Epnp.cpp

/******************************************************************************
 * Author:   Laurent Kneip                                                    *
 * Contact:  kneip.laurent@gmail.com                                          *
 * License:  Copyright (c) 2013 Laurent Kneip, ANU. All rights reserved.      *
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// Note: this code has been downloaded from the homepage of the "Computer 
// Vision Laboratory" at EPFL Lausanne, and was originally developped by the 
// authors of [4]. I only adapted it to Eigen.


#include <iostream>
using namespace std;

#include <opengv/absolute_pose/modules/Epnp.hpp>


opengv::absolute_pose::modules::Epnp::Epnp(void)
{
  maximum_number_of_correspondences = 0;
  number_of_correspondences = 0;

  pws = 0;
  us = 0;
  alphas = 0;
  pcs = 0;
  signs = 0; //added

  this->uc = 0.0;
  this->vc = 0.0;
  this->fu = 1.0;
  this->fv = 1.0;
}

opengv::absolute_pose::modules::Epnp::~Epnp()
{
  delete [] pws;
  delete [] us;
  delete [] alphas;
  delete [] pcs;
  delete [] signs; //added
}

void
opengv::absolute_pose::modules::Epnp::
    set_maximum_number_of_correspondences(int n)
{
  if (maximum_number_of_correspondences < n)
  {
    if (pws != 0) delete [] pws;
    if (us != 0) delete [] us;
    if (alphas != 0) delete [] alphas;
    if (pcs != 0) delete [] pcs;
    if (signs != 0) delete [] signs; //added

    maximum_number_of_correspondences = n;
    pws = new double[3 * maximum_number_of_correspondences];
    us = new double[2 * maximum_number_of_correspondences];
    alphas = new double[4 * maximum_number_of_correspondences];
    pcs = new double[3 * maximum_number_of_correspondences];
    signs = new int[maximum_number_of_correspondences];
  }
}

void
opengv::absolute_pose::modules::Epnp::reset_correspondences(void)
{
  number_of_correspondences = 0;
}

void
opengv::absolute_pose::modules::Epnp::add_correspondence(
    double X,
    double Y,
    double Z,
    double x,
    double y,
    double z) //changed this interface
{
  pws[3 * number_of_correspondences    ] = X;
  pws[3 * number_of_correspondences + 1] = Y;
  pws[3 * number_of_correspondences + 2] = Z;

  us[2 * number_of_correspondences    ] = x/z;
  us[2 * number_of_correspondences + 1] = y/z;

  //added the following
  if(z > 0.0)
    signs[number_of_correspondences] = 1;
  else
    signs[number_of_correspondences] = -1;

  number_of_correspondences++;
}

void
opengv::absolute_pose::modules::Epnp::choose_control_points(void)
{
  // Take C0 as the reference points centroid:
  cws[0][0] = cws[0][1] = cws[0][2] = 0;
  for(int i = 0; i < number_of_correspondences; i++)
    for(int j = 0; j < 3; j++)
      cws[0][j] += pws[3 * i + j];

  for(int j = 0; j < 3; j++)
    cws[0][j] /= number_of_correspondences;


  // Take C1, C2, and C3 from PCA on the reference points:
  Eigen::MatrixXd PW0(number_of_correspondences,3);

  for(int i = 0; i < number_of_correspondences; i++)
    for(int j = 0; j < 3; j++)
      PW0(i,j) = pws[3 * i + j] - cws[0][j];

