eigen.h

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 *  Point Cloud Library (PCL) - www.pointclouds.org
 *  Copyright (c) 2010-2012, Willow Garage, Inc.
 *  Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
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 * $Id: eigen.h 6126 2012-07-03 20:19:58Z aichim $
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// The computeRoots function included in this is based on materials
// covered by the following copyright and license:
//
// Geometric Tools, LLC
// Copyright (c) 1998-2010
// Distributed under the Boost Software License, Version 1.0.
//
// Permission is hereby granted, free of charge, to any person or organization
// obtaining a copy of the software and accompanying documentation covered by
// this license (the "Software") to use, reproduce, display, distribute,
// execute, and transmit the Software, and to prepare derivative works of the
// Software, and to permit third-parties to whom the Software is furnished to
// do so, all subject to the following:
//
// The copyright notices in the Software and this entire statement, including
// the above license grant, this restriction and the following disclaimer,
// must be included in all copies of the Software, in whole or in part, and
// all derivative works of the Software, unless such copies or derivative
// works are solely in the form of machine-executable object code generated by
// a source language processor.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
// SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
// FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
// ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.

#ifndef PCL_EIGEN_H_
#define PCL_EIGEN_H_

#ifndef NOMINMAX
#define NOMINMAX
#endif

#if defined __GNUC__
#  pragma GCC system_header
#elif defined __SUNPRO_CC
#  pragma disable_warn
#elif defined _MSC_VER
#  pragma warning(push, 1)
#endif

#include <Eigen/StdVector>
#include <Eigen/Core>
#include <Eigen/Eigenvalues>
#include <Eigen/Geometry>
#include <Eigen/SVD>
#include <Eigen/LU>
#include <Eigen/Dense>
#include <Eigen/Eigenvalues>

namespace pcl
{
  template <typename Scalar> inline Scalar
  sqrt (const Scalar &val)
  {
  }

  template<> inline double
  sqrt<double> (const double &val)
  {
    return (::sqrt (val));
  }

  template<> inline float
  sqrt<float> (const float &val)
  {
    return (::sqrtf (val));
  }

  template<> inline long double
  sqrt<long double> (const long double &val)
  {
    return (::sqrtl (val));
  }

  template<typename Scalar, typename Roots> inline void
  computeRoots2 (const Scalar& b, const Scalar& c, Roots& roots)
  {
    roots (0) = Scalar (0);
    Scalar d = Scalar (b * b - 4.0 * c);
    if (d < 0.0) // no real roots!!!! THIS SHOULD NOT HAPPEN!
      d = 0.0;

    Scalar sd = sqrt (d);

    roots (2) = 0.5f * (b + sd);
    roots (1) = 0.5f * (b - sd);
  }

  template<typename Matrix, typename Roots> inline void
  computeRoots (const Matrix& m, Roots& roots)
  {
    typedef typename Matrix::Scalar Scalar;

    // The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0.  The
    // eigenvalues are the roots to this equation, all guaranteed to be
    // real-valued, because the matrix is symmetric.
    Scalar c0 =            m (0, 0) * m (1, 1) * m (2, 2)
            + Scalar (2) * m (0, 1) * m (0, 2) * m (1, 2)
                         - m (0, 0) * m (1, 2) * m (1, 2)
                         - m (1, 1) * m (0, 2) * m (0, 2)
                         - m (2, 2) * m (0, 1) * m (0, 1);
    Scalar c1 = m (0, 0) * m (1, 1) -
                m (0, 1) * m (0, 1) +
                m (0, 0) * m (2, 2) -
                m (0, 2) * m (0, 2) +
                m (1, 1) * m (2, 2) -
                m (1, 2) * m (1, 2);
    Scalar c2 = m (0, 0) + m (1, 1) + m (2, 2);


    if (fabs (c0) < Eigen::NumTraits<Scalar>::epsilon ())// one root is 0 -> quadratic equation
      computeRoots2 (c2, c1, roots);
    else
    {
      const Scalar s_inv3 = Scalar (1.0 / 3.0);
      const Scalar s_sqrt3 = Eigen::internal::sqrt (Scalar (3.0));
      // Construct the parameters used in classifying the roots of the equation
      // and in solving the equation for the roots in closed form.
      Scalar c2_over_3 = c2*s_inv3;
      Scalar a_over_3 = (c1 - c2 * c2_over_3) * s_inv3;
      if (a_over_3 > Scalar (0))
        a_over_3 = Scalar (0);

