vpMeEllipse.cpp

/****************************************************************************
 *
 * ViSP, open source Visual Servoing Platform software.
 * Copyright (C) 2005 - 2019 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 http://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:
 * Moving edges.
 *
 * Authors:
 * Eric Marchand
 *
 *****************************************************************************/

#include <visp3/me/vpMeEllipse.h>

#include <visp3/core/vpDebug.h>
#include <visp3/core/vpException.h>
#include <visp3/core/vpImagePoint.h>
#include <visp3/core/vpRobust.h>
#include <visp3/me/vpMe.h>

#include <cmath>  // std::fabs
#include <limits> // numeric_limits
#include <vector>

void computeTheta(double &theta, vpColVector &K, const vpImagePoint &iP);

void computeTheta(double &theta, vpColVector &K, const vpImagePoint &iP)
{
  double i = iP.get_i();
  double j = iP.get_j();

  double A = 2 * i + 2 * K[1] * j + 2 * K[2];
  double B = 2 * K[0] * j + 2 * K[1] * i + 2 * K[3];

  theta = atan2(A, B); // Angle between the tangente and the i axis.

  while (theta > M_PI) {
    theta -= M_PI;
  }
  while (theta < 0) {
    theta += M_PI;
  }
}

vpMeEllipse::vpMeEllipse()
  : K(), iPc(), a(0.), b(0.), e(0.), iP1(), iP2(), alpha1(0), alpha2(2 * M_PI), ce(0.), se(0.), angle(), m00(0.),
    mu11(0.), mu20(0.), mu02(0.), m10(0.), m01(0.), m11(0.), m02(0.), m20(0.), thresholdWeight(0.2), expecteddensity(0.)
{
  // redimensionnement du vecteur de parametre
  // i^2 + K0 j^2 + 2 K1 i j + 2 K2 i + 2 K3 j + K4

  K.resize(5);

  // j1 = j2 = i1 = i2 = 0;
  iP1.set_i(0);
  iP1.set_j(0);
  iP2.set_i(0);
  iP2.set_j(0);
}

vpMeEllipse::vpMeEllipse(const vpMeEllipse &meellipse)
  : vpMeTracker(meellipse), K(meellipse.K), iPc(meellipse.iPc), a(0.), b(0.), e(0.), iP1(meellipse.iP1),
    iP2(meellipse.iP2), alpha1(0), alpha2(2 * M_PI), ce(0.), se(0.), angle(meellipse.angle), m00(0.), mu11(0.),
    mu20(0.), mu02(0.), m10(0.), m01(0.), m11(0.), m02(0.), m20(0.), thresholdWeight(0.2), expecteddensity(0.)
{
  a = meellipse.a;
  b = meellipse.b;
  e = meellipse.e;

  alpha1 = meellipse.alpha1;
  alpha2 = meellipse.alpha2;
  ce = meellipse.ce;
  se = meellipse.se;

  m00 = meellipse.m00;
  mu11 = meellipse.mu11;
  mu20 = meellipse.mu20;
  mu02 = meellipse.mu02;
  m10 = meellipse.m10;
  m01 = meellipse.m01;
  m11 = meellipse.m11;
  m02 = meellipse.m02;
  m20 = meellipse.m20;
  thresholdWeight = meellipse.thresholdWeight;

  expecteddensity = meellipse.expecteddensity;
}

vpMeEllipse::~vpMeEllipse()
{
  list.clear();
  angle.clear();
}

void vpMeEllipse::sample(const vpImage<unsigned char> &I, bool doNotTrack)
{
  (void)doNotTrack;
  if (!me) {
    throw(vpException(vpException::fatalError, "Moving edges on ellipse tracking not initialized"));
  }

  int height = (int)I.getHeight();
  int width = (int)I.getWidth();

