vpNurbs.cpp

/*
 * ViSP, open source Visual Servoing Platform software.
 * Copyright (C) 2005 - 2023 by Inria. All rights reserved.
 *
 * This software is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or
 * (at your option) any later version.
 * See the file LICENSE.txt at the root directory of this source
 * distribution for additional information about the GNU GPL.
 *
 * For using ViSP with software that can not be combined with the GNU
 * GPL, please contact Inria about acquiring a ViSP Professional
 * Edition License.
 *
 * See https://visp.inria.fr for more information.
 *
 * This software was developed at:
 * Inria Rennes - Bretagne Atlantique
 * Campus Universitaire de Beaulieu
 * 35042 Rennes Cedex
 * France
 *
 * If you have questions regarding the use of this file, please contact
 * Inria at visp@inria.fr
 *
 * This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
 * WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 *
 * Description:
 * This class implements the Non Uniform Rational B-Spline (NURBS)
 */
 
#include <cmath>  // std::fabs
#include <limits> // numeric_limits
#include <visp3/core/vpColVector.h>
#include <visp3/me/vpNurbs.h>
/*
  Compute the distance d = |Pw1-Pw2|
*/
inline double distance(const vpImagePoint &iP1, double w1, const vpImagePoint &iP2, double w2)
{
  double distancei = iP1.get_i() - iP2.get_i();
  double distancej = iP1.get_j() - iP2.get_j();
  double distancew = w1 - w2;
  return sqrt(vpMath::sqr(distancei) + vpMath::sqr(distancej) + vpMath::sqr(distancew));
}
 
vpNurbs::vpNurbs() : weights() { p = 3; }
 
vpNurbs::vpNurbs(const vpNurbs &nurbs) : vpBSpline(nurbs), weights(nurbs.weights) { }
 
vpImagePoint vpNurbs::computeCurvePoint(double l_u, unsigned int l_i, unsigned int l_p, std::vector<double> &l_knots,
                                        std::vector<vpImagePoint> &l_controlPoints, std::vector<double> &l_weights)
{
  vpBasisFunction *N = nullptr;
  N = computeBasisFuns(l_u, l_i, l_p, l_knots);
  vpImagePoint pt;
 
  double ic = 0;
  double jc = 0;
  double wc = 0;
  for (unsigned int j = 0; j <= l_p; j++) {
    ic = ic + N[j].value * (l_controlPoints[l_i - l_p + j]).get_i() * l_weights[l_i - l_p + j];
    jc = jc + N[j].value * (l_controlPoints[l_i - l_p + j]).get_j() * l_weights[l_i - l_p + j];
    wc = wc + N[j].value * l_weights[l_i - l_p + j];
  }
 
  pt.set_i(ic / wc);
  pt.set_j(jc / wc);
 
  if (N != nullptr)
    delete[] N;
 
  return pt;
}
 
vpImagePoint vpNurbs::computeCurvePoint(double u)
{
  vpBasisFunction *N = nullptr;
  N = computeBasisFuns(u);
  vpImagePoint pt;
 
  double ic = 0;
  double jc = 0;
  double wc = 0;
  for (unsigned int j = 0; j <= p; j++) {
    ic = ic + N[j].value * (controlPoints[N[0].i + j]).get_i() * weights[N[0].i + j]; // N[0].i = findSpan(u)-p
    jc = jc + N[j].value * (controlPoints[N[0].i + j]).get_j() * weights[N[0].i + j];
    wc = wc + N[j].value * weights[N[0].i + j];
  }
 
  pt.set_i(ic / wc);
  pt.set_j(jc / wc);
 
  if (N != nullptr)
    delete[] N;
 
  return pt;
}
 
vpMatrix vpNurbs::computeCurveDers(double l_u, unsigned int l_i, unsigned int l_p, unsigned int l_der,
                                   std::vector<double> &l_knots, std::vector<vpImagePoint> &l_controlPoints,
                                   std::vector<double> &l_weights)
{
  vpMatrix derivate(l_der + 1, 3);
  vpBasisFunction **N = nullptr;
  N = computeDersBasisFuns(l_u, l_i, l_p, l_der, l_knots);
 