      Scalar half_b = Scalar (0.5) * (c0 + c2_over_3 * (Scalar (2) * c2_over_3 * c2_over_3 - c1));

      Scalar q = half_b * half_b + a_over_3 * a_over_3*a_over_3;
      if (q > Scalar (0))
        q = Scalar (0);

      // Compute the eigenvalues by solving for the roots of the polynomial.
      Scalar rho = Eigen::internal::sqrt (-a_over_3);
      Scalar theta = std::atan2 (Eigen::internal::sqrt (-q), half_b) * s_inv3;
      Scalar cos_theta = Eigen::internal::cos (theta);
      Scalar sin_theta = Eigen::internal::sin (theta);
      roots (0) = c2_over_3 + Scalar (2) * rho * cos_theta;
      roots (1) = c2_over_3 - rho * (cos_theta + s_sqrt3 * sin_theta);
      roots (2) = c2_over_3 - rho * (cos_theta - s_sqrt3 * sin_theta);

      // Sort in increasing order.
      if (roots (0) >= roots (1))
        std::swap (roots (0), roots (1));
      if (roots (1) >= roots (2))
      {
        std::swap (roots (1), roots (2));
        if (roots (0) >= roots (1))
          std::swap (roots (0), roots (1));
      }

      if (roots (0) <= 0) // eigenval for symetric positive semi-definite matrix can not be negative! Set it to 0
        computeRoots2 (c2, c1, roots);
    }
  }

  template <typename Matrix, typename Vector> inline void
  eigen22 (const Matrix& mat, typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
  {
    // if diagonal matrix, the eigenvalues are the diagonal elements
    // and the eigenvectors are not unique, thus set to Identity
    if (fabs(mat.coeff (1)) <= std::numeric_limits<typename Matrix::Scalar>::min ())
    {
      if (mat.coeff (0) < mat.coeff (2))
      {
        eigenvalue = mat.coeff (0);
        eigenvector [0] = 1.0;
        eigenvector [1] = 0.0;
      }
      else
      {
        eigenvalue = mat.coeff (2);
        eigenvector [0] = 0.0;
        eigenvector [1] = 1.0;
      }
      return;
    }
    
    // 0.5 to optimize further calculations
    typename Matrix::Scalar trace = static_cast<typename Matrix::Scalar> (0.5) * (mat.coeff (0) + mat.coeff (3));
    typename Matrix::Scalar determinant = mat.coeff (0) * mat.coeff (3) - mat.coeff (1) * mat.coeff (1);

    typename Matrix::Scalar temp = trace * trace - determinant;

    if (temp < 0)
      temp = 0;

    eigenvalue = trace - sqrt (temp);
    
    eigenvector [0] = - mat.coeff (1);
    eigenvector [1] = mat.coeff (0) - eigenvalue;
    eigenvector.normalize ();
  }

  template <typename Matrix, typename Vector> inline void
  eigen22 (const Matrix& mat, Matrix& eigenvectors, Vector& eigenvalues)
  {
    // if diagonal matrix, the eigenvalues are the diagonal elements
    // and the eigenvectors are not unique, thus set to Identity
    if (fabs(mat.coeff (1)) <= std::numeric_limits<typename Matrix::Scalar>::min ())
    {
      if (mat.coeff (0) < mat.coeff (3))
      {
        eigenvalues.coeffRef (0) = mat.coeff (0);
        eigenvalues.coeffRef (1) = mat.coeff (3);
        eigenvectors.coeffRef (0) = 1.0;
        eigenvectors.coeffRef (1) = 0.0;
        eigenvectors.coeffRef (2) = 0.0;
        eigenvectors.coeffRef (3) = 1.0;        
      }
      else
      {
        eigenvalues.coeffRef (0) = mat.coeff (3);
        eigenvalues.coeffRef (1) = mat.coeff (0);
        eigenvectors.coeffRef (0) = 0.0;
        eigenvectors.coeffRef (1) = 1.0;
        eigenvectors.coeffRef (2) = 1.0;
        eigenvectors.coeffRef (3) = 0.0;        
      }
      return;
    }

