]
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+ (Real)16 * F[iy1][1]
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- (Real)4 * F[iy1][2]
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+ (Real)3 * F[iy2][0]
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- (Real)4 * F[iy2][1]
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+ F[iy2][2]);
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FXY[yBoundM1][xBoundM1] = (Real)0.25 * invDXDY * (
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(Real)9 * F[iy0][ix0]
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- (Real)12 * F[iy0][ix1]
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+ (Real)3 * F[iy0][ix2]
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- (Real)12 * F[iy1][ix0]
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+ (Real)16 * F[iy1][ix1]
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- (Real)4 * F[iy1][ix2]
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|
+ (Real)3 * F[iy2][ix0]
|
||
|
|
- (Real)4 * F[iy2][ix1]
|
||
|
|
+ F[iy2][ix2]);
|
||
|
|
|
||
|
|
// x-edges
|
||
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
||
|
|
{
|
||
|
|
FXY[0][ix] = (Real)0.25 * invDXDY * (
|
||
|
|
(Real)3 * (F[0][ix - 1] - F[0][ix + 1])
|
||
|
|
- (Real)4 * (F[1][ix - 1] - F[1][ix + 1])
|
||
|
|
+ (F[2][ix - 1] - F[2][ix + 1]));
|
||
|
|
|
||
|
|
FXY[yBoundM1][ix] = (Real)0.25 * invDXDY * (
|
||
|
|
(Real)3 * (F[iy0][ix - 1] - F[iy0][ix + 1])
|
||
|
|
- (Real)4 * (F[iy1][ix - 1] - F[iy1][ix + 1])
|
||
|
|
+ (F[iy2][ix - 1] - F[iy2][ix + 1]));
|
||
|
|
}
|
||
|
|
|
||
|
|
// y-edges
|
||
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
||
|
|
{
|
||
|
|
FXY[iy][0] = (Real)0.25 * invDXDY * (
|
||
|
|
(Real)3 * (F[iy - 1][0] - F[iy + 1][0])
|
||
|
|
- (Real)4 * (F[iy - 1][1] - F[iy + 1][1])
|
||
|
|
+ (F[iy - 1][2] - F[iy + 1][2]));
|
||
|
|
|
||
|
|
FXY[iy][xBoundM1] = (Real)0.25 * invDXDY * (
|
||
|
|
(Real)3 * (F[iy - 1][ix0] - F[iy + 1][ix0])
|
||
|
|
- (Real)4 * (F[iy - 1][ix1] - F[iy + 1][ix1])
|
||
|
|
+ (F[iy - 1][ix2] - F[iy + 1][ix2]));
|
||
|
|
}
|
||
|
|
|
||
|
|
// interior
|
||
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
||
|
|
{
|
||
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
||
|
|
{
|
||
|
|
FXY[iy][ix] = (Real)0.25 * invDXDY * (F[iy - 1][ix - 1] -
|
||
|
|
F[iy - 1][ix + 1] - F[iy + 1][ix - 1] + F[iy + 1][ix + 1]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
void GetPolynomials(Array2<Real> const& F, Array2<Real> const& FX,
|
||
|
|
Array2<Real> const& FY, Array2<Real> const& FXY)
|
||
|
|
{
|
||
|
|
int xBoundM1 = mXBound - 1;
|
||
|
|
int yBoundM1 = mYBound - 1;
|
||
|
|
for (int iy = 0; iy < yBoundM1; ++iy)
|
||
|
|
{
|
||
|
|
for (int ix = 0; ix < xBoundM1; ++ix)
|
||
|
|
{
|
||
|
|
// Note the 'transposing' of the 2x2 blocks (to match
|
||
|
|
// notation used in the polynomial definition).
|
||
|
|
Real G[2][2] =
|
||
|
|
{
|
||
|
|
{ F[iy][ix], F[iy + 1][ix] },
|
||
|
|
{ F[iy][ix + 1], F[iy + 1][ix + 1] }
|
||
|
|
};
|
||
|
|
|
||
|
|
Real GX[2][2] =
|
||
|
|
{
|
||
|
|
{ FX[iy][ix], FX[iy + 1][ix] },
|
||
|
|
{ FX[iy][ix + 1], FX[iy + 1][ix + 1] }
|
||
|
|
};
|
||
|
|
|
||
|
|
Real GY[2][2] =
|
||
|
|
{
|
||
|
|
{ FY[iy][ix], FY[iy + 1][ix] },
|
||
|
|
{ FY[iy][ix + 1], FY[iy + 1][ix + 1] }
|
||
|
|
};
|
||
|
|
|
||
|
|
Real GXY[2][2] =
|
||
|
|
{
|
||
|
|
{ FXY[iy][ix], FXY[iy + 1][ix] },
|
||
|
|
{ FXY[iy][ix + 1], FXY[iy + 1][ix + 1] }
|
||
|
|
};
|
||
|
|
|
||
|
|
Construct(mPoly[iy][ix], G, GX, GY, GXY);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
Real ComputeDerivative(Real const* slope) const
|
||
|
|
{
|
||
|
|
if (slope[1] != slope[2])
|
||
|
|
{
|
||
|
|