  for (unsigned int k = 0; k <= l_der; k++) {
    derivate[k][0] = 0.0;
    derivate[k][1] = 0.0;
    derivate[k][2] = 0.0;
 
    for (unsigned int j = 0; j <= l_p; j++) {
      derivate[k][0] = derivate[k][0] + N[k][j].value * (l_controlPoints[l_i - l_p + j]).get_i();
      derivate[k][1] = derivate[k][1] + N[k][j].value * (l_controlPoints[l_i - l_p + j]).get_j();
      derivate[k][2] = derivate[k][2] + N[k][j].value * (l_weights[l_i - l_p + j]);
    }
  }
 
  if (N != nullptr) {
    for (unsigned int i = 0; i <= l_der; i++)
      delete[] N[i];
    delete[] N;
  }
  return derivate;
}
 
vpMatrix vpNurbs::computeCurveDers(double u, unsigned int der)
{
  vpMatrix derivate(der + 1, 3);
  vpBasisFunction **N = nullptr;
  N = computeDersBasisFuns(u, der);
 
  for (unsigned int k = 0; k <= der; k++) {
    derivate[k][0] = 0.0;
    derivate[k][1] = 0.0;
    derivate[k][2] = 0.0;
    for (unsigned int j = 0; j <= p; j++) {
      derivate[k][0] = derivate[k][0] + N[k][j].value * (controlPoints[N[0][0].i - p + j]).get_i();
      derivate[k][1] = derivate[k][1] + N[k][j].value * (controlPoints[N[0][0].i - p + j]).get_j();
      derivate[k][2] = derivate[k][2] + N[k][j].value * (weights[N[0][0].i - p + j]);
    }
  }
 
  if (N != nullptr)
    delete[] N;
 
  return derivate;
}
 
vpImagePoint *vpNurbs::computeCurveDersPoint(double l_u, unsigned int l_i, unsigned int l_p, unsigned int l_der,
                                             std::vector<double> &l_knots, std::vector<vpImagePoint> &l_controlPoints,
                                             std::vector<double> &l_weights)
{
  std::vector<vpImagePoint> A;
  vpImagePoint pt;
  for (unsigned int j = 0; j < l_controlPoints.size(); j++) {
    pt = l_controlPoints[j];
    pt.set_i(pt.get_i() * l_weights[j]);
    pt.set_j(pt.get_j() * l_weights[j]);
    A.push_back(pt);
  }
 
  vpMatrix Awders = computeCurveDers(l_u, l_i, l_p, l_der, l_knots, A, l_weights);
 
  vpImagePoint *CK = new vpImagePoint[l_der + 1];
 
  for (unsigned int k = 0; k <= l_der; k++) {
    double ic = Awders[k][0];
    double jc = Awders[k][1];
    for (unsigned int j = 1; j <= k; j++) {
      double tmpComb = static_cast<double>(vpMath::comb(k, j));
      ic = ic - tmpComb * Awders[k][2] * (CK[k - j].get_i());
      jc = jc - tmpComb * Awders[j][2] * (CK[k - j].get_j());
    }
    CK[k].set_ij(ic / Awders[0][2], jc / Awders[0][2]);
  }
  return CK;
}
 
 
vpImagePoint *vpNurbs::computeCurveDersPoint(double u, unsigned int der)
{
  unsigned int i = findSpan(u);
  return computeCurveDersPoint(u, i, p, der, knots, controlPoints, weights);
}
 
 
void vpNurbs::curveKnotIns(double l_u, unsigned int l_k, unsigned int l_s, unsigned int l_r, unsigned int l_p,
                           std::vector<double> &l_knots, std::vector<vpImagePoint> &l_controlPoints,
                           std::vector<double> &l_weights)
{
  vpMatrix Rw(l_p + 1, 3);
  std::vector<vpImagePoint>::iterator it1;
  std::vector<double>::iterator it2;
  vpImagePoint pt;
  double w = 0;
 
  for (unsigned int j = 0; j <= l_p - l_s; j++) {
    Rw[j][0] = (l_controlPoints[l_k - l_p + j]).get_i() * l_weights[l_k - l_p + j];
    Rw[j][1] = (l_controlPoints[l_k - l_p + j]).get_j() * l_weights[l_k - l_p + j];
    Rw[j][2] = l_weights[l_k - l_p + j];
  }
 
