bsplinebasis1d.C

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/*
 * This file is part of the SPLINTER library.
 * Copyright (C) 2012 Bjarne Grimstad (bjarne.grimstad@gmail.com).
 *
 * This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/.
*/
 
#include <bsplinebasis1d.h>
#include <knots.h>
#include <algorithm>
#include <utilities.h>
#include <iostream>
 
namespace SPLINTER
{
 
BSplineBasis1D::BSplineBasis1D()
{
}
 
BSplineBasis1D::BSplineBasis1D(const std::vector<double>& knots,
                               unsigned int degree)
    : degree(degree),
      knots(knots),
      targetNumBasisfunctions((degree + 1) + 2 * degree + 1) // Minimum p+1
{
    //    if (degree <= 0)
    //        throw Exception("BSplineBasis1D::BSplineBasis1D: Cannot create B-spline basis functions of degree <= 0.");
    if (!isKnotVectorRegular(knots, degree))
    {
        throw Exception("BSplineBasis1D::BSplineBasis1D: Knot vector is not regular.");
    }
}
 
SparseVector BSplineBasis1D::eval(double x) const
{
    SparseVector values(getNumBasisFunctions());
 
    if (!insideSupport(x))
    {
        return values;
    }
 
    supportHack(x);
    std::vector<int> indexSupported = indexSupportedBasisfunctions(x);
    values.reserve(indexSupported.size());
 
    // Evaluate nonzero basis functions
    for (auto it = indexSupported.begin(); it != indexSupported.end(); ++it)
    {
        double val = deBoorCox(x, * it, degree);
 
        if (fabs(val) > 1e-12)
        {
            values.insert(* it) = val;
        }
    }
 
    // Alternative evaluation using basis matrix
    //    int knotIndex = indexHalfopenInterval(x); // knot index
    //    SparseMatrix basisvalues2 = buildBsplineMatrix(x, knotIndex, 1);
    //    for (int i = 2; i <= basisDegree; i++)
    //    {
    //        SparseMatrix Ri = buildBsplineMatrix(x, knotIndex, i);
    //        basisvalues2 = basisvalues2*Ri;
    //    }
    //    basisvalues2.makeCompressed();
    //    assert(basisvalues2.rows() == 1);
    //    assert(basisvalues2.cols() == basisDegree + 1);
    return values;
}
 
SparseVector BSplineBasis1D::evalDerivative(double x, int r) const
{
    // Evaluate rth derivative of basis functions at x
    // Returns vector [D^(r)B_(u-p,p)(x) ... D^(r)B_(u,p)(x)]
    // where u is the knot index and p is the degree
    int p = degree;
 
    // Continuity requirement
    //assert(p > r);
    if (p <= r)
    {
        // Return zero-gradient
        SparseVector DB(getNumBasisFunctions());
        return DB;
    }
 
    // Check for knot multiplicity here!
    supportHack(x);
    int knotIndex = indexHalfopenInterval(x);
    // Algorithm 3.18 from Lyche and Moerken (2011)
    SparseMatrix B(1, 1);
    B.insert(0, 0) = 1;
 
    for (int i = 1; i <= p - r; i++)
    {
        SparseMatrix R = buildBasisMatrix(x, knotIndex, i);
        B = B * R;
    }
 
    for (int i = p - r + 1; i <= p; i++)
    {
        SparseMatrix DR = buildBasisMatrix(x, knotIndex, i, true);
        B = B * DR;
    }
 
    double factorial = std::tgamma(p + 1) / std::tgamma(p - r + 1);
    B = B * factorial;
 
    if (B.cols() != p + 1)
    {
        throw Exception("BSplineBasis1D::evalDerivative: Wrong number of columns of B matrix.");
    }
 