    // 0.5 to optimize further calculations
    typename Matrix::Scalar trace = static_cast<typename Matrix::Scalar> (0.5) * (mat.coeff (0) + mat.coeff (3));
    typename Matrix::Scalar determinant = mat.coeff (0) * mat.coeff (3) - mat.coeff (1) * mat.coeff (1);

    typename Matrix::Scalar temp = trace * trace - determinant;

    if (temp < 0)
      temp = 0;
    else
      temp = sqrt (temp);

    eigenvalues.coeffRef (0) = trace - temp;
    eigenvalues.coeffRef (1) = trace + temp;

    // either this is in a row or column depending on RowMajor or ColumnMajor
    eigenvectors.coeffRef (0) = - mat.coeff (1);
    eigenvectors.coeffRef (2) = mat.coeff (0) - eigenvalues.coeff (0);
    typename Matrix::Scalar norm = static_cast<typename Matrix::Scalar> (1.0) / 
                                   static_cast<typename Matrix::Scalar> (sqrt (eigenvectors.coeffRef (0) * eigenvectors.coeffRef (0) + eigenvectors.coeffRef (2) * eigenvectors.coeffRef (2)));
    eigenvectors.coeffRef (0) *= norm;
    eigenvectors.coeffRef (2) *= norm;
    eigenvectors.coeffRef (1) = eigenvectors.coeffRef (2);
    eigenvectors.coeffRef (3) = -eigenvectors.coeffRef (0);
  }

  template<typename Matrix, typename Vector> inline void
  computeCorrespondingEigenVector (const Matrix& mat, const typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
  {
    typedef typename Matrix::Scalar Scalar;
    // Scale the matrix so its entries are in [-1,1].  The scaling is applied
    // only when at least one matrix entry has magnitude larger than 1.

    Scalar scale = mat.cwiseAbs ().maxCoeff ();
    if (scale <= std::numeric_limits<Scalar>::min ())
      scale = Scalar (1.0);

    Matrix scaledMat = mat / scale;

    scaledMat.diagonal ().array () -= eigenvalue / scale;

    Vector vec1 = scaledMat.row (0).cross (scaledMat.row (1));
    Vector vec2 = scaledMat.row (0).cross (scaledMat.row (2));
    Vector vec3 = scaledMat.row (1).cross (scaledMat.row (2));

    Scalar len1 = vec1.squaredNorm ();
    Scalar len2 = vec2.squaredNorm ();
    Scalar len3 = vec3.squaredNorm ();

    if (len1 >= len2 && len1 >= len3)
      eigenvector = vec1 / Eigen::internal::sqrt (len1);
    else if (len2 >= len1 && len2 >= len3)
      eigenvector = vec2 / Eigen::internal::sqrt (len2);
    else
      eigenvector = vec3 / Eigen::internal::sqrt (len3);
  }
  
  template<typename Matrix, typename Vector> inline void
  eigen33 (const Matrix& mat, typename Matrix::Scalar& eigenvalue, Vector& eigenvector)
  {
    typedef typename Matrix::Scalar Scalar;
    // Scale the matrix so its entries are in [-1,1].  The scaling is applied
    // only when at least one matrix entry has magnitude larger than 1.