  // if (me->getSampleStep()==0)
  if (std::fabs(me->getSampleStep()) <= std::numeric_limits<double>::epsilon()) {
    std::cout << "In vpMeEllipse::sample: ";
    std::cout << "function called with sample step = 0";
    // return fatalError;
  }

  vpImagePoint iP11;

  double incr = vpMath::rad(me->getSampleStep()); // angle increment en degree
  vpColor col = vpColor::red;
  getParameters();

  // Delete old list
  list.clear();

  angle.clear();

  // sample positions
  double k = alpha1;
  while (k < alpha2) {
    double j = a * sin(k); // equation of an ellipse
    double i = b * cos(k); // equation of an ellipse

    // (i,j) are the coordinates on the origin centered ellipse;
    // a rotation by "e" and a translation by (xci,jc) are done
    // to get the coordinates of the point on the shifted ellipse
    iP11.set_j(iPc.get_j() + ce * j + se * i);
    iP11.set_i(iPc.get_i() - se * j + ce * i);

    vpDisplay::displayCross(I, iP11, 5, col);

    double theta;
    computeTheta(theta, K, iP11);

    // If point is in the image, add to the sample list
    if (!outOfImage(vpMath::round(iP11.get_i()), vpMath::round(iP11.get_j()), 0, height, width)) {
      vpMeSite pix;
      pix.init((int)iP11.get_i(), (int)iP11.get_j(), theta);
      pix.setDisplay(selectDisplay);
      pix.setState(vpMeSite::NO_SUPPRESSION);

      if (vpDEBUG_ENABLE(3)) {
        vpDisplay::displayCross(I, iP11, 5, vpColor::blue);
      }
      list.push_back(pix);
      angle.push_back(k);
    }
    k += incr;
  }
  vpMeTracker::initTracking(I);
}

void vpMeEllipse::reSample(const vpImage<unsigned char> &I)
{
  if (!me) {
    throw(vpException(vpException::fatalError, "Moving edges on ellipse tracking not initialized"));
  }

  unsigned int n = numberOfSignal();
  expecteddensity = (alpha2 - alpha1) / vpMath::rad((double)me->getSampleStep());
  if ((double)n < 0.5 * expecteddensity) {
    sample(I);
    vpMeTracker::track(I);
  }
}

void vpMeEllipse::getParameters()
{
  double k[6];
  for (unsigned int i = 0; i < 5; i++)
    k[i + 1] = K[i];
  k[0] = 1;

  double d = k[2] * k[2] - k[0] * k[1];

  iPc.set_i((k[1] * k[3] - k[2] * k[4]) / d);
  iPc.set_j((k[0] * k[4] - k[2] * k[3]) / d);

  double sq = sqrt(vpMath::sqr(k[1] - k[0]) + 4.0 * vpMath::sqr(k[2]));

  if (std::fabs(k[2]) <= std::numeric_limits<double>::epsilon()) {
    e = 0;
  } else {
    e = (k[1] - k[0] + sq) / (2.0 * k[2]);
    e = (-1 / e);
  }

  e = atan(e);

  if (e < 0.0)
    e += M_PI;

  ce = cos(e);
  se = sin(e);

  double num = 2.0 * (k[0] * iPc.get_i() * iPc.get_i() + 2.0 * k[2] * iPc.get_j() * iPc.get_i() +
                      k[1] * iPc.get_j() * iPc.get_j() - k[5]);
  double a2 = num / (k[0] + k[1] + sq);
  double b2 = num / (k[0] + k[1] - sq);

  a = sqrt(a2);
  b = sqrt(b2);
}

void vpMeEllipse::printParameters()
{
  std::cout << "K" << std::endl;
  std::cout << K.t();
  std::cout << iPc << std::endl;
}

void vpMeEllipse::computeAngle(const vpImagePoint &pt1, const vpImagePoint &pt2)
{
  getParameters();

  int number_of_points = 2000;
  double incr = 2 * M_PI / number_of_points; // angle increment

  double dmin1 = 1e6;
  double dmin2 = 1e6;

  double k = 0;
  while (k < 2 * M_PI) {

    double j1 = a * sin(k); // equation of an ellipse
    double i1 = b * cos(k); // equation of an ellipse