  Eigen::MatrixXd PW0tPW0 = PW0.transpose() * PW0;
  Eigen::JacobiSVD< Eigen::MatrixXd > SVD(
      PW0tPW0,
      Eigen::ComputeFullV | Eigen::ComputeFullU );
  Eigen::MatrixXd D = SVD.singularValues();
  Eigen::MatrixXd Ut = SVD.matrixU().transpose();

  for(int i = 1; i < 4; i++)
  {
    double k = sqrt(D(i - 1,0) / number_of_correspondences);
    for(int j = 0; j < 3; j++)
      cws[i][j] = cws[0][j] + k * Ut((i - 1),j);
  }
}

void
opengv::absolute_pose::modules::Epnp::
    compute_barycentric_coordinates(void)
{
  Eigen::Matrix3d CC;

  for(int i = 0; i < 3; i++)
    for(int j = 1; j < 4; j++)
      CC(i,j-1) = cws[j][i] - cws[0][i];

  Eigen::Matrix3d CC_inv = CC.inverse();

  for(int i = 0; i < number_of_correspondences; i++)
  {
    double * pi = pws + 3 * i;
    double * a = alphas + 4 * i;

    for(int j = 0; j < 3; j++)
      a[1 + j] =
        CC_inv(j,0) * (pi[0] - cws[0][0]) +
        CC_inv(j,1) * (pi[1] - cws[0][1]) +
        CC_inv(j,2) * (pi[2] - cws[0][2]);
    a[0] = 1.0f - a[1] - a[2] - a[3];
  }
}

void
opengv::absolute_pose::modules::Epnp::fill_M(
    Eigen::MatrixXd & M,
    const int row,
    const double * as,
    const double u,
    const double v)
{
  for(int i = 0; i < 4; i++)
  {
    M(row,3*i) = as[i] * fu;
    M(row,3*i+1) = 0.0;
    M(row,3*i+2) = as[i] * (uc - u);

    M(row+1,3*i) = 0.0;
    M(row+1,3*i+1) = as[i] * fv;
    M(row+1,3*i+2) = as[i] * (vc - v);
  }
}

void
opengv::absolute_pose::modules::Epnp::compute_ccs(
    const double * betas,
    const Eigen::MatrixXd & ut)
{
  for(int i = 0; i < 4; i++)
    ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0f;

  for(int i = 0; i < 4; i++)
  {
    for(int j = 0; j < 4; j++)
      for(int k = 0; k < 3; k++)
        ccs[j][k] += betas[i] * ut(11-i,3 * j + k);
  }
}

void
opengv::absolute_pose::modules::Epnp::compute_pcs(void)
{
  for(int i = 0; i < number_of_correspondences; i++)
  {
    double * a = alphas + 4 * i;
    double * pc = pcs + 3 * i;

    for(int j = 0; j < 3; j++)
      pc[j] =
          a[0] * ccs[0][j] +
          a[1] * ccs[1][j] +
          a[2] * ccs[2][j] +
          a[3] * ccs[3][j];
  }
}

double
opengv::absolute_pose::modules::Epnp::compute_pose(
    double R[3][3],
    double t[3])
{
  choose_control_points();
  compute_barycentric_coordinates();

  Eigen::MatrixXd M(2*number_of_correspondences,12);

  for(int i = 0; i < number_of_correspondences; i++)
    fill_M(M, 2 * i, alphas + 4 * i, us[2 * i], us[2 * i + 1]);

  Eigen::MatrixXd MtM = M.transpose() * M;
  Eigen::JacobiSVD< Eigen::MatrixXd > SVD(
      MtM,
      Eigen::ComputeFullV | Eigen::ComputeFullU );
  Eigen::MatrixXd Ut = SVD.matrixU().transpose();

  Eigen::Matrix<double,6,10> L_6x10;
  Eigen::Matrix<double,6,1> Rho;

  compute_L_6x10(Ut,L_6x10);
  compute_rho(Rho);

  double Betas[4][4], rep_errors[4];
  double Rs[4][3][3], ts[4][3];

  find_betas_approx_1(L_6x10, Rho, Betas[1]);
  gauss_newton(L_6x10, Rho, Betas[1]);
  rep_errors[1] = compute_R_and_t(Ut, Betas[1], Rs[1], ts[1]);