    Scalar scale = mat.cwiseAbs ().maxCoeff ();
    if (scale <= std::numeric_limits<Scalar>::min ())
      scale = Scalar (1.0);

    Matrix scaledMat = mat / scale;

    Vector eigenvalues;
    computeRoots (scaledMat, eigenvalues);

    eigenvalue = eigenvalues (0) * scale;

    scaledMat.diagonal ().array () -= eigenvalues (0);

    Vector vec1 = scaledMat.row (0).cross (scaledMat.row (1));
    Vector vec2 = scaledMat.row (0).cross (scaledMat.row (2));
    Vector vec3 = scaledMat.row (1).cross (scaledMat.row (2));

    Scalar len1 = vec1.squaredNorm ();
    Scalar len2 = vec2.squaredNorm ();
    Scalar len3 = vec3.squaredNorm ();

    if (len1 >= len2 && len1 >= len3)
      eigenvector = vec1 / Eigen::internal::sqrt (len1);
    else if (len2 >= len1 && len2 >= len3)
      eigenvector = vec2 / Eigen::internal::sqrt (len2);
    else
      eigenvector = vec3 / Eigen::internal::sqrt (len3);
  }

  template<typename Matrix, typename Vector> inline void
  eigen33 (const Matrix& mat, Vector& evals)
  {
    typedef typename Matrix::Scalar Scalar;
    Scalar scale = mat.cwiseAbs ().maxCoeff ();
    if (scale <= std::numeric_limits<Scalar>::min ())
      scale = Scalar (1.0);

    Matrix scaledMat = mat / scale;
    computeRoots (scaledMat, evals);
    evals *= scale;
  }

  template<typename Matrix, typename Vector> inline void
  eigen33 (const Matrix& mat, Matrix& evecs, Vector& evals)
  {
    typedef typename Matrix::Scalar Scalar;
    // Scale the matrix so its entries are in [-1,1].  The scaling is applied
    // only when at least one matrix entry has magnitude larger than 1.

    Scalar scale = mat.cwiseAbs ().maxCoeff ();
    if (scale <= std::numeric_limits<Scalar>::min ())
      scale = Scalar (1.0);

    Matrix scaledMat = mat / scale;

    // Compute the eigenvalues
    computeRoots (scaledMat, evals);

    if ((evals (2) - evals (0)) <= Eigen::NumTraits<Scalar>::epsilon ())
    {
      // all three equal
      evecs.setIdentity ();
    }
    else if ((evals (1) - evals (0)) <= Eigen::NumTraits<Scalar>::epsilon () )
    {
      // first and second equal
      Matrix tmp;
      tmp = scaledMat;
      tmp.diagonal ().array () -= evals (2);

      Vector vec1 = tmp.row (0).cross (tmp.row (1));
      Vector vec2 = tmp.row (0).cross (tmp.row (2));
      Vector vec3 = tmp.row (1).cross (tmp.row (2));

      Scalar len1 = vec1.squaredNorm ();
      Scalar len2 = vec2.squaredNorm ();
      Scalar len3 = vec3.squaredNorm ();

      if (len1 >= len2 && len1 >= len3)
        evecs.col (2) = vec1 / Eigen::internal::sqrt (len1);
      else if (len2 >= len1 && len2 >= len3)
        evecs.col (2) = vec2 / Eigen::internal::sqrt (len2);
      else
        evecs.col (2) = vec3 / Eigen::internal::sqrt (len3);

      evecs.col (1) = evecs.col (2).unitOrthogonal ();
      evecs.col (0) = evecs.col (1).cross (evecs.col (2));
    }
    else if ((evals (2) - evals (1)) <= Eigen::NumTraits<Scalar>::epsilon () )
    {
      // second and third equal
      Matrix tmp;
      tmp = scaledMat;
      tmp.diagonal ().array () -= evals (0);

      Vector vec1 = tmp.row (0).cross (tmp.row (1));
      Vector vec2 = tmp.row (0).cross (tmp.row (2));
      Vector vec3 = tmp.row (1).cross (tmp.row (2));

      Scalar len1 = vec1.squaredNorm ();
      Scalar len2 = vec2.squaredNorm ();
      Scalar len3 = vec3.squaredNorm ();

      if (len1 >= len2 && len1 >= len3)
        evecs.col (0) = vec1 / Eigen::internal::sqrt (len1);
      else if (len2 >= len1 && len2 >= len3)
        evecs.col (0) = vec2 / Eigen::internal::sqrt (len2);
      else
        evecs.col (0) = vec3 / Eigen::internal::sqrt (len3);

      evecs.col (1) = evecs.col (0).unitOrthogonal ();
      evecs.col (2) = evecs.col (0).cross (evecs.col (1));
    }
    else
    {
      Matrix tmp;
      tmp = scaledMat;
      tmp.diagonal ().array () -= evals (2);