    // (i1,j1) are the coordinates on the origin centered ellipse;
    // a rotation by "e" and a translation by (xci,jc) are done
    // to get the coordinates of the point on the shifted ellipse

    double j11 = iPc.get_j() + ce * j1 + se * i1;
    double i11 = iPc.get_i() - se * j1 + ce * i1;

    double d = vpMath::sqr(pt1.get_i() - i11) + vpMath::sqr(pt1.get_j() - j11);
    if (d < dmin1) {
      dmin1 = d;
      alpha1 = k;
    }
    d = vpMath::sqr(pt2.get_i() - i11) + vpMath::sqr(pt2.get_j() - j11);
    if (d < dmin2) {
      dmin2 = d;
      alpha2 = k;
    }
    k += incr;
  }

  if (alpha2 < alpha1)
    alpha2 += 2 * M_PI;
  // else if (alpha2 == alpha1)
  else if (std::fabs(alpha2 - alpha1) < std::fabs(alpha1) * std::numeric_limits<double>::epsilon())
    alpha2 += 2 * M_PI;
}

void vpMeEllipse::updateTheta()
{
  vpMeSite p_me;
  double theta;
  for (std::list<vpMeSite>::iterator it = list.begin(); it != list.end(); ++it) {
    p_me = *it;
    vpImagePoint iP;
    iP.set_i(p_me.ifloat);
    iP.set_j(p_me.jfloat);
    computeTheta(theta, K, iP);
    p_me.alpha = theta;
    *it = p_me;
  }
}

void vpMeEllipse::suppressPoints()
{
  // Loop through list of sites to track
  std::list<vpMeSite>::iterator itList = list.begin();
  for (std::list<double>::iterator it = angle.begin(); it != angle.end();) {
    vpMeSite s = *itList; // current reference pixel
    if (s.getState() != vpMeSite::NO_SUPPRESSION) {
      itList = list.erase(itList);
      it = angle.erase(it);
    } else {
      ++itList;
      ++it;
    }
  }
}

void vpMeEllipse::seekExtremities(const vpImage<unsigned char> &I)
{
  if (!me) {
    throw(vpException(vpException::fatalError, "Moving edges on ellipse tracking not initialized"));
  }

  int rows = (int)I.getHeight();
  int cols = (int)I.getWidth();

  vpImagePoint ip;

  unsigned int memory_range = me->getRange();
  me->setRange(2);

  double memory_mu1 = me->getMu1();
  me->setMu1(0.5);

  double memory_mu2 = me->getMu2();
  me->setMu2(0.5);

  double incr = vpMath::rad(2.0);

  if (alpha2 - alpha1 < 2 * M_PI - vpMath::rad(6.0)) {
    vpMeSite P;
    double k = alpha1;
    double i1, j1;

    for (unsigned int i = 0; i < 3; i++) {
      k -= incr;
      // while ( k < -M_PI ) { k+=2*M_PI; }

      i1 = b * cos(k); // equation of an ellipse
      j1 = a * sin(k); // equation of an ellipse
      P.ifloat = iPc.get_i() - se * j1 + ce * i1;
      P.i = (int)P.ifloat;
      P.jfloat = iPc.get_j() + ce * j1 + se * i1;
      P.j = (int)P.jfloat;

      if (!outOfImage(P.i, P.j, 5, rows, cols)) {
        P.track(I, me, false);

        if (P.getState() == vpMeSite::NO_SUPPRESSION) {
          list.push_back(P);
          angle.push_back(k);
          if (vpDEBUG_ENABLE(3)) {
            ip.set_i(P.i);
            ip.set_j(P.j);