  find_betas_approx_2(L_6x10, Rho, Betas[2]);
  gauss_newton(L_6x10, Rho, Betas[2]);
  rep_errors[2] = compute_R_and_t(Ut, Betas[2], Rs[2], ts[2]);

  find_betas_approx_3(L_6x10, Rho, Betas[3]);
  gauss_newton(L_6x10, Rho, Betas[3]);
  rep_errors[3] = compute_R_and_t(Ut, Betas[3], Rs[3], ts[3]);

  int N = 1;
  if (rep_errors[2] < rep_errors[1]) N = 2;
  if (rep_errors[3] < rep_errors[N]) N = 3;

  copy_R_and_t(Rs[N], ts[N], R, t);

  return rep_errors[N];
}

void
opengv::absolute_pose::modules::Epnp::copy_R_and_t(
    const double R_src[3][3],
    const double t_src[3],
    double R_dst[3][3],
    double t_dst[3])
{
  for(int i = 0; i < 3; i++)
  {
    for(int j = 0; j < 3; j++)
      R_dst[i][j] = R_src[i][j];
    t_dst[i] = t_src[i];
  }
}

double
opengv::absolute_pose::modules::Epnp::dist2(
    const double * p1,
    const double * p2)
{
  return
    (p1[0] - p2[0]) * (p1[0] - p2[0]) +
    (p1[1] - p2[1]) * (p1[1] - p2[1]) +
    (p1[2] - p2[2]) * (p1[2] - p2[2]);
}

double
opengv::absolute_pose::modules::Epnp::dot(
    const double * v1,
    const double * v2)
{
  return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}

double
opengv::absolute_pose::modules::Epnp::reprojection_error(
    const double R[3][3],
    const double t[3])
{
  double sum2 = 0.0;

  for(int i = 0; i < number_of_correspondences; i++)
  {
    double * pw = pws + 3 * i;
    double Xc = dot(R[0], pw) + t[0];
    double Yc = dot(R[1], pw) + t[1];
    double inv_Zc = 1.0 / (dot(R[2], pw) + t[2]);
    double ue = uc + fu * Xc * inv_Zc;
    double ve = vc + fv * Yc * inv_Zc;
    double u = us[2 * i], v = us[2 * i + 1];

    sum2 += sqrt( (u - ue) * (u - ue) + (v - ve) * (v - ve) );
  }

  return sum2 / number_of_correspondences;
}

void
opengv::absolute_pose::modules::Epnp::estimate_R_and_t(
    double R[3][3],
    double t[3])
{
  double pc0[3], pw0[3];

  pc0[0] = pc0[1] = pc0[2] = 0.0;
  pw0[0] = pw0[1] = pw0[2] = 0.0;

  for(int i = 0; i < number_of_correspondences; i++)
  {
    const double * pc = pcs + 3 * i;
    const double * pw = pws + 3 * i;

    for(int j = 0; j < 3; j++)
    {
      pc0[j] += pc[j];
      pw0[j] += pw[j];
    }
  }
  for(int j = 0; j < 3; j++)
  {
    pc0[j] /= number_of_correspondences;
    pw0[j] /= number_of_correspondences;
  }

  Eigen::MatrixXd Abt(3,3);

  for(int i = 0; i < 3; i++)
  {
    for(int j = 0; j < 3; j++)
      Abt(i,j) = 0.0;
  }

  for(int i = 0; i < number_of_correspondences; i++)
  {
    double * pc = pcs + 3 * i;
    double * pw = pws + 3 * i;

    for(int j = 0; j < 3; j++)
    {
      Abt(j,0) += (pc[j] - pc0[j]) * (pw[0] - pw0[0]);
      Abt(j,1) += (pc[j] - pc0[j]) * (pw[1] - pw0[1]);
      Abt(j,2) += (pc[j] - pc0[j]) * (pw[2] - pw0[2]);
    }
  }