      Vector vec1 = tmp.row (0).cross (tmp.row (1));
      Vector vec2 = tmp.row (0).cross (tmp.row (2));
      Vector vec3 = tmp.row (1).cross (tmp.row (2));

      Scalar len1 = vec1.squaredNorm ();
      Scalar len2 = vec2.squaredNorm ();
      Scalar len3 = vec3.squaredNorm ();
#ifdef _WIN32
      Scalar *mmax = new Scalar[3];
#else
      Scalar mmax[3];
#endif
      unsigned int min_el = 2;
      unsigned int max_el = 2;
      if (len1 >= len2 && len1 >= len3)
      {
        mmax[2] = len1;
        evecs.col (2) = vec1 / Eigen::internal::sqrt (len1);
      }
      else if (len2 >= len1 && len2 >= len3)
      {
        mmax[2] = len2;
        evecs.col (2) = vec2 / Eigen::internal::sqrt (len2);
      }
      else
      {
        mmax[2] = len3;
        evecs.col (2) = vec3 / Eigen::internal::sqrt (len3);
      }

      tmp = scaledMat;
      tmp.diagonal ().array () -= evals (1);

      vec1 = tmp.row (0).cross (tmp.row (1));
      vec2 = tmp.row (0).cross (tmp.row (2));
      vec3 = tmp.row (1).cross (tmp.row (2));

      len1 = vec1.squaredNorm ();
      len2 = vec2.squaredNorm ();
      len3 = vec3.squaredNorm ();
      if (len1 >= len2 && len1 >= len3)
      {
        mmax[1] = len1;
        evecs.col (1) = vec1 / Eigen::internal::sqrt (len1);
        min_el = len1 <= mmax[min_el] ? 1 : min_el;
        max_el = len1 > mmax[max_el] ? 1 : max_el;
      }
      else if (len2 >= len1 && len2 >= len3)
      {
        mmax[1] = len2;
        evecs.col (1) = vec2 / Eigen::internal::sqrt (len2);
        min_el = len2 <= mmax[min_el] ? 1 : min_el;
        max_el = len2 > mmax[max_el] ? 1 : max_el;
      }
      else
      {
        mmax[1] = len3;
        evecs.col (1) = vec3 / Eigen::internal::sqrt (len3);
        min_el = len3 <= mmax[min_el] ? 1 : min_el;
        max_el = len3 > mmax[max_el] ? 1 : max_el;
      }

      tmp = scaledMat;
      tmp.diagonal ().array () -= evals (0);

      vec1 = tmp.row (0).cross (tmp.row (1));
      vec2 = tmp.row (0).cross (tmp.row (2));
      vec3 = tmp.row (1).cross (tmp.row (2));

      len1 = vec1.squaredNorm ();
      len2 = vec2.squaredNorm ();
      len3 = vec3.squaredNorm ();
      if (len1 >= len2 && len1 >= len3)
      {
        mmax[0] = len1;
        evecs.col (0) = vec1 / Eigen::internal::sqrt (len1);
        min_el = len3 <= mmax[min_el] ? 0 : min_el;
        max_el = len3 > mmax[max_el] ? 0 : max_el;
      }
      else if (len2 >= len1 && len2 >= len3)
      {
        mmax[0] = len2;
        evecs.col (0) = vec2 / Eigen::internal::sqrt (len2);
        min_el = len3 <= mmax[min_el] ? 0 : min_el;
        max_el = len3 > mmax[max_el] ? 0 : max_el;
      }
      else
      {
        mmax[0] = len3;
        evecs.col (0) = vec3 / Eigen::internal::sqrt (len3);
        min_el = len3 <= mmax[min_el] ? 0 : min_el;
        max_el = len3 > mmax[max_el] ? 0 : max_el;
      }