            vpDisplay::displayCross(I, ip, 5, vpColor::green);
          }
        } else {
          if (vpDEBUG_ENABLE(3)) {
            ip.set_i(P.i);
            ip.set_j(P.j);
            vpDisplay::displayCross(I, ip, 10, vpColor::blue);
          }
        }
      }
    }

    k = alpha2;

    for (unsigned int i = 0; i < 3; i++) {
      k += incr;
      // while ( k > M_PI ) { k-=2*M_PI; }

      i1 = b * cos(k); // equation of an ellipse
      j1 = a * sin(k); // equation of an ellipse
      P.ifloat = iPc.get_i() - se * j1 + ce * i1;
      P.i = (int)P.ifloat;
      P.jfloat = iPc.get_j() + ce * j1 + se * i1;
      P.j = (int)P.jfloat;

      if (!outOfImage(P.i, P.j, 5, rows, cols)) {
        P.track(I, me, false);

        if (P.getState() == vpMeSite::NO_SUPPRESSION) {
          list.push_back(P);
          angle.push_back(k);
          if (vpDEBUG_ENABLE(3)) {
            ip.set_i(P.i);
            ip.set_j(P.j);

            vpDisplay::displayCross(I, ip, 5, vpColor::green);
          }
        } else {
          if (vpDEBUG_ENABLE(3)) {
            ip.set_i(P.i);
            ip.set_j(P.j);
            vpDisplay::displayCross(I, ip, 10, vpColor::blue);
          }
        }
      }
    }
  }

  suppressPoints();

  me->setRange(memory_range);
  me->setMu1(memory_mu1);
  me->setMu2(memory_mu2);
}

void vpMeEllipse::setExtremities()
{
  double alphamin = +1e6;
  double alphamax = -1e6;
  double imin = 0;
  double jmin = 0;
  double imax = 0;
  double jmax = 0;

  // Loop through list of sites to track
  std::list<double>::const_iterator itAngle = angle.begin();

  for (std::list<vpMeSite>::const_iterator itList = list.begin(); itList != list.end(); ++itList) {
    vpMeSite s = *itList; // current reference pixel
    double alpha = *itAngle;
    if (alpha < alphamin) {
      alphamin = alpha;
      imin = s.ifloat;
      jmin = s.jfloat;
    }

    if (alpha > alphamax) {
      alphamax = alpha;
      imax = s.ifloat;
      jmax = s.jfloat;
    }
    ++itAngle;
  }

  alpha1 = alphamin;
  alpha2 = alphamax;
  iP1.set_ij(imin, jmin);
  iP2.set_ij(imax, jmax);
}

void vpMeEllipse::leastSquare()
{
  // Construction du systeme Ax=b
  // i^2 + K0 j^2 + 2 K1 i j + 2 K2 i + 2 K3 j + K4
  // A = (j^2 2ij 2i 2j 1)   x = (K0 K1 K2 K3 K4)^T  b = (-i^2 )
  unsigned int i;

  vpMeSite p_me;

  unsigned int iter = 0;
  vpColVector b_(numberOfSignal());
  vpRobust r(numberOfSignal());
  r.setThreshold(2);
  r.setIteration(0);
  vpMatrix D(numberOfSignal(), numberOfSignal());
  D.eye();
  vpMatrix DA, DAmemory;
  vpColVector DAx;
  vpColVector w(numberOfSignal());
  w = 1;
  unsigned int nos_1 = numberOfSignal();

  if (list.size() < 3) {
    throw(vpException(vpException::dimensionError, "Not enought moving edges to track the ellipse"));
  }

  vpMatrix A(numberOfSignal(), 5);
  vpColVector x(5);

  unsigned int k = 0;
  for (std::list<vpMeSite>::const_iterator it = list.begin(); it != list.end(); ++it) {
    p_me = *it;
    if (p_me.getState() == vpMeSite::NO_SUPPRESSION) {
      A[k][0] = vpMath::sqr(p_me.jfloat);
      A[k][1] = 2 * p_me.ifloat * p_me.jfloat;
      A[k][2] = 2 * p_me.ifloat;
      A[k][3] = 2 * p_me.jfloat;
      A[k][4] = 1;

