// 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/BasisFunction.h>
#include <Mathematics/Vector3.h>
// The algorithm implemented here is based on the document
// https://www.geometrictools.com/Documentation/BSplineSurfaceLeastSquaresFit.pdf
namespace WwiseGTE
{
template <typename Real>
class BSplineSurfaceFit
{
public:
// Construction. The preconditions for calling the constructor are
// 1 <= degree0 && degree0 + 1 < numControls0 <= numSamples0
// 1 <= degree1 && degree1 + 1 < numControls1 <= numSamples1
// The sample data must be in row-major order. The control data is
// also stored in row-major order.
BSplineSurfaceFit(int degree0, int numControls0, int numSamples0,
int degree1, int numControls1, int numSamples1, Vector3<Real> const* sampleData)
:
mSampleData(sampleData),
mControlData(numControls0 * numControls1)
{
LogAssert(1 <= degree0 && degree0 + 1 < numControls0, "Invalid degree.");
LogAssert(numControls0 <= numSamples0, "Invalid number of controls.");
LogAssert(1 <= degree1 && degree1 + 1 < numControls1, "Invalid degree.");
LogAssert(numControls1 <= numSamples1, "Invalid number of controls.");
LogAssert(sampleData, "Invalid sample data.");
mDegree[0] = degree0;
mNumSamples[0] = numSamples0;
mNumControls[0] = numControls0;
mDegree[1] = degree1;
mNumSamples[1] = numSamples1;
mNumControls[1] = numControls1;
BasisFunctionInput<Real> input;
Real tMultiplier[2];
int dim;
for (dim = 0; dim < 2; ++dim)
{
input.numControls = mNumControls[dim];
input.degree = mDegree[dim];
input.uniform = true;
input.periodic = false;
input.numUniqueKnots = mNumControls[dim] - mDegree[dim] + 1;
input.uniqueKnots.resize(input.numUniqueKnots);
input.uniqueKnots[0].t = (Real)0;
input.uniqueKnots[0].multiplicity = mDegree[dim] + 1;
int last = input.numUniqueKnots - 1;
Real factor = (Real)1 / (Real)last;
for (int i = 1; i < last; ++i)
{
input.uniqueKnots[i].t = factor * (Real)i;
input.uniqueKnots[i].multiplicity = 1;
}
input.uniqueKnots[last].t = (Real)1;
input.uniqueKnots[last].multiplicity = mDegree[dim] + 1;
mBasis[dim].Create(input);
tMultiplier[dim] = ((Real)1) / (Real)(mNumSamples[dim] - 1);
}
// Fit the data points with a B-spline surface using a
// least-squares error metric. The problem is of the form
// A0^T*A0*Q*A1^T*A1 = A0^T*P*A1, where A0^T*A0 and A1^T*A1 are
// banded matrices, P contains the sample data, and Q is the
// unknown matrix of control points.
Real t;
int i0, i1, i2, imin, imax;
// Construct the matrices A0^T*A0 and A1^T*A1.
BandedMatrix<Real> ATAMat[2] =
{
BandedMatrix<Real>(mNumControls[0], mDegree[0] + 1, mDegree[0] + 1),
BandedMatrix<Real>(mNumControls[1], mDegree[1] + 1, mDegree[1] + 1)
};
for (dim = 0; dim < 2; ++dim)
{
for (i0 = 0; i0 < mNumControls[dim]; ++i0)
{
for (i1 = 0; i1 < i0; ++i1)
{
ATAMat[dim](i0, i1) = ATAMat[dim](i1, i0);
}
int i1Max = i0 + mDegree[dim];
if (i1Max >= mNumControls[dim])
{
i1Max = mNumControls[dim] - 1;
}
for (i1 = i0; i1 <= i1Max; ++i1)
{
Real value = (Real)0;
for (i2 = 0; i2 < mNumSamples[dim]; ++i2)
{
t = tMultiplier[dim] * (Real)i2;
mBasis[dim].Evaluate(t, 0, imin, imax);
if (imin <= i0 && i0 <= imax && imin <= i1 && i1 <= imax)
{
Real b0 = mBasis[dim].GetValue(0, i0);
Real b1 = mBasis[dim].GetValue(0, i1);
value += b0 * b1;
}
}
ATAMat[dim](i0, i1) = value;
}
}
}
// Construct the matrices A0^T and A1^T. A[d]^T has
// mNumControls[d] rows and mNumSamples[d] columns.