      unsigned mid_el = 3 - min_el - max_el;
      evecs.col (min_el) = evecs.col ((min_el + 1) % 3).cross ( evecs.col ((min_el + 2) % 3) ).normalized ();
      evecs.col (mid_el) = evecs.col ((mid_el + 1) % 3).cross ( evecs.col ((mid_el + 2) % 3) ).normalized ();
#ifdef _WIN32
      delete [] mmax;
#endif
    }
    // Rescale back to the original size.
    evals *= scale;
  }

  template<typename Matrix> inline typename Matrix::Scalar
  invert2x2 (const Matrix& matrix, Matrix& inverse)
  {
    typedef typename Matrix::Scalar Scalar;
    Scalar det = matrix.coeff (0) * matrix.coeff (3) - matrix.coeff (1) * matrix.coeff (2) ;

    if (det != 0)
    {
      //Scalar inv_det = Scalar (1.0) / det;
      inverse.coeffRef (0) =   matrix.coeff (3);
      inverse.coeffRef (1) = - matrix.coeff (1);
      inverse.coeffRef (2) = - matrix.coeff (2);
      inverse.coeffRef (3) =   matrix.coeff (0);
      inverse /= det;
    }
    return det;
  }

  template<typename Matrix> inline typename Matrix::Scalar
  invert3x3SymMatrix (const Matrix& matrix, Matrix& inverse)
  {
    typedef typename Matrix::Scalar Scalar;
    // elements
    // a b c
    // b d e
    // c e f
    //| a b c |-1             |   fd-ee    ce-bf   be-cd  |
    //| b d e |    =  1/det * |   ce-bf    af-cc   bc-ae  |
    //| c e f |               |   be-cd    bc-ae   ad-bb  |

    //det = a(fd-ee) + b(ec-fb) + c(eb-dc)

    Scalar fd_ee = matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (7) * matrix.coeff (5);
    Scalar ce_bf = matrix.coeff (2) * matrix.coeff (5) - matrix.coeff (1) * matrix.coeff (8);
    Scalar be_cd = matrix.coeff (1) * matrix.coeff (5) - matrix.coeff (2) * matrix.coeff (4);

    Scalar det = matrix.coeff (0) * fd_ee + matrix.coeff (1) * ce_bf + matrix.coeff (2) * be_cd;

    if (det != 0)
    {
      //Scalar inv_det = Scalar (1.0) / det;
      inverse.coeffRef (0) = fd_ee;
      inverse.coeffRef (1) = inverse.coeffRef (3) = ce_bf;
      inverse.coeffRef (2) = inverse.coeffRef (6) = be_cd;
      inverse.coeffRef (4) = (matrix.coeff (0) * matrix.coeff (8) - matrix.coeff (2) * matrix.coeff (2));
      inverse.coeffRef (5) = inverse.coeffRef (7) = (matrix.coeff (1) * matrix.coeff (2) - matrix.coeff (0) * matrix.coeff (5));
      inverse.coeffRef (8) = (matrix.coeff (0) * matrix.coeff (4) - matrix.coeff (1) * matrix.coeff (1));
      inverse /= det;
    }
    return det;
  }

  template<typename Matrix> inline typename Matrix::Scalar
  invert3x3Matrix (const Matrix& matrix, Matrix& inverse)
  {
    typedef typename Matrix::Scalar Scalar;

    //| a b c |-1             |   ie-hf    hc-ib   fb-ec  |
    //| d e f |    =  1/det * |   gf-id    ia-gc   dc-fa  |
    //| g h i |               |   hd-ge    gb-ha   ea-db  |
    //det = a(ie-hf) + d(hc-ib) + g(fb-ec)