      b_[k] = -vpMath::sqr(p_me.ifloat);
      k++;
    }
  }

  while (iter < 4) {
    DA = D * A;
    vpMatrix DAp;

    x = DA.pseudoInverse(1e-26) * D * b_;

    vpColVector residu(nos_1);
    residu = b_ - A * x;
    r.setIteration(iter);
    r.MEstimator(vpRobust::TUKEY, residu, w);

    k = 0;
    for (i = 0; i < nos_1; i++) {
      D[k][k] = w[k];
      k++;
    }
    iter++;
  }

  k = 0;
  for (std::list<vpMeSite>::iterator it = list.begin(); it != list.end(); ++it) {
    p_me = *it;
    if (p_me.getState() == vpMeSite::NO_SUPPRESSION) {
      if (w[k] < thresholdWeight) {
        p_me.setState(vpMeSite::M_ESTIMATOR);

        *it = p_me;
      }
      k++;
    }
  }
  for (i = 0; i < 5; i++)
    K[i] = x[i];

  getParameters();
}

void vpMeEllipse::display(const vpImage<unsigned char> &I, vpColor col)
{
  vpMeEllipse::display(I, iPc, a, b, e, alpha1, alpha2, col);
}

void vpMeEllipse::initTracking(const vpImage<unsigned char> &I)
{
  const unsigned int n = 5;
  vpImagePoint iP[n];

  for (unsigned int k = 0; k < n; k++) {
    std::cout << "Click points " << k + 1 << "/" << n;
    std::cout << " on the ellipse in the trigonometric order" << std::endl;
    vpDisplay::getClick(I, iP[k], true);
    vpDisplay::displayCross(I, iP[k], 7, vpColor::red);
    vpDisplay::flush(I);
    std::cout << iP[k] << std::endl;
  }

  iP1 = iP[0];
  iP2 = iP[n - 1];

  initTracking(I, n, iP);
}

void vpMeEllipse::initTracking(const vpImage<unsigned char> &I, unsigned int n, vpImagePoint *iP)
{
  vpMatrix A(n, 5);
  vpColVector b_(n);
  vpColVector x(5);

  // Construction du systeme Ax=b
  // i^2 + K0 j^2 + 2 K1 i j + 2 K2 i + 2 K3 j + K4
  // A = (j^2 2ij 2i 2j 1)   x = (K0 K1 K2 K3 K4)^T  b = (-i^2 )

  for (unsigned int k = 0; k < n; k++) {
    A[k][0] = vpMath::sqr(iP[k].get_j());
    A[k][1] = 2 * iP[k].get_i() * iP[k].get_j();
    A[k][2] = 2 * iP[k].get_i();
    A[k][3] = 2 * iP[k].get_j();
    A[k][4] = 1;

    b_[k] = -vpMath::sqr(iP[k].get_i());
  }

  K = A.pseudoInverse(1e-26) * b_;

  iP1 = iP[0];
  iP2 = iP[n - 1];

  getParameters();

  computeAngle(iP1, iP2);

  expecteddensity = (alpha2 - alpha1) / vpMath::rad((double)me->getSampleStep());

  display(I, vpColor::green);
  sample(I);

  vpMeTracker::initTracking(I);

  track(I);

  vpMeTracker::display(I);
  vpDisplay::flush(I);
}

void vpMeEllipse::initTracking(const vpImage<unsigned char> &I, const std::vector<vpImagePoint> &iP)
{
  unsigned int n = (unsigned int)(iP.size());
  vpMatrix A(n, 5);
  vpColVector b_(n);
  vpColVector x(5);