  Eigen::JacobiSVD< Eigen::MatrixXd > SVD(
      Abt,
      Eigen::ComputeFullV | Eigen::ComputeFullU );
  Eigen::MatrixXd Abt_u = SVD.matrixU();
  Eigen::MatrixXd Abt_v = SVD.matrixV();

  for(int i = 0; i < 3; i++)
    for(int j = 0; j < 3; j++)
      R[i][j] = Abt_u.row(i) * Abt_v.row(j).transpose();

  const double det =
      R[0][0] * R[1][1] * R[2][2] +
      R[0][1] * R[1][2] * R[2][0] +
      R[0][2] * R[1][0] * R[2][1] -
      R[0][2] * R[1][1] * R[2][0] -
      R[0][1] * R[1][0] * R[2][2] -
      R[0][0] * R[1][2] * R[2][1];

  //change 1: negative determinant problem is solved by changing Abt_v, not R

  if (det < 0)
  {
    //R[2][0] = -R[2][0];
    //R[2][1] = -R[2][1];
    //R[2][2] = -R[2][2];
    Eigen::MatrixXd Abt_v_prime = Abt_v;
    Abt_v_prime.col(2) = -Abt_v.col(2);
    for(int i = 0; i < 3; i++)
      for(int j = 0; j < 3; j++)
        R[i][j] = Abt_u.row(i) * Abt_v_prime.row(j).transpose();
  }

  t[0] = pc0[0] - dot(R[0], pw0);
  t[1] = pc0[1] - dot(R[1], pw0);
  t[2] = pc0[2] - dot(R[2], pw0);
}

void
opengv::absolute_pose::modules::Epnp::print_pose(
    const double R[3][3],
    const double t[3])
{
  cout << R[0][0] << " " << R[0][1] << " " << R[0][2] << " " << t[0] << endl;
  cout << R[1][0] << " " << R[1][1] << " " << R[1][2] << " " << t[1] << endl;
  cout << R[2][0] << " " << R[2][1] << " " << R[2][2] << " " << t[2] << endl;
}

void
opengv::absolute_pose::modules::Epnp::solve_for_sign(void)
{
  //change2: the following method is not independent of the optical system
  /*if (pcs[2] < 0.0) {
    for(int i = 0; i < 4; i++)
      for(int j = 0; j < 3; j++)
  ccs[i][j] = -ccs[i][j];

    for(int i = 0; i < number_of_correspondences; i++) {
      pcs[3 * i    ] = -pcs[3 * i];
      pcs[3 * i + 1] = -pcs[3 * i + 1];
      pcs[3 * i + 2] = -pcs[3 * i + 2];
    }
  }*/

  //change to this (using original depths)
  if( (pcs[2] < 0.0 && signs[0] > 0) || (pcs[2] > 0.0 && signs[0] < 0) )
  {
    for(int i = 0; i < 4; i++)
      for(int j = 0; j < 3; j++)
        ccs[i][j] = -ccs[i][j];

    for(int i = 0; i < number_of_correspondences; i++)
    {
      pcs[3 * i    ] = -pcs[3 * i];
      pcs[3 * i + 1] = -pcs[3 * i + 1];
      pcs[3 * i + 2] = -pcs[3 * i + 2];
    }
  }
}

double
opengv::absolute_pose::modules::Epnp::compute_R_and_t(
    const Eigen::MatrixXd & Ut,
    const double * betas,
    double R[3][3],
    double t[3])
{
  compute_ccs(betas, Ut);
  compute_pcs();

  solve_for_sign();

  estimate_R_and_t(R, t);

  return reprojection_error(R, t);
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_1 = [B11 B12     B13         B14]

void
opengv::absolute_pose::modules::Epnp::find_betas_approx_1(
    const Eigen::Matrix<double,6,10> & L_6x10,
    const Eigen::Matrix<double,6,1> & Rho,
    double * betas)
{
  Eigen::MatrixXd L_6x4(6,4);

  for(int i = 0; i < 6; i++)
  {
    L_6x4(i,0) = L_6x10(i,0);
    L_6x4(i,1) = L_6x10(i,1);
    L_6x4(i,2) = L_6x10(i,3);
    L_6x4(i,3) = L_6x10(i,6);
  }