    Scalar ie_hf = matrix.coeff (8) * matrix.coeff (4) - matrix.coeff (7) * matrix.coeff (5);
    Scalar hc_ib = matrix.coeff (7) * matrix.coeff (2) - matrix.coeff (8) * matrix.coeff (1);
    Scalar fb_ec = matrix.coeff (5) * matrix.coeff (1) - matrix.coeff (4) * matrix.coeff (2);
    Scalar det = matrix.coeff (0) * (ie_hf) + matrix.coeff (3) * (hc_ib) + matrix.coeff (6) * (fb_ec) ;

    if (det != 0)
    {
      inverse.coeffRef (0) = ie_hf;
      inverse.coeffRef (1) = hc_ib;
      inverse.coeffRef (2) = fb_ec;
      inverse.coeffRef (3) = matrix.coeff (6) * matrix.coeff (5) - matrix.coeff (8) * matrix.coeff (3);
      inverse.coeffRef (4) = matrix.coeff (8) * matrix.coeff (0) - matrix.coeff (6) * matrix.coeff (2);
      inverse.coeffRef (5) = matrix.coeff (3) * matrix.coeff (2) - matrix.coeff (5) * matrix.coeff (0);
      inverse.coeffRef (6) = matrix.coeff (7) * matrix.coeff (3) - matrix.coeff (6) * matrix.coeff (4);
      inverse.coeffRef (7) = matrix.coeff (6) * matrix.coeff (1) - matrix.coeff (7) * matrix.coeff (0);
      inverse.coeffRef (8) = matrix.coeff (4) * matrix.coeff (0) - matrix.coeff (3) * matrix.coeff (1);

      inverse /= det;
    }
    return det;
  }

  template<typename Matrix> inline typename Matrix::Scalar
  determinant3x3Matrix (const Matrix& matrix)
  {
    // result is independent of Row/Col Major storage!
    return matrix.coeff (0) * (matrix.coeff (4) * matrix.coeff (8) - matrix.coeff (5) * matrix.coeff (7)) +
           matrix.coeff (1) * (matrix.coeff (5) * matrix.coeff (6) - matrix.coeff (3) * matrix.coeff (8)) +
           matrix.coeff (2) * (matrix.coeff (3) * matrix.coeff (7) - matrix.coeff (4) * matrix.coeff (6)) ;
  }
  
  inline void
  getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis, const Eigen::Vector3f& y_direction,
                             Eigen::Affine3f& transformation);

  inline Eigen::Affine3f
  getTransFromUnitVectorsZY (const Eigen::Vector3f& z_axis, const Eigen::Vector3f& y_direction);

  inline void
  getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis, const Eigen::Vector3f& y_direction,
                             Eigen::Affine3f& transformation);

  inline Eigen::Affine3f
  getTransFromUnitVectorsXY (const Eigen::Vector3f& x_axis, const Eigen::Vector3f& y_direction);

  inline void
  getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction, const Eigen::Vector3f& z_axis,
                                       Eigen::Affine3f& transformation);

  inline Eigen::Affine3f
  getTransformationFromTwoUnitVectors (const Eigen::Vector3f& y_direction, const Eigen::Vector3f& z_axis);

  inline void
  getTransformationFromTwoUnitVectorsAndOrigin (const Eigen::Vector3f& y_direction, const Eigen::Vector3f& z_axis,
                                                const Eigen::Vector3f& origin, Eigen::Affine3f& transformation);

  inline void
  getEulerAngles (const Eigen::Affine3f& t, float& roll, float& pitch, float& yaw);

  inline void
  getTranslationAndEulerAngles (const Eigen::Affine3f& t, float& x, float& y, float& z,
                                float& roll, float& pitch, float& yaw);

  inline void
  getTransformation (float x, float y, float z, float roll, float pitch, float yaw, Eigen::Affine3f& t);

  inline Eigen::Affine3f
  getTransformation (float x, float y, float z, float roll, float pitch, float yaw);

  template <typename Derived> void
  saveBinary (const Eigen::MatrixBase<Derived>& matrix, std::ostream& file);

  template <typename Derived> void
  loadBinary (Eigen::MatrixBase<Derived> const& matrix, std::istream& file);
}

#include <pcl/common/impl/eigen.hpp>

#if defined __SUNPRO_CC
#  pragma enable_warn
#elif defined _MSC_VER
#  pragma warning(pop)
#endif

#endif  //#ifndef PCL_EIGEN_H_