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