  // Construction du systeme Ax=b
  // i^2 + K0 j^2 + 2 K1 i j + 2 K2 i + 2 K3 j + K4
  // A = (j^2 2ij 2i 2j 1)   x = (K0 K1 K2 K3 K4)^T  b = (-i^2 )

  for (unsigned int k = 0; k < n; k++) {
    A[k][0] = vpMath::sqr(iP[k].get_j());
    A[k][1] = 2 * iP[k].get_i() * iP[k].get_j();
    A[k][2] = 2 * iP[k].get_i();
    A[k][3] = 2 * iP[k].get_j();
    A[k][4] = 1;

    b_[k] = -vpMath::sqr(iP[k].get_i());
  }

  K = A.pseudoInverse(1e-26) * b_;

  iP1 = iP[0];
  iP2 = iP[n - 1];

  getParameters();

  computeAngle(iP1, iP2);

  expecteddensity = (alpha2 - alpha1) / vpMath::rad((double)me->getSampleStep());

  display(I, vpColor::green);
  sample(I);

  vpMeTracker::initTracking(I);

  track(I);

  vpMeTracker::display(I);
  vpDisplay::flush(I);
}

void vpMeEllipse::initTracking(const vpImage<unsigned char> &I, const vpImagePoint &ic, double a_p, double b_p,
                               double e_p, double low_alpha, double high_alpha)
{
  iPc = ic;
  a = a_p;
  b = b_p;
  e = e_p;
  alpha1 = low_alpha;
  alpha2 = high_alpha;

  if (alpha2 < alpha1)
    alpha2 += 2 * M_PI;

  ce = cos(e);
  se = sin(e);

  display(I, vpColor::green);
  sample(I);

  vpMeTracker::initTracking(I);

  track(I);

  vpMeTracker::display(I);
  vpDisplay::flush(I);
}

void vpMeEllipse::track(const vpImage<unsigned char> &I)
{
  vpMeTracker::track(I);

  // Estimation des parametres de la droite aux moindres carre
  suppressPoints();
  setExtremities();

  leastSquare();
  seekExtremities(I);
  setExtremities();
  leastSquare();

  // suppression des points rejetes par la regression robuste
  suppressPoints();
  setExtremities();

  // reechantillonage si necessaire
  reSample(I);

  // remet a jour l'angle delta pour chaque  point de la liste

  updateTheta();

  computeMoments();

  // Remise a jour de delta dans la liste de site me
  if (vpDEBUG_ENABLE(2)) {
    display(I, vpColor::red);
    vpMeTracker::display(I);
    vpDisplay::flush(I);
  }
}

void vpMeEllipse::computeMoments()
{
  double tane = tan(-1 / e);
  m00 = M_PI * a * b;
  m10 = m00 * iPc.get_i();
  m01 = m00 * iPc.get_j();
  m20 = m00 * (a * a + b * b * tane * tane) / (4 * (1 + tane * tane)) + m00 * iPc.get_i() * iPc.get_i();
  m02 = m00 * (a * a * tane * tane + b * b) / (4 * (1 + tane * tane)) + m00 * iPc.get_j() * iPc.get_j();
  m11 = m00 * tane * (a * a - b * b) / (4 * (1 + tane * tane)) + m00 * iPc.get_i() * iPc.get_j();
  mu11 = m11 - iPc.get_j() * m10;
  mu02 = m02 - iPc.get_j() * m01;
  mu20 = m20 - iPc.get_i() * m10;
}

#ifdef VISP_BUILD_DEPRECATED_FUNCTIONS

void vpMeEllipse::computeAngle(int ip1, int jp1, double &_alpha1, int ip2, int jp2, double &_alpha2)
{
  getParameters();

  int number_of_points = 2000;
  double incr = 2 * M_PI / number_of_points; // angle increment

  double dmin1 = 1e6;
  double dmin2 = 1e6;

  double k = -M_PI;
  while (k < M_PI) {

    double j1 = a * cos(k); // equation of an ellipse
    double i1 = b * sin(k); // equation of an ellipse