  it1 = l_controlPoints.begin();
  l_controlPoints.insert(it1 + (int)l_k - (int)l_s, l_r, pt);
  it2 = l_weights.begin();
  l_weights.insert(it2 + (int)l_k - (int)l_s, l_r, w);
 
  unsigned int L = 0;
  double alpha;
  for (unsigned int j = 1; j <= l_r; j++) {
    L = l_k - l_p + j;
 
    for (unsigned int i = 0; i <= l_p - j - l_s; i++) {
      alpha = (l_u - l_knots[L + i]) / (l_knots[i + l_k + 1] - l_knots[L + i]);
      Rw[i][0] = alpha * Rw[i + 1][0] + (1.0 - alpha) * Rw[i][0];
      Rw[i][1] = alpha * Rw[i + 1][1] + (1.0 - alpha) * Rw[i][1];
      Rw[i][2] = alpha * Rw[i + 1][2] + (1.0 - alpha) * Rw[i][2];
    }
 
    pt.set_ij(Rw[0][0] / Rw[0][2], Rw[0][1] / Rw[0][2]);
    l_controlPoints[L] = pt;
    l_weights[L] = Rw[0][2];
 
    pt.set_ij(Rw[l_p - j - l_s][0] / Rw[l_p - j - l_s][2], Rw[l_p - j - l_s][1] / Rw[l_p - j - l_s][2]);
    l_controlPoints[l_k + l_r - j - l_s] = pt;
    l_weights[l_k + l_r - j - l_s] = Rw[l_p - j - l_s][2];
  }
 
  for (unsigned int j = L + 1; j < l_k - l_s; j++) {
    pt.set_ij(Rw[j - L][0] / Rw[j - L][2], Rw[j - L][1] / Rw[j - L][2]);
    l_controlPoints[j] = pt;
    l_weights[j] = Rw[j - L][2];
  }
 
  it2 = l_knots.begin();
  l_knots.insert(it2 + (int)l_k, l_r, l_u);
}
 
void vpNurbs::curveKnotIns(double u, unsigned int s, unsigned int r)
{
  unsigned int i = findSpan(u);
  curveKnotIns(u, i, s, r, p, knots, controlPoints, weights);
}
 
 
void vpNurbs::refineKnotVectCurve(double *l_x, unsigned int l_r, unsigned int l_p, std::vector<double> &l_knots,
                                  std::vector<vpImagePoint> &l_controlPoints, std::vector<double> &l_weights)
{
  unsigned int a = findSpan(l_x[0], l_p, l_knots);
  unsigned int b = findSpan(l_x[l_r], l_p, l_knots);
  b++;
 
  unsigned int n = (unsigned int)l_controlPoints.size();
  unsigned int m = (unsigned int)l_knots.size();
 
  for (unsigned int j = 0; j < n; j++) {
    l_controlPoints[j].set_ij(l_controlPoints[j].get_i() * l_weights[j], l_controlPoints[j].get_j() * l_weights[j]);
  }
 
  std::vector<double> l_knots_tmp(l_knots);
  std::vector<vpImagePoint> l_controlPoints_tmp(l_controlPoints);
  std::vector<double> l_weights_tmp(l_weights);
 
  vpImagePoint pt;
  double w = 0;
 
  for (unsigned int j = 0; j <= l_r; j++) {
    l_controlPoints.push_back(pt);
    l_weights.push_back(w);
    l_knots.push_back(w);
  }
 
  for (unsigned int j = b + l_p; j <= m - 1; j++)
    l_knots[j + l_r + 1] = l_knots_tmp[j];
 
  for (unsigned int j = b - 1; j <= n - 1; j++) {
    l_controlPoints[j + l_r + 1] = l_controlPoints_tmp[j];
    l_weights[j + l_r + 1] = l_weights_tmp[j];
  }
 
  unsigned int i = b + l_p - 1;
  unsigned int k = b + l_p + l_r;
 
