// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2020
// 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: 4.0.2019.08.13
#pragma once
#include <Mathematics/BandedMatrix.h>
#include <Mathematics/GMatrix.h>
#include <Mathematics/Integration.h>
#include <Mathematics/IntrIntervals.h>
#include <Mathematics/Vector.h>
// The BSplineReduction class is an implementation of the algorithm in
// https://www.geometrictools.com/Documentation/BSplineReduction.pdf
// for least-squares fitting of points in the continuous sense by
// an L2 integral norm. The least-squares fitting implemented in the
// file GteBSplineCurveFit.h is in the discrete sense by an L2 summation.
// The intended use for this class is to take an open B-spline curve,
// defined by its control points and degree, and reducing the number of
// control points dramatically to obtain another curve that is close to
// the original one.
namespace WwiseGTE
{
// The input numCtrlPoints must be 2 or larger. The input degree must
// satisfy the condition 1 <= degree <= inControls.size()-1. The degree
// of the output curve is the same as that of the input curve. The input
// fraction must be in [0,1]. If the fraction is 1, the output curve
// is identical to the input curve. If the fraction is too small to
// produce a valid number of control points, outControls.size() is chosen
// to be degree+1.
template <int N, typename Real>
class BSplineReduction
{
public:
void operator()(std::vector<Vector<N, Real>> const& inControls,
int degree, Real fraction, std::vector<Vector<N, Real>>& outControls)
{
int numInControls = static_cast<int>(inControls.size());
LogAssert(numInControls >= 2 && 1 <= degree && degree < numInControls, "Invalid input.");
// Clamp the number of control points to [degree+1,quantity-1].
int numOutControls = static_cast<int>(fraction * numInControls);
if (numOutControls >= numInControls)
{
outControls = inControls;
return;
}
if (numOutControls < degree + 1)
{
numOutControls = degree + 1;
}
// Allocate output control points.
outControls.resize(numOutControls);
// Set up basis function parameters. Function 0 corresponds to
// the output curve. Function 1 corresponds to the input curve.
mDegree = degree;
mQuantity[0] = numOutControls;
mQuantity[1] = numInControls;
for (int j = 0; j <= 1; ++j)
{
mNumKnots[j] = mQuantity[j] + mDegree + 1;
mKnot[j].resize(mNumKnots[j]);
int i;
for (i = 0; i <= mDegree; ++i)
{
mKnot[j][i] = (Real)0;
}
Real factor = (Real)1 / static_cast<Real>(mQuantity[j] - mDegree);
for (/**/; i < mQuantity[j]; ++i)
{
mKnot[j][i] = (i - mDegree) * factor;
}
for (/**/; i < mNumKnots[j]; ++i)
{
mKnot[j][i] = (Real)1;
}
}
// Construct matrix A (depends only on the output basis function).
Real value, tmin, tmax;
int i0, i1;
mBasis[0] = 0;
mBasis[1] = 0;
std::function<Real(Real)> integrand = [this](Real t)
{
Real value0 = F(mBasis[0], mIndex[0], mDegree, t);
Real value1 = F(mBasis[1], mIndex[1], mDegree, t);
Real result = value0 * value1;
return result;
};
BandedMatrix<Real> A(mQuantity[0], mDegree, mDegree);
for (i0 = 0; i0 < mQuantity[0]; ++i0)
{
mIndex[0] = i0;
tmax = MaxSupport(0, i0);
for (i1 = i0; i1 <= i0 + mDegree && i1 < mQuantity[0]; ++i1)
{
mIndex[1] = i1;
tmin = MinSupport(0, i1);
value = Integration<Real>::Romberg(8, tmin, tmax, integrand);
A(i0, i1) = value;
A(i1, i0) = value;
}
}
// Construct A^{-1}. TODO: This is inefficient. Use an iterative
// scheme to invert A?
GMatrix<Real> invA(mQuantity[0], mQuantity[0]);
bool invertible = A.template ComputeInverse<true>(&invA[0]);
LogAssert(invertible, "Failed to invert matrix.");
// Construct B (depends on both input and output basis functions).
mBasis[1] = 1;
GMatrix<Real> B(mQuantity[0], mQuantity[1]);
FIQuery<Real, std::array<Real, 2>, std::array<Real, 2>> query;
for (i0 = 0; i0 < mQuantity[0]; ++i0)
{
mIndex[0] = i0;
Real tmin0 = MinSupport(0, i0);
Real tmax0 = MaxSupport(0, i0);
for (i1 = 0; i1 < mQuantity[1]; ++i1)
{
mIndex[1] = i1;
Real tmin1 = MinSupport(1, i1);
Real tmax1 = MaxSupport(1, i1);
std::array<Real, 2> interval0 = { tmin0, tmax0 };
std::array<Real, 2> interval1 = { tmin1, tmax1 };
auto result = query(interval0, interval1);
if (result.numIntersections == 2)
{
value = Integration<Real>::Romberg(8, result.overlap[0],
result.overlap[1], integrand);
B(i0, i1) = value;
}
else
{
B(i0, i1) = (Real)0;
}
}
}
// Construct A^{-1}*B.
GMatrix<Real> prod = invA * B;
// Construct the control points for the least-squares curve.
std::fill(outControls.begin(), outControls.end(), Vector<N, Real>::Zero());
for (i0 = 0; i0 < mQuantity[0]; ++i0)
{
for (i1 = 0; i1 < mQuantity[1]; ++i1)
{
outControls[i0] += inControls[i1] * prod(i0, i1);
}
}
}
private:
inline Real MinSupport(int basis, int i) const
{
return mKnot[basis][i];
}
inline Real MaxSupport(int basis, int i) const
{
return mKnot[basis][i + 1 + mDegree];
}
Real F(int basis, int i, int j, Real t)
{
if (j > 0)
{
Real result = (Real)0;
Real denom = mKnot[basis][i + j] - mKnot[basis][i];
if (denom > (Real)0)
{
result += (t - mKnot[basis][i]) *
F(basis, i, j - 1, t) / denom;
}
denom = mKnot[basis][i + j + 1] - mKnot[basis][i + 1];
if (denom > (Real)0)
{
result += (mKnot[basis][i + j + 1] - t) *
F(basis, i + 1, j - 1, t) / denom;
}
return result;
}
if (mKnot[basis][i] <= t && t < mKnot[basis][i + 1])
{
return (Real)1;
}
else
{
return (Real)0;
}
}
int mDegree;
std::array<int, 2> mQuantity;
std::array<int, 2> mNumKnots; // N+D+2
std::array<std::vector<Real>, 2> mKnot;
// For the integration-based least-squares fitting.
std::array<int, 2> mBasis, mIndex;
};
}