    // From row vector to extended column vector
    SparseVector DB(getNumBasisFunctions());
    DB.reserve(p + 1);
    int i = knotIndex - p; // First insertion index
 
    for (int k = 0; k < B.outerSize(); ++k)
        for (SparseMatrix::InnerIterator it(B, k); it; ++it)
        {
            DB.insert(i + it.col()) = it.value();
        }
 
    return DB;
}
 
// Old implementation of first derivative of basis functions
SparseVector BSplineBasis1D::evalFirstDerivative(double x) const
{
    SparseVector values(getNumBasisFunctions());
    supportHack(x);
    std::vector<int> supportedBasisFunctions = indexSupportedBasisfunctions(x);
 
    for (int i : supportedBasisFunctions)
    {
        // Differentiate basis function
        // Equation 3.35 in Lyche & Moerken (2011)
        double b1 = deBoorCox(x, i, degree - 1);
        double b2 = deBoorCox(x, i + 1, degree - 1);
        double t11 = knots.at(i);
        double t12 = knots.at(i + degree);
        double t21 = knots.at(i + 1);
        double t22 = knots.at(i + degree + 1);
        (t12 == t11) ? b1 = 0 : b1 = b1 / (t12 - t11);
        (t22 == t21) ? b2 = 0 : b2 = b2 / (t22 - t21);
        values.insert(i) = degree * (b1 - b2);
    }
 
    return values;
}
 
// Used to evaluate basis functions - alternative to the recursive deBoorCox
SparseMatrix BSplineBasis1D::buildBasisMatrix(double x, unsigned int u,
        unsigned int k, bool diff) const
{
    /* Build B-spline Matrix
     * R_k in R^(k,k+1)
     * or, if diff = true, the differentiated basis matrix
     * DR_k in R^(k,k+1)
     */
    if (!(k >= 1 && k <= getBasisDegree()))
    {
        throw Exception("BSplineBasis1D::buildBasisMatrix: Incorrect input paramaters!");
    }
 
    //    assert(u >= basisDegree + 1);
    //    assert(u < ks.size() - basisDegree);
    unsigned int rows = k;
    unsigned int cols = k + 1;
    SparseMatrix R(rows, cols);
    R.reserve(Eigen::VectorXi::Constant(cols, 2));
 
    for (unsigned int i = 0; i < rows; i++)
    {
        double dk = knots.at(u + 1 + i) - knots.at(u + 1 + i - k);
 
        if (dk == 0)
        {
            continue;
        }
        else
        {
            if (diff)
            {
                // Insert diagonal element
                R.insert(i, i) = -1 / dk;
                // Insert super-diagonal element
                R.insert(i, i + 1) = 1 / dk;
            }
            else
            {
                // Insert diagonal element
                double a = (knots.at(u + 1 + i) - x) / dk;
 
                if (a != 0)
                {
                    R.insert(i, i) = a;
                }
 
                // Insert super-diagonal element
                double b = (x - knots.at(u + 1 + i - k)) / dk;
 
                if (b != 0)
                {
                    R.insert(i, i + 1) = b;
                }
            }
        }
    }
 
    R.makeCompressed();
    return R;
}
 
double BSplineBasis1D::deBoorCox(double x, int i, int k) const
{
    if (k == 0)
    {
        if (inHalfopenInterval(x, knots.at(i), knots.at(i + 1)))
        {
            return 1;
        }
        else
        {
            return 0;
        }
    }
    else
    {
        double s1, s2, r1, r2;
        s1 = deBoorCoxCoeff(x, knots.at(i),   knots.at(i + k));
        s2 = deBoorCoxCoeff(x, knots.at(i + 1), knots.at(i + k + 1));
        r1 = deBoorCox(x, i,   k - 1);
        r2 = deBoorCox(x, i + 1, k - 1);
        return s1 * r1 + (1 - s2) * r2;
    }
}
 
double BSplineBasis1D::deBoorCoxCoeff(double x, double x_min,
                                      double x_max) const
{
    if (x_min < x_max && x_min <= x && x <= x_max)
    {
        return (x - x_min) / (x_max - x_min);
    }
 
    return 0;
}
 
// Insert knots and compute knot insertion matrix (to update control points)
SparseMatrix BSplineBasis1D::insertKnots(double tau, unsigned int multiplicity)
{
    if (!insideSupport(tau))
    {
        throw Exception("BSplineBasis1D::insertKnots: Cannot insert knot outside domain!");
    }
 
    if (knotMultiplicity(tau) + multiplicity > degree + 1)
    {
        throw Exception("BSplineBasis1D::insertKnots: Knot multiplicity is too high!");
    }
 