  {
    unsigned int j = l_r + 1;
    do {
      j--;
      while (l_x[j] <= l_knots[i] && i > a) {
        l_controlPoints[k - l_p - 1] = l_controlPoints_tmp[i - l_p - 1];
        l_weights[k - l_p - 1] = l_weights_tmp[i - l_p - 1];
        l_knots[k] = l_knots_tmp[i];
        k--;
        i--;
      }
 
      l_controlPoints[k - l_p - 1] = l_controlPoints[k - l_p];
      l_weights[k - l_p - 1] = l_weights[k - l_p];
 
      for (unsigned int l = 1; l <= l_p; l++) {
        unsigned int ind = k - l_p + l;
        double alpha = l_knots[k + l] - l_x[j];
        // if (vpMath::abs(alpha) == 0.0)
        if (std::fabs(alpha) <= std::numeric_limits<double>::epsilon()) {
          l_controlPoints[ind - 1] = l_controlPoints[ind];
          l_weights[ind - 1] = l_weights[ind];
        }
        else {
          alpha = alpha / (l_knots[k + l] - l_knots_tmp[i - l_p + l]);
          l_controlPoints[ind - 1].set_i(alpha * l_controlPoints[ind - 1].get_i() +
                                         (1.0 - alpha) * l_controlPoints[ind].get_i());
          l_controlPoints[ind - 1].set_j(alpha * l_controlPoints[ind - 1].get_j() +
                                         (1.0 - alpha) * l_controlPoints[ind].get_j());
          l_weights[ind - 1] = alpha * l_weights[ind - 1] + (1.0 - alpha) * l_weights[ind];
        }
      }
      l_knots[k] = l_x[j];
      k--;
    } while (j != 0);
  }
 
  for (unsigned int j = 0; j < n; j++) {
    l_controlPoints[j].set_ij(l_controlPoints[j].get_i() / l_weights[j], l_controlPoints[j].get_j() / l_weights[j]);
  }
}
 
void vpNurbs::refineKnotVectCurve(double *x, unsigned int r)
{
  refineKnotVectCurve(x, r, p, knots, controlPoints, weights);
}
 
unsigned int vpNurbs::removeCurveKnot(double l_u, unsigned int l_r, unsigned int l_num, double l_TOL, unsigned int l_s,
                                      unsigned int l_p, std::vector<double> &l_knots,
                                      std::vector<vpImagePoint> &l_controlPoints, std::vector<double> &l_weights)
{
  unsigned int n = (unsigned int)l_controlPoints.size();
  unsigned int m = n + l_p + 1;
 
  for (unsigned int j = 0; j < n; j++) {
    l_controlPoints[j].set_ij(l_controlPoints[j].get_i() * l_weights[j], l_controlPoints[j].get_j() * l_weights[j]);
  }
 
  unsigned int ord = l_p + 1;
  double fout = (2 * l_r - l_s - l_p) / 2.;
  unsigned int last = l_r - l_s;
  unsigned int first = l_r - l_p;
  unsigned int tblSize = 2 * l_p + 1;
  vpImagePoint *tempP = new vpImagePoint[tblSize];
  double *tempW = new double[tblSize];
  vpImagePoint pt;
  unsigned int t = 0;
  double alfi = 0;
  double alfj = 0;
  unsigned int i, j;
 
  for (t = 0; t < l_num; t++) {
    unsigned int off = first - 1;
    tempP[0] = l_controlPoints[off];
    tempW[0] = l_weights[off];
    tempP[last + 1 - off] = l_controlPoints[last + 1];
    tempW[last + 1 - off] = l_weights[last + 1];
    i = first;
    j = last;
    unsigned int ii = 1;
    unsigned int jj = last - off;
    int remflag = 0;
    while (j - i > t) {
      alfi = (l_u - l_knots[i]) / (l_knots[i + ord + t] - l_knots[i]);
      alfj = (l_u - l_knots[j - t]) / (l_knots[j + ord] - l_knots[j - t]);
      pt.set_i((l_controlPoints[i].get_i() - (1.0 - alfi) * tempP[ii - 1].get_i()) / alfi);
      tempP[ii].set_i((l_controlPoints[i].get_i() - (1.0 - alfi) * tempP[ii - 1].get_i()) / alfi);
      tempP[ii].set_j((l_controlPoints[i].get_j() - (1.0 - alfi) * tempP[ii - 1].get_j()) / alfi);
      tempW[ii] = ((l_weights[i] - (1.0 - alfi) * tempW[ii - 1]) / alfi);
      tempP[jj].set_i((l_controlPoints[j].get_i() - alfj * tempP[jj + 1].get_i()) / (1.0 - alfj));
      tempP[jj].set_j((l_controlPoints[j].get_j() - alfj * tempP[jj + 1].get_j()) / (1.0 - alfj));
      tempW[jj] = ((l_weights[j] - alfj * tempW[jj + 1]) / (1.0 - alfj));
      i++;
      j--;
      ii++;
      jj--;
    }
 