  Eigen::JacobiSVD<Eigen::MatrixXd> SVD(
      L_6x4,
      Eigen::ComputeFullV | Eigen::ComputeFullU);
  Eigen::VectorXd Rho_temp = Rho;
  Eigen::VectorXd b4 = SVD.solve(Rho_temp);

  if (b4[0] < 0)
  {
    betas[0] = sqrt(-b4[0]);
    betas[1] = -b4[1] / betas[0];
    betas[2] = -b4[2] / betas[0];
    betas[3] = -b4[3] / betas[0];
  }
  else
  {
    betas[0] = sqrt(b4[0]);
    betas[1] = b4[1] / betas[0];
    betas[2] = b4[2] / betas[0];
    betas[3] = b4[3] / betas[0];
  }
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_2 = [B11 B12 B22                            ]

void
opengv::absolute_pose::modules::Epnp::find_betas_approx_2(
    const Eigen::Matrix<double,6,10> & L_6x10,
    const Eigen::Matrix<double,6,1> & Rho,
    double * betas)
{
  Eigen::MatrixXd L_6x3(6,3);

  for(int i = 0; i < 6; i++)
  {
    L_6x3(i,0) = L_6x10(i,0);
    L_6x3(i,1) = L_6x10(i,1);
    L_6x3(i,2) = L_6x10(i,2);
  }

  Eigen::JacobiSVD<Eigen::MatrixXd> SVD(
      L_6x3,
      Eigen::ComputeFullV | Eigen::ComputeFullU);
  Eigen::VectorXd Rho_temp = Rho;
  Eigen::VectorXd b3 = SVD.solve(Rho_temp);

  if (b3[0] < 0)
  {
    betas[0] = sqrt(-b3[0]);
    betas[1] = (b3[2] < 0) ? sqrt(-b3[2]) : 0.0;
  }
  else
  {
    betas[0] = sqrt(b3[0]);
    betas[1] = (b3[2] > 0) ? sqrt(b3[2]) : 0.0;
  }

  if (b3[1] < 0) betas[0] = -betas[0];

  betas[2] = 0.0;
  betas[3] = 0.0;
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_3 = [B11 B12 B22 B13 B23                    ]

void
opengv::absolute_pose::modules::Epnp::find_betas_approx_3(
    const Eigen::Matrix<double,6,10> & L_6x10,
    const Eigen::Matrix<double,6,1> & Rho,
    double * betas)
{
  Eigen::MatrixXd L_6x5(6,5);

  for(int i = 0; i < 6; i++)
  {
    L_6x5(i,0) = L_6x10(i,0);
    L_6x5(i,1) = L_6x10(i,1);
    L_6x5(i,2) = L_6x10(i,2);
    L_6x5(i,3) = L_6x10(i,3);
    L_6x5(i,4) = L_6x10(i,4);
  }

  Eigen::JacobiSVD<Eigen::MatrixXd> SVD(
      L_6x5,
      Eigen::ComputeFullV | Eigen::ComputeFullU);
  Eigen::VectorXd Rho_temp = Rho;
  Eigen::VectorXd b5 = SVD.solve(Rho_temp);

  if (b5[0] < 0)
  {
    betas[0] = sqrt(-b5[0]);
    betas[1] = (b5[2] < 0) ? sqrt(-b5[2]) : 0.0;
  }
  else
  {
    betas[0] = sqrt(b5[0]);
    betas[1] = (b5[2] > 0) ? sqrt(b5[2]) : 0.0;
  }
  if (b5[1] < 0) betas[0] = -betas[0];
  betas[2] = b5[3] / betas[0];
  betas[3] = 0.0;
}

