RBFSpline::RBFSpline(const DataTable &samples, RadialBasisFunctionType type, bool normalized)
: samples(samples),
normalized(normalized),
precondition(false),
dim(samples.getNumVariables()),
numSamples(samples.getNumSamples())
{
if (type == RadialBasisFunctionType::THIN_PLATE_SPLINE)
{
fn = std::shared_ptr<RadialBasisFunction>(new ThinPlateSpline());
}
else if (type == RadialBasisFunctionType::MULTIQUADRIC)
{
fn = std::shared_ptr<RadialBasisFunction>(new Multiquadric());
}
else if (type == RadialBasisFunctionType::INVERSE_QUADRIC)
{
fn = std::shared_ptr<RadialBasisFunction>(new InverseQuadric());
}
else if (type == RadialBasisFunctionType::INVERSE_MULTIQUADRIC)
{
fn = std::shared_ptr<RadialBasisFunction>(new InverseMultiquadric());
}
else if (type == RadialBasisFunctionType::GAUSSIAN)
{
fn = std::shared_ptr<RadialBasisFunction>(new Gaussian());
}
else
{
fn = std::shared_ptr<RadialBasisFunction>(new ThinPlateSpline());
}
/* Want to solve the linear system A*w = b,
* where w is the vector of weights.
* NOTE: the system is dense and by default badly conditioned.
* It should be solved by a specialized solver such as GMRES
* with preconditioning (e.g. ACBF) as in matlab.
* NOTE: Consider trying the Łukaszyk–Karmowski metric (for two variables)
*/
//SparseMatrix A(numSamples,numSamples);
//A.reserve(numSamples*numSamples);
DenseMatrix A; A.setZero(numSamples, numSamples);
DenseMatrix b; b.setZero(numSamples,1);
int i=0;
for(auto it1 = samples.cbegin(); it1 != samples.cend(); ++it1, ++i)
{
double sum = 0;
int j=0;
for(auto it2 = samples.cbegin(); it2 != samples.cend(); ++it2, ++j)
{
double val = fn->eval(dist(*it1, *it2));
if(val != 0)
{
//A.insert(i,j) = val;
A(i,j) = val;
sum += val;
}
}
double y = it1->getY();
if(normalized) b(i) = sum*y;
else b(i) = y;
}
//A.makeCompressed();
if(precondition)
{
// Calcualte precondition matrix P
DenseMatrix P = computePreconditionMatrix();
// Preconditioned A and b
DenseMatrix Ap = P*A;
DenseMatrix bp = P*b;
A = Ap;
b = bp;
}
#ifndef NDEBUG
std::cout << "Computing RBF weights using dense solver." << std::endl;
#endif // NDEBUG
// SVD analysis
Eigen::JacobiSVD<DenseMatrix> svd(A, Eigen::ComputeThinU | Eigen::ComputeThinV);
auto svals = svd.singularValues();
double svalmax = svals(0);
double svalmin = svals(svals.rows()-1);
double rcondnum = (svalmax <= 0.0 || svalmin <= 0.0) ? 0.0 : svalmin/svalmax;
#ifndef NDEBUG
std::cout << "The reciprocal of the condition number is: " << rcondnum << std::endl;
std::cout << "Largest/smallest singular value: " << svalmax << " / " << svalmin << std::endl;
#endif // NDEBUG
// Solve for weights
weights = svd.solve(b);
#ifndef NDEBUG
// Compute error. If it is used later on, move this statement above the NDEBUG
double err = (A*weights - b).norm() / b.norm();
std::cout << "Error: " << std::setprecision(10) << err << std::endl;
#endif // NDEBUG
// // Alternative solver
// DenseQR s;
// bool success = s.solve(A,b,weights);
// assert(success);
// NOTE: Tried using experimental GMRES solver in Eigen, but it did not work very well.
}