    // New knot vector
    int index = indexHalfopenInterval(tau);
    std::vector<double> extKnots = knots;
 
    for (unsigned int i = 0; i < multiplicity; i++)
    {
        extKnots.insert(extKnots.begin() + index + 1, tau);
    }
 
    if (!isKnotVectorRegular(extKnots, degree))
    {
        throw Exception("BSplineBasis1D::insertKnots: New knot vector is not regular!");
    }
 
    // Return knot insertion matrix
    SparseMatrix A = buildKnotInsertionMatrix(extKnots);
    // Update knots
    knots = extKnots;
    return A;
}
 
SparseMatrix BSplineBasis1D::refineKnots()
{
    // Build refine knot vector
    std::vector<double> refinedKnots = knots;
    unsigned int targetNumKnots = targetNumBasisfunctions + degree + 1;
 
    while (refinedKnots.size() < targetNumKnots)
    {
        int index = indexLongestInterval(refinedKnots);
        double newKnot = (refinedKnots.at(index) + refinedKnots.at(index + 1)) / 2.0;
        refinedKnots.insert(std::lower_bound(refinedKnots.begin(), refinedKnots.end(),
                                             newKnot), newKnot);
    }
 
    if (!isKnotVectorRegular(refinedKnots, degree))
    {
        throw Exception("BSplineBasis1D::refineKnots: New knot vector is not regular!");
    }
 
    if (!isKnotVectorRefinement(knots, refinedKnots))
    {
        throw Exception("BSplineBasis1D::refineKnots: New knot vector is not a proper refinement!");
    }
 
    // Return knot insertion matrix
    SparseMatrix A = buildKnotInsertionMatrix(refinedKnots);
    // Update knots
    knots = refinedKnots;
    return A;
}
 
SparseMatrix BSplineBasis1D::refineKnotsLocally(double x)
{
    if (!insideSupport(x))
    {
        throw Exception("BSplineBasis1D::refineKnotsLocally: Cannot refine outside support!");
    }
 
    if (getNumBasisFunctions() >= getNumBasisFunctionsTarget()
            || assertNear(knots.front(), knots.back()))
    {
        unsigned int n = getNumBasisFunctions();
        DenseMatrix A = DenseMatrix::Identity(n, n);
        return A.sparseView();
    }
 
    // Refined knot vector
    std::vector<double> refinedKnots = knots;
    auto upper = std::lower_bound(refinedKnots.begin(), refinedKnots.end(), x);
 
    // Check left boundary
    if (upper == refinedKnots.begin())
    {
        std::advance(upper, degree + 1);
    }
 
    // Get previous iterator
    auto lower = std::prev(upper);
 
    // Do not insert if upper and lower bounding knot are close
    if (assertNear(* upper, * lower))
    {
        unsigned int n = getNumBasisFunctions();
        DenseMatrix A = DenseMatrix::Identity(n, n);
        return A.sparseView();
    }
 
    // Insert knot at x
    double insertVal = x;
 
    // Adjust x if it is on or close to a knot
    if (knotMultiplicity(x) > 0
            || assertNear(* upper, x, 1e-6, 1e-6)
            || assertNear(* lower, x, 1e-6, 1e-6))
    {
        insertVal = (* upper + * lower) / 2.0;
    }
 
    // Insert new knot
    refinedKnots.insert(upper, insertVal);
 
    if (!isKnotVectorRegular(refinedKnots, degree))
    {
        throw Exception("BSplineBasis1D::refineKnotsLocally: New knot vector is not regular!");
    }
 
    if (!isKnotVectorRefinement(knots, refinedKnots))
    {
        throw Exception("BSplineBasis1D::refineKnotsLocally: New knot vector is not a proper refinement!");
    }
 