void
opengv::absolute_pose::modules::Epnp::compute_L_6x10(
    const Eigen::MatrixXd & Ut,
    Eigen::Matrix<double,6,10> & L_6x10 )
{
  double dv[4][6][3];

  for(int i = 0; i < 4; i++)
  {
    int a = 0, b = 1;
    for(int j = 0; j < 6; j++)
    {
      dv[i][j][0] = Ut(11-i,3 * a    ) - Ut(11-i,3 * b);
      dv[i][j][1] = Ut(11-i,3 * a + 1) - Ut(11-i,3 * b + 1);
      dv[i][j][2] = Ut(11-i,3 * a + 2) - Ut(11-i,3 * b + 2);

      b++;
      if (b > 3)
      {
        a++;
        b = a + 1;
      }
    }
  }

  for(int i = 0; i < 6; i++)
  {
    L_6x10(i,0) =        dot(dv[0][i], dv[0][i]);
    L_6x10(i,1) = 2.0f * dot(dv[0][i], dv[1][i]);
    L_6x10(i,2) =        dot(dv[1][i], dv[1][i]);
    L_6x10(i,3) = 2.0f * dot(dv[0][i], dv[2][i]);
    L_6x10(i,4) = 2.0f * dot(dv[1][i], dv[2][i]);
    L_6x10(i,5) =        dot(dv[2][i], dv[2][i]);
    L_6x10(i,6) = 2.0f * dot(dv[0][i], dv[3][i]);
    L_6x10(i,7) = 2.0f * dot(dv[1][i], dv[3][i]);
    L_6x10(i,8) = 2.0f * dot(dv[2][i], dv[3][i]);
    L_6x10(i,9) =        dot(dv[3][i], dv[3][i]);
  }
}

void
opengv::absolute_pose::modules::Epnp::compute_rho(
    Eigen::Matrix<double,6,1> & Rho)
{
  Rho[0] = dist2(cws[0], cws[1]);
  Rho[1] = dist2(cws[0], cws[2]);
  Rho[2] = dist2(cws[0], cws[3]);
  Rho[3] = dist2(cws[1], cws[2]);
  Rho[4] = dist2(cws[1], cws[3]);
  Rho[5] = dist2(cws[2], cws[3]);
}

void
opengv::absolute_pose::modules::Epnp::compute_A_and_b_gauss_newton(
    const Eigen::Matrix<double,6,10> & L_6x10,
    const Eigen::Matrix<double,6,1> & Rho,
    double betas[4],
    Eigen::Matrix<double,6,4> & A,
    Eigen::Matrix<double,6,1> & b)
{
  for(int i = 0; i < 6; i++)
  {
    A(i,0) = 2*L_6x10(i,0)*betas[0] +   L_6x10(i,1)*betas[1] +
               L_6x10(i,3)*betas[2] +   L_6x10(i,6)*betas[3];
    A(i,1) =   L_6x10(i,1)*betas[0] + 2*L_6x10(i,2)*betas[1] +
               L_6x10(i,4)*betas[2] +   L_6x10(i,7)*betas[3];
    A(i,2) =   L_6x10(i,3)*betas[0] +   L_6x10(i,4)*betas[1] +
             2*L_6x10(i,5)*betas[2] +   L_6x10(i,8)*betas[3];
    A(i,3) =   L_6x10(i,6)*betas[0] +   L_6x10(i,7)*betas[1] +
               L_6x10(i,8)*betas[2] + 2*L_6x10(i,9)*betas[3];

    b(i,0) = Rho[i] - (
               L_6x10(i,0) * betas[0] * betas[0] +
               L_6x10(i,1) * betas[0] * betas[1] +
               L_6x10(i,2) * betas[1] * betas[1] +
               L_6x10(i,3) * betas[0] * betas[2] +
               L_6x10(i,4) * betas[1] * betas[2] +
               L_6x10(i,5) * betas[2] * betas[2] +
               L_6x10(i,6) * betas[0] * betas[3] +
               L_6x10(i,7) * betas[1] * betas[3] +
               L_6x10(i,8) * betas[2] * betas[3] +
               L_6x10(i,9) * betas[3] * betas[3]);
  }
}

void
opengv::absolute_pose::modules::Epnp::gauss_newton(
    const Eigen::Matrix<double,6,10> & L_6x10,
    const Eigen::Matrix<double,6,1> & Rho,
    double betas[4])
{
  const int iterations_number = 5;