    if (j - i < t) {
      double distancei = tempP[ii - 1].get_i() - tempP[jj + 1].get_i();
      double distancej = tempP[ii - 1].get_j() - tempP[jj + 1].get_j();
      double distancew = tempW[ii - 1] - tempW[jj + 1];
      double distance = sqrt(vpMath::sqr(distancei) + vpMath::sqr(distancej) + vpMath::sqr(distancew));
      if (distance <= l_TOL)
        remflag = 1;
    }
    else {
      alfi = (l_u - l_knots[i]) / (l_knots[i + ord + t] - l_knots[i]);
      double distancei =
        l_controlPoints[i].get_i() - (alfi * tempP[ii + t + 1].get_i() + (1.0 - alfi) * tempP[ii - 1].get_i());
      double distancej =
        l_controlPoints[i].get_j() - (alfi * tempP[ii + t + 1].get_j() + (1.0 - alfi) * tempP[ii - 1].get_j());
      double distancew = l_weights[i] - (alfi * tempW[ii + t + 1] + (1.0 - alfi) * tempW[ii - 1]);
      double distance = sqrt(vpMath::sqr(distancei) + vpMath::sqr(distancej) + vpMath::sqr(distancew));
      if (distance <= l_TOL)
        remflag = 1;
    }
    if (remflag == 0)
      break;
    else {
      i = first;
      j = last;
      while (j - i > t) {
        l_controlPoints[i].set_i(tempP[i - off].get_i());
        l_controlPoints[i].set_j(tempP[i - off].get_j());
        l_weights[i] = tempW[i - off];
        l_controlPoints[j].set_i(tempP[j - off].get_i());
        l_controlPoints[j].set_j(tempP[j - off].get_j());
        l_weights[j] = tempW[j - off];
        i++;
        j--;
      }
    }
    first--;
    last++;
  }
  if (t == 0) {
    delete[] tempP;
    delete[] tempW;
    return t;
  }
  for (unsigned int k = l_r + 1; k <= m; k++)
    l_knots[k - t] = l_knots[k];
  j = (unsigned int)fout;
  i = j;
  for (unsigned int k = 1; k < t; k++) {
    if (k % 2)
      i++;
    else
      j--;
  }
  for (unsigned int k = i + 1; k <= n; k++) {
    l_controlPoints[j].set_i(l_controlPoints[k].get_i());
    l_controlPoints[j].set_j(l_controlPoints[k].get_j());
    l_weights[j] = l_weights[k];
    j++;
  }
  for (unsigned int k = 0; k < t; k++) {
    l_knots.erase(l_knots.end() - 1);
    l_controlPoints.erase(l_controlPoints.end() - 1);
  }
 
  for (unsigned int k = 0; k < l_controlPoints.size(); k++)
    l_controlPoints[k].set_ij(l_controlPoints[k].get_i() / l_weights[k], l_controlPoints[k].get_j() / l_weights[k]);
 
  delete[] tempP;
  delete[] tempW;
  return t;
}
 
unsigned int vpNurbs::removeCurveKnot(double l_u, unsigned int l_r, unsigned int l_num, double l_TOL)
{
  return removeCurveKnot(l_u, l_r, l_num, l_TOL, 0, p, knots, controlPoints, weights);
}
 