    // Build knot insertion matrix
    SparseMatrix A = buildKnotInsertionMatrix(refinedKnots);
    // Update knots
    knots = refinedKnots;
    return A;
}
 
SparseMatrix BSplineBasis1D::decomposeToBezierForm()
{
    // Build refine knot vector
    std::vector<double> refinedKnots = knots;
    // Start at first knot and add knots until all knots have multiplicity degree + 1
    std::vector<double>::iterator knoti = refinedKnots.begin();
 
    while (knoti != refinedKnots.end())
    {
        // Insert new knots
        int mult = degree + 1 - knotMultiplicity(* knoti);
 
        if (mult > 0)
        {
            std::vector<double> newKnots(mult, * knoti);
            refinedKnots.insert(knoti, newKnots.begin(), newKnots.end());
        }
 
        // Advance to next knot
        knoti = std::upper_bound(refinedKnots.begin(), refinedKnots.end(), * knoti);
    }
 
    if (!isKnotVectorRegular(refinedKnots, degree))
    {
        throw Exception("BSplineBasis1D::refineKnots: New knot vector is not regular!");
    }
 
    if (!isKnotVectorRefinement(knots, refinedKnots))
    {
        throw Exception("BSplineBasis1D::refineKnots: New knot vector is not a proper refinement!");
    }
 
    // Return knot insertion matrix
    SparseMatrix A = buildKnotInsertionMatrix(refinedKnots);
    // Update knots
    knots = refinedKnots;
    return A;
}
 
SparseMatrix BSplineBasis1D::buildKnotInsertionMatrix(const std::vector<double>
        &
        refinedKnots) const
{
    if (!isKnotVectorRegular(refinedKnots, degree))
    {
        throw Exception("BSplineBasis1D::buildKnotInsertionMatrix: New knot vector is not regular!");
    }
 
    if (!isKnotVectorRefinement(knots, refinedKnots))
    {
        throw Exception("BSplineBasis1D::buildKnotInsertionMatrix: New knot vector is not a proper refinement!");
    }
 
    std::vector<double> knotsAug = refinedKnots;
    unsigned int n = knots.size() - degree - 1;
    unsigned int m = knotsAug.size() - degree - 1;
    SparseMatrix A(m, n);
    //A.resize(m,n);
    A.reserve(Eigen::VectorXi::Constant(n, degree + 1));
 
    // Build A row-by-row
    for (unsigned int i = 0; i < m; i++)
    {
        int u = indexHalfopenInterval(knotsAug.at(i));
        SparseMatrix R(1, 1);
        R.insert(0, 0) = 1;
 
        // For p > 0
        for (unsigned int j = 1; j <= degree; j++)
        {
            SparseMatrix Ri = buildBasisMatrix(knotsAug.at(i + j), u, j);
            R = R * Ri;
        }
 
        // Size check
        if (R.rows() != 1 || R.cols() != (int)degree + 1)
        {
            throw Exception("BSplineBasis1D::buildKnotInsertionMatrix: Incorrect matrix dimensions!");
        }
 
        // Insert row values
        int j = u - degree; // First insertion index
 
        for (int k = 0; k < R.outerSize(); ++k)
            for (SparseMatrix::InnerIterator it(R, k); it; ++it)
            {
                A.insert(i, j + it.col()) = it.value();
            }
    }
 
    A.makeCompressed();
    return A;
}
 
/*
 * The B-spline domain is the half-open domain [ knots.first(), knots.end() ).
 * The hack checks if x is at the right boundary (if x = knots.end()), if so,
 * a small number is subtracted from x, moving x into the half-open domain.
 */
void BSplineBasis1D::supportHack(double& x) const
{
    if (x == knots.back())
    {
        x = std::nextafter(x, std::numeric_limits<double>::lowest());
    }
}
 
/*
 * Finds index i such that knots.at(i) <= x < knots.at(i+1).
 * Returns false if x is outside support.
 */
int BSplineBasis1D::indexHalfopenInterval(double x) const
{
    if (x < knots.front() || x > knots.back())
    {
        throw Exception("BSplineBasis1D::indexHalfopenInterval: x outside knot interval!");
    }
 