  Eigen::Matrix<double,6,4> A;
  Eigen::Matrix<double,6,1> B;
  Eigen::Matrix<double,4,1> X;

  for(int k = 0; k < iterations_number; k++)
  {
    compute_A_and_b_gauss_newton(L_6x10,Rho,betas,A,B);
    qr_solve(A,B,X);

    for(int i = 0; i < 4; i++)
      betas[i] += X[i];
  }
}

void
opengv::absolute_pose::modules::Epnp::qr_solve(
    Eigen::Matrix<double,6,4> & A_orig,
    Eigen::Matrix<double,6,1> & b,
    Eigen::Matrix<double,4,1> & X)
{
  Eigen::Matrix<double,4,6> A = A_orig.transpose();

  static int max_nr = 0;
  static double * A1, * A2;

  const int nr = A_orig.rows();
  const int nc = A_orig.cols();

  if (max_nr != 0 && max_nr < nr)
  {
    delete [] A1;
    delete [] A2;
  }
  if (max_nr < nr)
  {
    max_nr = nr;
    A1 = new double[nr];
    A2 = new double[nr];
  }

  double * pA = A.data(), * ppAkk = pA;
  for(int k = 0; k < nc; k++)
  {
    double * ppAik = ppAkk, eta = fabs(*ppAik);
    for(int i = k + 1; i < nr; i++)
    {
      double elt = fabs(*ppAik);
      if (eta < elt) eta = elt;
      ppAik += nc;
    }

    if (eta == 0)
    {
      A1[k] = A2[k] = 0.0;
      cerr << "God damnit, A is singular, this shouldn't happen." << endl;
      return;
    }
    else
    {
      double * ppAik = ppAkk, sum = 0.0, inv_eta = 1. / eta;
      for(int i = k; i < nr; i++)
      {
        *ppAik *= inv_eta;
        sum += *ppAik * *ppAik;
        ppAik += nc;
      }
      double sigma = sqrt(sum);
      if (*ppAkk < 0)
        sigma = -sigma;
      *ppAkk += sigma;
      A1[k] = sigma * *ppAkk;
      A2[k] = -eta * sigma;
      for(int j = k + 1; j < nc; j++)
      {
        double * ppAik = ppAkk, sum = 0;
        for(int i = k; i < nr; i++)
        {
          sum += *ppAik * ppAik[j - k];
          ppAik += nc;
        }
        double tau = sum / A1[k];
        ppAik = ppAkk;
        for(int i = k; i < nr; i++)
        {
          ppAik[j - k] -= tau * *ppAik;
          ppAik += nc;
        }
      }
    }
    ppAkk += nc + 1;
  }

  // b <- Qt b
  double * ppAjj = pA, * pb = b.data();
  for(int j = 0; j < nc; j++)
  {
    double * ppAij = ppAjj, tau = 0;
    for(int i = j; i < nr; i++)
    {
      tau += *ppAij * pb[i];
      ppAij += nc;
    }
    tau /= A1[j];
    ppAij = ppAjj;
    for(int i = j; i < nr; i++)
    {
      pb[i] -= tau * *ppAij;
      ppAij += nc;
    }
    ppAjj += nc + 1;
  }