    // (i1,j1) are the coordinates on the origin centered ellipse;
    // a rotation by "e" and a translation by (xci,jc) are done
    // to get the coordinates of the point on the shifted ellipse
    double j11 = iPc.get_j() + ce * j1 - se * i1;
    double i11 = iPc.get_i() - (se * j1 + ce * i1);

    double d = vpMath::sqr(ip1 - i11) + vpMath::sqr(jp1 - j11);
    if (d < dmin1) {
      dmin1 = d;
      alpha1 = k;
      _alpha1 = k;
    }
    d = vpMath::sqr(ip2 - i11) + vpMath::sqr(jp2 - j11);
    if (d < dmin2) {
      dmin2 = d;
      alpha2 = k;
      _alpha2 = k;
    }
    k += incr;
  }

  if (alpha2 < alpha1)
    alpha2 += 2 * M_PI;

  vpCDEBUG(1) << "end vpMeEllipse::computeAngle(..)" << alpha1 << "  " << alpha2 << std::endl;
}

void vpMeEllipse::computeAngle(int ip1, int jp1, int ip2, int jp2)
{

  double a1, a2;
  computeAngle(ip1, jp1, a1, ip2, jp2, a2);
}

void vpMeEllipse::initTracking(const vpImage<unsigned char> &I, unsigned int n, unsigned *i, unsigned *j)
{
  vpMatrix A(n, 5);
  vpColVector b_(n);
  vpColVector x(5);

  // Construction du systeme Ax=b
  // A = (j^2 2ij 2i 2j 1)   x = (K0 K1 K2 K3 K4)^T  b = (-i^2 )

  for (unsigned int k = 0; k < n; k++) {
    A[k][0] = vpMath::sqr(j[k]);
    A[k][1] = 2 * i[k] * j[k];
    A[k][2] = 2 * i[k];
    A[k][3] = 2 * j[k];
    A[k][4] = 1;

    b_[k] = -vpMath::sqr(i[k]);
  }

  K = A.pseudoInverse(1e-26) * b_;
  // std::cout << K << std::endl;

  iP1.set_i(i[0]);
  iP1.set_j(j[0]);
  iP2.set_i(i[n - 1]);
  iP2.set_j(j[n - 1]);

  getParameters();
  computeAngle(iP1, iP2);
  display(I, vpColor::green);

  sample(I);

  //  2. On appelle ce qui n'est pas specifique
  {
    vpMeTracker::initTracking(I);
  }

  track(I);

  vpMeTracker::display(I);
  vpDisplay::flush(I);
}
#endif // Deprecated

void vpMeEllipse::display(const vpImage<unsigned char> &I, const vpImagePoint &center, const double &A, const double &B,
                          const double &E, const double &smallalpha, const double &highalpha, const vpColor &color,
                          unsigned int thickness)
{
  double j1, i1;
  vpImagePoint iP11;
  double j2, i2;
  vpImagePoint iP22;
  j1 = j2 = i1 = i2 = 0;

  double incr = vpMath::rad(2); // angle increment

  vpDisplay::displayCross(I, center, 20, vpColor::red, thickness);

  double k = smallalpha;
  while (k + incr < highalpha) {
    j1 = A * cos(k); // equation of an ellipse
    i1 = B * sin(k); // equation of an ellipse

    j2 = A * cos(k + incr); // equation of an ellipse
    i2 = B * sin(k + incr); // equation of an ellipse