Array2<Real> ATMat[2];
for (dim = 0; dim < 2; dim++)
{
ATMat[dim] = Array2<Real>(mNumSamples[dim], mNumControls[dim]);
size_t numBytes = mNumControls[dim] * mNumSamples[dim] * sizeof(Real);
std::memset(ATMat[dim][0], 0, numBytes);
for (i0 = 0; i0 < mNumControls[dim]; ++i0)
{
for (i1 = 0; i1 < mNumSamples[dim]; ++i1)
{
t = tMultiplier[dim] * (Real)i1;
mBasis[dim].Evaluate(t, 0, imin, imax);
if (imin <= i0 && i0 <= imax)
{
ATMat[dim][i0][i1] = mBasis[dim].GetValue(0, i0);
}
}
}
}
// Compute X0 = (A0^T*A0)^{-1}*A0^T and X1 = (A1^T*A1)^{-1}*A1^T
// by solving the linear systems A0^T*A0*X0 = A0^T and
// A1^T*A1*X1 = A1^T.
for (dim = 0; dim < 2; ++dim)
{
bool solved = ATAMat[dim].template SolveSystem<true>(ATMat[dim][0], mNumSamples[dim]);
LogAssert(solved, "Failed to solve linear system in BSplineSurfaceFit constructor.");
}
// The control points for the fitted surface are stored in the matrix
// Q = X0*P*X1^T, where P is the matrix of sample data.
for (i1 = 0; i1 < mNumControls[1]; ++i1)
{
for (i0 = 0; i0 < mNumControls[0]; ++i0)
{
Vector3<Real> sum = Vector3<Real>::Zero();
for (int j1 = 0; j1 < mNumSamples[1]; ++j1)
{
Real x1Value = ATMat[1][i1][j1];
for (int j0 = 0; j0 < mNumSamples[0]; ++j0)
{
Real x0Value = ATMat[0][i0][j0];
Vector3<Real> sample =
mSampleData[j0 + mNumSamples[0] * j1];
sum += (x0Value * x1Value) * sample;
}
}
mControlData[i0 + mNumControls[0] * i1] = sum;
}
}
}
// Access to input sample information.
inline int GetNumSamples(int dimension) const
{
return mNumSamples[dimension];
}
inline Vector3<Real> const* GetSampleData() const
{
return mSampleData;
}
// Access to output control point and surface information.
inline int GetDegree(int dimension) const
{
return mDegree[dimension];
}
inline int GetNumControls(int dimension) const
{
return mNumControls[dimension];
}
inline Vector3<Real> const* GetControlData() const
{
return &mControlData[0];
}
inline BasisFunction<Real> const& GetBasis(int dimension) const
{
return mBasis[dimension];
}
// Evaluation of the B-spline surface. It is defined for
// 0 <= u <= 1 and 0 <= v <= 1. If a parameter value is outside
// [0,1], it is clamped to [0,1].
Vector3<Real> GetPosition(Real u, Real v) const
{
int iumin, iumax, ivmin, ivmax;
mBasis[0].Evaluate(u, 0, iumin, iumax);
mBasis[1].Evaluate(v, 0, ivmin, ivmax);
Vector3<Real> position = Vector3<Real>::Zero();
for (int iv = ivmin; iv <= ivmax; ++iv)
{
Real value1 = mBasis[1].GetValue(0, iv);
for (int iu = iumin; iu <= iumax; ++iu)
{
Real value0 = mBasis[0].GetValue(0, iu);
Vector3<Real> control = mControlData[iu + mNumControls[0] * iv];
position += (value0 * value1) * control;
}
}
return position;
}
private:
// Input sample information.
int mNumSamples[2];
Vector3<Real> const* mSampleData;
// The fitted B-spline surface, open and with uniform knots.
int mDegree[2];
int mNumControls[2];
std::vector<Vector3<Real>> mControlData;
BasisFunction<Real> mBasis[2];
};
}