    // Find first knot that is larger than x
    std::vector<double>::const_iterator it = std::upper_bound(knots.begin(),
        knots.end(), x);
    // Return index
    int index = it - knots.begin();
    return index - 1;
}
 
SparseMatrix BSplineBasis1D::reduceSupport(double lb, double ub)
{
    // Check bounds
    if (lb < knots.front() || ub > knots.back())
    {
        throw Exception("BSplineBasis1D::reduceSupport: Cannot increase support!");
    }
 
    unsigned int k = degree + 1;
    int index_lower = indexSupportedBasisfunctions(lb).front();
    int index_upper = indexSupportedBasisfunctions(ub).back();
 
    // Check lower bound index
    if (k != knotMultiplicity(knots.at(index_lower)))
    {
        int suggested_index = index_lower - 1;
 
        if (0 <= suggested_index)
        {
            index_lower = suggested_index;
        }
        else
        {
            throw Exception("BSplineBasis1D::reduceSupport: Suggested index is negative!");
        }
    }
 
    // Check upper bound index
    if (knotMultiplicity(ub) == k && knots.at(index_upper) == ub)
    {
        index_upper -= k;
    }
 
    // New knot vector
    std::vector<double> si;
    si.insert(si.begin(), knots.begin() + index_lower,
              knots.begin() + index_upper + k + 1);
    // Construct selection matrix A
    int numOld = knots.size() - k; // Current number of basis functions
    int numNew = si.size() - k; // Number of basis functions after update
 
    if (numOld < numNew)
    {
        throw Exception("BSplineBasis1D::reduceSupport: Number of basis functions is increased instead of reduced!");
    }
 
    DenseMatrix Ad = DenseMatrix::Zero(numOld, numNew);
    Ad.block(index_lower, 0, numNew, numNew) = DenseMatrix::Identity(numNew,
        numNew);
    SparseMatrix A = Ad.sparseView();
    // Update knots
    knots = si;
    return A;
}
 
double BSplineBasis1D::getKnotValue(unsigned int index) const
{
    return knots.at(index);
}
 
unsigned int BSplineBasis1D::knotMultiplicity(double tau) const
{
    return std::count(knots.begin(), knots.end(), tau);
}
 
bool BSplineBasis1D::inHalfopenInterval(double x, double x_min,
                                        double x_max) const
{
    return (x_min <= x) && (x < x_max);
}
 
bool BSplineBasis1D::insideSupport(double x) const
{
    return (knots.front() <= x) && (x <= knots.back());
}
 
unsigned int BSplineBasis1D::getNumBasisFunctions() const
{
    return knots.size() - (degree + 1);
}
 
unsigned int BSplineBasis1D::getNumBasisFunctionsTarget() const
{
    return targetNumBasisfunctions;
}
 
// Return indices of supporting basis functions at x
std::vector<int> BSplineBasis1D::indexSupportedBasisfunctions(double x) const
{
    std::vector<int> ret;
 
    if (insideSupport(x))
    {
        int last = indexHalfopenInterval(x);
 
        if (last < 0)
        {
            // NOTE: can this happen?
            last = knots.size() - 1 - (degree + 1);
        }
 
        int first = std::max((int)(last - degree), 0);
 
        for (int i = first; i <= last; i++)
        {
            ret.push_back(i);
        }
    }
 
    return ret;
}
 
unsigned int BSplineBasis1D::indexLongestInterval() const
{
    return indexLongestInterval(knots);
}
 
unsigned int BSplineBasis1D::indexLongestInterval(const std::vector<double>&
        vec) const
{
    double longest = 0;
    double interval = 0;
    unsigned int index = 0;
 
    for (unsigned int i = 0; i < vec.size() - 1; i++)
    {
        interval = vec.at(i + 1) - vec.at(i);
 
        if (longest < interval)
        {
            longest = interval;
            index = i;
        }
    }
 
    return index;
}
 
} // namespace SPLINTER