    // (i1,j1) are the coordinates on the origin centered ellipse;
    // a rotation by "e" and a translation by (xci,jc) are done
    // to get the coordinates of the point on the shifted ellipse
    iP11.set_j(center.get_j() + cos(E) * j1 - sin(E) * i1);
    iP11.set_i(center.get_i() - (sin(E) * j1 + cos(E) * i1));
    // to get the coordinates of the point on the shifted ellipse
    iP22.set_j(center.get_j() + cos(E) * j2 - sin(E) * i2);
    iP22.set_i(center.get_i() - (sin(E) * j2 + cos(E) * i2));

    vpDisplay::displayLine(I, iP11, iP22, color, thickness);

    k += incr;
  }

  j1 = A * cos(smallalpha); // equation of an ellipse
  i1 = B * sin(smallalpha); // equation of an ellipse

  j2 = A * cos(highalpha); // equation of an ellipse
  i2 = B * sin(highalpha); // equation of an ellipse

  // (i1,j1) are the coordinates on the origin centered ellipse;
  // a rotation by "e" and a translation by (xci,jc) are done
  // to get the coordinates of the point on the shifted ellipse
  iP11.set_j(center.get_j() + cos(E) * j1 - sin(E) * i1);
  iP11.set_i(center.get_i() - (sin(E) * j1 + cos(E) * i1));
  // to get the coordinates of the point on the shifted ellipse
  iP22.set_j(center.get_j() + cos(E) * j2 - sin(E) * i2);
  iP22.set_i(center.get_i() - (sin(E) * j2 + cos(E) * i2));

  vpDisplay::displayLine(I, center, iP11, vpColor::red, thickness);
  vpDisplay::displayLine(I, center, iP22, vpColor::blue, thickness);
}

void vpMeEllipse::display(const vpImage<vpRGBa> &I, const vpImagePoint &center, const double &A, const double &B,
                          const double &E, const double &smallalpha, const double &highalpha, const vpColor &color,
                          unsigned int thickness)
{
  double j1, i1;
  vpImagePoint iP11;
  double j2, i2;
  vpImagePoint iP22;
  j1 = j2 = i1 = i2 = 0;

  double incr = vpMath::rad(2); // angle increment

  vpDisplay::displayCross(I, center, 20, vpColor::red, thickness);

  double k = smallalpha;
  while (k + incr < highalpha) {
    j1 = A * cos(k); // equation of an ellipse
    i1 = B * sin(k); // equation of an ellipse

    j2 = A * cos(k + incr); // equation of an ellipse
    i2 = B * sin(k + incr); // equation of an ellipse

    // (i1,j1) are the coordinates on the origin centered ellipse;
    // a rotation by "e" and a translation by (xci,jc) are done
    // to get the coordinates of the point on the shifted ellipse
    iP11.set_j(center.get_j() + cos(E) * j1 - sin(E) * i1);
    iP11.set_i(center.get_i() - (sin(E) * j1 + cos(E) * i1));
    // to get the coordinates of the point on the shifted ellipse
    iP22.set_j(center.get_j() + cos(E) * j2 - sin(E) * i2);
    iP22.set_i(center.get_i() - (sin(E) * j2 + cos(E) * i2));

    vpDisplay::displayLine(I, iP11, iP22, color, thickness);

    k += incr;
  }

  j1 = A * cos(smallalpha); // equation of an ellipse
  i1 = B * sin(smallalpha); // equation of an ellipse

  j2 = A * cos(highalpha); // equation of an ellipse
  i2 = B * sin(highalpha); // equation of an ellipse

  // (i1,j1) are the coordinates on the origin centered ellipse;
  // a rotation by "e" and a translation by (xci,jc) are done
  // to get the coordinates of the point on the shifted ellipse
  iP11.set_j(center.get_j() + cos(E) * j1 - sin(E) * i1);
  iP11.set_i(center.get_i() - (sin(E) * j1 + cos(E) * i1));
  // to get the coordinates of the point on the shifted ellipse
  iP22.set_j(center.get_j() + cos(E) * j2 - sin(E) * i2);
  iP22.set_i(center.get_i() - (sin(E) * j2 + cos(E) * i2));

  vpDisplay::displayLine(I, center, iP11, vpColor::red, thickness);
  vpDisplay::displayLine(I, center, iP22, vpColor::blue, thickness);
}