void vpNurbs::globalCurveInterp(std::vector<vpImagePoint> &l_crossingPoints, unsigned int l_p,
                                std::vector<double> &l_knots, std::vector<vpImagePoint> &l_controlPoints,
                                std::vector<double> &l_weights)
{
  if (l_p == 0) {
    // vpERROR_TRACE("Bad degree of the NURBS basis functions");
    throw(vpException(vpException::badValue, "Bad degree of the NURBS basis functions"));
  }
 
  l_knots.clear();
  l_controlPoints.clear();
  l_weights.clear();
  unsigned int n = (unsigned int)l_crossingPoints.size() - 1;
  unsigned int m = n + l_p + 1;
 
  double d = 0;
  for (unsigned int k = 1; k <= n; k++)
    d = d + distance(l_crossingPoints[k], 1, l_crossingPoints[k - 1], 1);
 
  // Compute ubar
  std::vector<double> ubar;
  ubar.push_back(0.0);
  for (unsigned int k = 1; k < n; k++) {
    ubar.push_back(ubar[k - 1] + distance(l_crossingPoints[k], 1, l_crossingPoints[k - 1], 1) / d);
  }
  ubar.push_back(1.0);
 
  // Compute the knot vector
  for (unsigned int k = 0; k <= l_p; k++)
    l_knots.push_back(0.0);
 
  double sum = 0;
  for (unsigned int k = 1; k <= l_p; k++)
    sum = sum + ubar[k];
 
  // Centripetal method
  for (unsigned int k = 1; k <= n - l_p; k++) {
    l_knots.push_back(sum / l_p);
    sum = sum - ubar[k - 1] + ubar[l_p + k - 1];
  }
 
  for (unsigned int k = m - l_p; k <= m; k++)
    l_knots.push_back(1.0);
 
  vpMatrix A(n + 1, n + 1);
  vpBasisFunction *N;
 
  for (unsigned int i = 0; i <= n; i++) {
    unsigned int span = findSpan(ubar[i], l_p, l_knots);
    N = computeBasisFuns(ubar[i], span, l_p, l_knots);
    for (unsigned int k = 0; k <= l_p; k++)
      A[i][span - l_p + k] = N[k].value;
    delete[] N;
  }
  // vpMatrix Ainv = A.inverseByLU();
  vpMatrix Ainv;
  A.pseudoInverse(Ainv);
  vpColVector Qi(n + 1);
  vpColVector Qj(n + 1);
  vpColVector Qw(n + 1);
  for (unsigned int k = 0; k <= n; k++) {
    Qi[k] = l_crossingPoints[k].get_i();
    Qj[k] = l_crossingPoints[k].get_j();
  }
  Qw = 1;
  vpColVector Pi = Ainv * Qi;
  vpColVector Pj = Ainv * Qj;
  vpColVector Pw = Ainv * Qw;
 
  vpImagePoint pt;
  for (unsigned int k = 0; k <= n; k++) {
    pt.set_ij(Pi[k], Pj[k]);
    l_controlPoints.push_back(pt);
    l_weights.push_back(Pw[k]);
  }
}
 
void vpNurbs::globalCurveInterp(vpList<vpMeSite> &l_crossingPoints)
{
  std::vector<vpImagePoint> v_crossingPoints;
  l_crossingPoints.front();
  vpMeSite s = l_crossingPoints.value();
  vpImagePoint pt(s.ifloat, s.jfloat);
  vpImagePoint pt_1 = pt;
  v_crossingPoints.push_back(pt);
  l_crossingPoints.next();
  while (!l_crossingPoints.outside()) {
    s = l_crossingPoints.value();
    pt.set_ij(s.ifloat, s.jfloat);
    if (vpImagePoint::distance(pt_1, pt) >= 10) {
      v_crossingPoints.push_back(pt);
      pt_1 = pt;
    }
    l_crossingPoints.next();
  }
  globalCurveInterp(v_crossingPoints, p, knots, controlPoints, weights);
}
 
void vpNurbs::globalCurveInterp(const std::list<vpImagePoint> &l_crossingPoints)
{
  std::vector<vpImagePoint> v_crossingPoints;
  for (std::list<vpImagePoint>::const_iterator it = l_crossingPoints.begin(); it != l_crossingPoints.end(); ++it) {
    v_crossingPoints.push_back(*it);
  }
  globalCurveInterp(v_crossingPoints, p, knots, controlPoints, weights);
}
 