if (slope[0] != slope[1])
|
||
|
|
{
|
||
|
|
if (slope[2] != slope[3])
|
||
|
|
{
|
||
|
|
Real ad0 = std::fabs(slope[3] - slope[2]);
|
||
|
|
Real ad1 = std::fabs(slope[0] - slope[1]);
|
||
|
|
return (ad0 * slope[1] + ad1 * slope[2]) / (ad0 + ad1);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
return slope[2];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
if (slope[2] != slope[3])
|
||
|
|
{
|
||
|
|
return slope[1];
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
return (Real)0.5 * (slope[1] + slope[2]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
return slope[1];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
void Construct(Polynomial& poly, Real const F[2][2], Real const FX[2][2],
|
||
|
|
Real const FY[2][2], Real const FXY[2][2])
|
||
|
|
{
|
||
|
|
Real dx = mXSpacing;
|
||
|
|
Real dy = mYSpacing;
|
||
|
|
Real invDX = (Real)1 / dx, invDX2 = invDX * invDX;
|
||
|
|
Real invDY = (Real)1 / dy, invDY2 = invDY * invDY;
|
||
|
|
Real b0, b1, b2, b3;
|
||
|
|
|
||
|
|
poly.A(0, 0) = F[0][0];
|
||
|
|
poly.A(1, 0) = FX[0][0];
|
||
|
|
poly.A(0, 1) = FY[0][0];
|
||
|
|
poly.A(1, 1) = FXY[0][0];
|
||
|
|
|
||
|
|
b0 = (F[1][0] - poly(0, 0, dx, (Real)0)) * invDX2;
|
||
|
|
b1 = (FX[1][0] - poly(1, 0, dx, (Real)0)) * invDX;
|
||
|
|
poly.A(2, 0) = (Real)3 * b0 - b1;
|
||
|
|
poly.A(3, 0) = ((Real)-2 * b0 + b1) * invDX;
|
||
|
|
|
||
|
|
b0 = (F[0][1] - poly(0, 0, (Real)0, dy)) * invDY2;
|
||
|
|
b1 = (FY[0][1] - poly(0, 1, (Real)0, dy)) * invDY;
|
||
|
|
poly.A(0, 2) = (Real)3 * b0 - b1;
|
||
|
|
poly.A(0, 3) = ((Real)-2 * b0 + b1) * invDY;
|
||
|
|
|
||
|
|
b0 = (FY[1][0] - poly(0, 1, dx, (Real)0)) * invDX2;
|
||
|
|
b1 = (FXY[1][0] - poly(1, 1, dx, (Real)0)) * invDX;
|
||
|
|
poly.A(2, 1) = (Real)3 * b0 - b1;
|
||
|
|
poly.A(3, 1) = ((Real)-2 * b0 + b1) * invDX;
|
||
|
|
|
||
|
|
b0 = (FX[0][1] - poly(1, 0, (Real)0, dy)) * invDY2;
|
||
|
|
b1 = (FXY[0][1] - poly(1, 1, (Real)0, dy)) * invDY;
|
||
|
|
poly.A(1, 2) = (Real)3 * b0 - b1;
|
||
|
|
poly.A(1, 3) = ((Real)-2 * b0 + b1) * invDY;
|
||
|
|
|
||
|
|
b0 = (F[1][1] - poly(0, 0, dx, dy)) * invDX2 * invDY2;
|
||
|
|
b1 = (FX[1][1] - poly(1, 0, dx, dy)) * invDX * invDY2;
|
||
|
|
b2 = (FY[1][1] - poly(0, 1, dx, dy)) * invDX2 * invDY;
|
||
|
|
b3 = (FXY[1][1] - poly(1, 1, dx, dy)) * invDX * invDY;
|
||
|
|
poly.A(2, 2) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
||
|
|
poly.A(3, 2) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDX;
|
||
|
|
poly.A(2, 3) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDY;
|
||
|
|
poly.A(3, 3) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDX * invDY;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Support for evaluation.
|
||
|
|
void XLookup(Real x, int& xIndex, Real& dx) const
|
||
|
|
{
|
||
|
|
for (xIndex = 0; xIndex + 1 < mXBound; ++xIndex)
|
||
|
|
{
|
||
|
|
if (x < mXMin + mXSpacing * (xIndex + 1))
|
||
|
|
{
|
||
|
|
dx = x - (mXMin + mXSpacing * xIndex);
|
||
|
|
return;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
--xIndex;
|
||
|
|
dx = x - (mXMin + mXSpacing * xIndex);
|
||
|
|
}
|
||
|
|
|
||
|
|
void YLookup(Real y, int& yIndex, Real& dy) const
|
||
|
|
{
|
||
|
|
for (yIndex = 0; yIndex + 1 < mYBound; ++yIndex)
|
||
|
|
{
|
||
|
|
if (y < mYMin + mYSpacing * (yIndex + 1))
|
||
|
|
{
|
||
|
|
dy = y - (mYMin + mYSpacing * yIndex);
|
||
|
|
return;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
yIndex--;
|
||
|
|
dy = y - (mYMin + mYSpacing * yIndex);
|
||
|
|
}
|
||
|
|
|
||
|
|
int mXBound, mYBound, mQuantity;
|
||
|
|
Real mXMin, mXMax, mXSpacing;
|
||
|
|
Real mYMin, mYMax, mYSpacing;
|
||
|
|
Real const* mF;
|
||
|
|
Array2<Polynomial> mPoly;
|
||
|
|
};
|
||
|
|
}
|