  // X = R-1 b
  double * pX = X.data();
  pX[nc - 1] = pb[nc - 1] / A2[nc - 1];
  for(int i = nc - 2; i >= 0; i--)
  {
    double * ppAij = pA + i * nc + (i + 1), sum = 0;

    for(int j = i + 1; j < nc; j++)
    {
      sum += *ppAij * pX[j];
      ppAij++;
    }
    pX[i] = (pb[i] - sum) / A2[i];
  }
}



void
opengv::absolute_pose::modules::Epnp::relative_error(
    double & rot_err,
    double & transl_err,
    const double Rtrue[3][3],
    const double ttrue[3],
    const double Rest[3][3],
    const double test[3])
{
  double qtrue[4], qest[4];

  mat_to_quat(Rtrue, qtrue);
  mat_to_quat(Rest, qest);

  double rot_err1 = sqrt((qtrue[0] - qest[0]) * (qtrue[0] - qest[0]) +
                         (qtrue[1] - qest[1]) * (qtrue[1] - qest[1]) +
                         (qtrue[2] - qest[2]) * (qtrue[2] - qest[2]) +
                         (qtrue[3] - qest[3]) * (qtrue[3] - qest[3])) /
                    sqrt(qtrue[0] * qtrue[0] +
                         qtrue[1] * qtrue[1] +
                         qtrue[2] * qtrue[2] +
                         qtrue[3] * qtrue[3]);

  double rot_err2 = sqrt((qtrue[0] + qest[0]) * (qtrue[0] + qest[0]) +
                         (qtrue[1] + qest[1]) * (qtrue[1] + qest[1]) +
                         (qtrue[2] + qest[2]) * (qtrue[2] + qest[2]) +
                         (qtrue[3] + qest[3]) * (qtrue[3] + qest[3]) ) /
                    sqrt(qtrue[0] * qtrue[0] +
                         qtrue[1] * qtrue[1] +
                         qtrue[2] * qtrue[2] +
                         qtrue[3] * qtrue[3]);

  rot_err = min(rot_err1,rot_err2);

  transl_err = sqrt((ttrue[0] - test[0]) * (ttrue[0] - test[0]) +
                    (ttrue[1] - test[1]) * (ttrue[1] - test[1]) +
                    (ttrue[2] - test[2]) * (ttrue[2] - test[2])) /
               sqrt(ttrue[0] * ttrue[0] +
                    ttrue[1] * ttrue[1] +
                    ttrue[2] * ttrue[2]);
}

void
opengv::absolute_pose::modules::Epnp::mat_to_quat(
    const double R[3][3],
    double q[4])
{
  double tr = R[0][0] + R[1][1] + R[2][2];
  double n4;

  if (tr > 0.0f)
  {
    q[0] = R[1][2] - R[2][1];
    q[1] = R[2][0] - R[0][2];
    q[2] = R[0][1] - R[1][0];
    q[3] = tr + 1.0f;
    n4 = q[3];
  }
  else if ( (R[0][0] > R[1][1]) && (R[0][0] > R[2][2]) )
  {
    q[0] = 1.0f + R[0][0] - R[1][1] - R[2][2];
    q[1] = R[1][0] + R[0][1];
    q[2] = R[2][0] + R[0][2];
    q[3] = R[1][2] - R[2][1];
    n4 = q[0];
  }
  else if (R[1][1] > R[2][2])
  {
    q[0] = R[1][0] + R[0][1];
    q[1] = 1.0f + R[1][1] - R[0][0] - R[2][2];
    q[2] = R[2][1] + R[1][2];
    q[3] = R[2][0] - R[0][2];
    n4 = q[1];
  }
  else
  {
    q[0] = R[2][0] + R[0][2];
    q[1] = R[2][1] + R[1][2];
    q[2] = 1.0f + R[2][2] - R[0][0] - R[1][1];
    q[3] = R[0][1] - R[1][0];
    n4 = q[2];
  }
  double scale = 0.5f / double(sqrt(n4));

  q[0] *= scale;
  q[1] *= scale;
  q[2] *= scale;
  q[3] *= scale;
}