 
void vpNurbs::globalCurveInterp(const std::list<vpMeSite> &l_crossingPoints)
{
  std::vector<vpImagePoint> v_crossingPoints;
  vpMeSite s = l_crossingPoints.front();
  vpImagePoint pt(s.ifloat, s.jfloat);
  vpImagePoint pt_1 = pt;
  v_crossingPoints.push_back(pt);
  std::list<vpMeSite>::const_iterator it = l_crossingPoints.begin();
  ++it;
  for (; it != l_crossingPoints.end(); ++it) {
    vpImagePoint pt_tmp(it->ifloat, it->jfloat);
    if (vpImagePoint::distance(pt_1, pt_tmp) >= 10) {
      v_crossingPoints.push_back(pt_tmp);
      pt_1 = pt_tmp;
    }
  }
  globalCurveInterp(v_crossingPoints, p, knots, controlPoints, weights);
}
 
void vpNurbs::globalCurveInterp() { globalCurveInterp(crossingPoints, p, knots, controlPoints, weights); }
 
void vpNurbs::globalCurveApprox(std::vector<vpImagePoint> &l_crossingPoints, unsigned int l_p, unsigned int l_n,
                                std::vector<double> &l_knots, std::vector<vpImagePoint> &l_controlPoints,
                                std::vector<double> &l_weights)
{
  l_knots.clear();
  l_controlPoints.clear();
  l_weights.clear();
  unsigned int m = (unsigned int)l_crossingPoints.size() - 1;
 
  double d = 0;
  for (unsigned int k = 1; k <= m; k++)
    d = d + distance(l_crossingPoints[k], 1, l_crossingPoints[k - 1], 1);
 
  // Compute ubar
  std::vector<double> ubar;
  ubar.push_back(0.0);
  for (unsigned int k = 1; k < m; k++)
    ubar.push_back(ubar[k - 1] + distance(l_crossingPoints[k], 1, l_crossingPoints[k - 1], 1) / d);
  ubar.push_back(1.0);
 
  // Compute the knot vector
  for (unsigned int k = 0; k <= l_p; k++)
    l_knots.push_back(0.0);
 
  d = (double)(m + 1) / (double)(l_n - l_p + 1);
 
  for (unsigned int j = 1; j <= l_n - l_p; j++) {
    double i = floor(j * d);
    double alpha = j * d - i;
    l_knots.push_back((1.0 - alpha) * ubar[(unsigned int)i - 1] + alpha * ubar[(unsigned int)i]);
  }
 
  for (unsigned int k = 0; k <= l_p; k++)
    l_knots.push_back(1.0);
 
  // Compute Rk
  std::vector<vpImagePoint> Rk;
  vpBasisFunction *N;
  for (unsigned int k = 1; k <= m - 1; k++) {
    unsigned int span = findSpan(ubar[k], l_p, l_knots);
    if (span == l_p && span == l_n) {
      N = computeBasisFuns(ubar[k], span, l_p, l_knots);
      vpImagePoint pt(l_crossingPoints[k].get_i() - N[0].value * l_crossingPoints[0].get_i() -
                          N[l_p].value * l_crossingPoints[m].get_i(),
                      l_crossingPoints[k].get_j() - N[0].value * l_crossingPoints[0].get_j() -
                          N[l_p].value * l_crossingPoints[m].get_j());
      Rk.push_back(pt);
      delete[] N;
    }
    else if (span == l_p) {
      N = computeBasisFuns(ubar[k], span, l_p, l_knots);
      vpImagePoint pt(l_crossingPoints[k].get_i() - N[0].value * l_crossingPoints[0].get_i(),
                      l_crossingPoints[k].get_j() - N[0].value * l_crossingPoints[0].get_j());
      Rk.push_back(pt);
      delete[] N;
    }
    else if (span == l_n) {
      N = computeBasisFuns(ubar[k], span, l_p, l_knots);
      vpImagePoint pt(l_crossingPoints[k].get_i() - N[l_p].value * l_crossingPoints[m].get_i(),
                      l_crossingPoints[k].get_j() - N[l_p].value * l_crossingPoints[m].get_j());
      Rk.push_back(pt);
      delete[] N;
    }
    else {
      Rk.push_back(l_crossingPoints[k]);
    }
  }
 
  vpMatrix A(m - 1, l_n - 1);
  // Compute A
  for (unsigned int i = 1; i <= m - 1; i++) {
    unsigned int span = findSpan(ubar[i], l_p, l_knots);
    N = computeBasisFuns(ubar[i], span, l_p, l_knots);
    for (unsigned int k = 0; k <= l_p; k++) {
      if (N[k].i > 0 && N[k].i < l_n)
        A[i - 1][N[k].i - 1] = N[k].value;
    }
    delete[] N;
  }
 
  vpColVector Ri(l_n - 1);
  vpColVector Rj(l_n - 1);
  vpColVector Rw(l_n - 1);
  for (unsigned int i = 0; i < l_n - 1; i++) {
    double sum = 0;
    for (unsigned int k = 0; k < m - 1; k++)
      sum = sum + A[k][i] * Rk[k].get_i();
    Ri[i] = sum;
    sum = 0;
    for (unsigned int k = 0; k < m - 1; k++)
      sum = sum + A[k][i] * Rk[k].get_j();
    Rj[i] = sum;
    sum = 0;
    for (unsigned int k = 0; k < m - 1; k++)
      sum = sum + A[k][i]; // The crossing points weigths are equal to 1.
    Rw[i] = sum;
  }
 
  vpMatrix AtA = A.AtA();
  vpMatrix AtAinv;
  AtA.pseudoInverse(AtAinv);
 
  vpColVector Pi = AtAinv * Ri;
  vpColVector Pj = AtAinv * Rj;
  vpColVector Pw = AtAinv * Rw;
 
  vpImagePoint pt;
  l_controlPoints.push_back(l_crossingPoints[0]);
  l_weights.push_back(1.0);
  for (unsigned int k = 0; k < l_n - 1; k++) {
    pt.set_ij(Pi[k], Pj[k]);
    l_controlPoints.push_back(pt);
    l_weights.push_back(Pw[k]);
  }
  l_controlPoints.push_back(l_crossingPoints[m]);
  l_weights.push_back(1.0);
}
 
 
void vpNurbs::globalCurveApprox(vpList<vpMeSite> &l_crossingPoints, unsigned int n)
{
  std::vector<vpImagePoint> v_crossingPoints;
  l_crossingPoints.front();
  while (!l_crossingPoints.outside()) {
    vpMeSite s = l_crossingPoints.value();
    vpImagePoint pt(s.ifloat, s.jfloat);
    v_crossingPoints.push_back(pt);
    l_crossingPoints.next();
  }
  globalCurveApprox(v_crossingPoints, p, n, knots, controlPoints, weights);
}
 
 
void vpNurbs::globalCurveApprox(const std::list<vpImagePoint> &l_crossingPoints, unsigned int n)
{
  std::vector<vpImagePoint> v_crossingPoints;
  for (std::list<vpImagePoint>::const_iterator it = l_crossingPoints.begin(); it != l_crossingPoints.end(); ++it) {
    v_crossingPoints.push_back(*it);
  }
  globalCurveApprox(v_crossingPoints, p, n, knots, controlPoints, weights);
}
 
void vpNurbs::globalCurveApprox(const std::list<vpMeSite> &l_crossingPoints, unsigned int n)
{
  std::vector<vpImagePoint> v_crossingPoints;
  for (std::list<vpMeSite>::const_iterator it = l_crossingPoints.begin(); it != l_crossingPoints.end(); ++it) {
    vpImagePoint pt(it->ifloat, it->jfloat);
    v_crossingPoints.push_back(pt);
  }
  globalCurveApprox(v_crossingPoints, p, n, knots, controlPoints, weights);
}
 
void vpNurbs::globalCurveApprox(unsigned int n)
{
  globalCurveApprox(crossingPoints, p, n, knots, controlPoints, weights);
}