polynomial_calculations.hpp

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#pragma once
 
#include <pcl/common/polynomial_calculations.h>
 
 
namespace pcl
{
 
template <typename real>
inline void
PolynomialCalculationsT<real>::Parameters::setZeroValue (real new_zero_value)
{
  zero_value = new_zero_value;
  sqr_zero_value = zero_value*zero_value;
}
 
 
template <typename real>
inline void
PolynomialCalculationsT<real>::solveLinearEquation (real a, real b, std::vector<real>& roots) const
{
  //std::cout << "Trying to solve "<<a<<"x + "<<b<<" = 0\n";
 
  if (isNearlyZero (b))
  {
    roots.push_back (0.0);
  }
  if (!isNearlyZero (a/b))
  {
    roots.push_back (-b/a);
  }
 
#if 0
  std::cout << __PRETTY_FUNCTION__ << ": Found "<<roots.size ()<<" roots.\n";
  for (unsigned int i=0; i<roots.size (); i++)
  {
    real x=roots[i];
    real result = a*x + b;
    if (!isNearlyZero (result))
    {
      std::cout << "Something went wrong during solving of polynomial "<<a<<"x + "<<b<<" = 0\n";
      //roots.clear ();
    }
    std::cout << "Root "<<i<<" = "<<roots[i]<<". ("<<a<<"x^ + "<<b<<" = "<<result<<")\n";
  }
#endif
}
 
 
template <typename real>
inline void
PolynomialCalculationsT<real>::solveQuadraticEquation (real a, real b, real c, std::vector<real>& roots) const
{
  //std::cout << "Trying to solve "<<a<<"x^2 + "<<b<<"x + "<<c<<" = 0\n";
 
  if (isNearlyZero (a))
  {
    //std::cout << "Highest order element is 0 => Calling solveLineaqrEquation.\n";
    solveLinearEquation (b, c, roots);
    return;
  }
 
  if (isNearlyZero (c))
  {
    roots.push_back (0.0);
    //std::cout << "Constant element is 0 => Adding root 0 and calling solveLinearEquation.\n";
    std::vector<real> tmpRoots;
    solveLinearEquation (a, b, tmpRoots);
    for (const auto& tmpRoot: tmpRoots)
      if (!isNearlyZero (tmpRoot))
        roots.push_back (tmpRoot);
    return;
  }
 
  real tmp = b*b - 4*a*c;
  if (tmp>0)
  {
    tmp = sqrt (tmp);
    real tmp2 = 1.0/ (2*a);
    roots.push_back ( (-b + tmp)*tmp2);
    roots.push_back ( (-b - tmp)*tmp2);
  }
  else if (sqrtIsNearlyZero (tmp))
  {
    roots.push_back (-b/ (2*a));
  }
 
#if 0
  std::cout << __PRETTY_FUNCTION__ << ": Found "<<roots.size ()<<" roots.\n";
  for (unsigned int i=0; i<roots.size (); i++)
  {
    real x=roots[i], x2=x*x;
    real result = a*x2 + b*x + c;
    if (!isNearlyZero (result))
    {
      std::cout << "Something went wrong during solving of polynomial "<<a<<"x^2 + "<<b<<"x + "<<c<<" = 0\n";
      //roots.clear ();
    }
    //std::cout << "Root "<<i<<" = "<<roots[i]<<". ("<<a<<"x^2 + "<<b<<"x + "<<c<<" = "<<result<<")\n";
  }
#endif
}
 
 
template<typename real>
inline void
PolynomialCalculationsT<real>::solveCubicEquation (real a, real b, real c, real d, std::vector<real>& roots) const
{
  //std::cout << "Trying to solve "<<a<<"x^3 + "<<b<<"x^2 + "<<c<<"x + "<<d<<" = 0\n";
 
  if (isNearlyZero (a))
  {
    //std::cout << "Highest order element is 0 => Calling solveQuadraticEquation.\n";
    solveQuadraticEquation (b, c, d, roots);
    return;
  }
 
  if (isNearlyZero (d))
  {
    roots.push_back (0.0);
    //std::cout << "Constant element is 0 => Adding root 0 and calling solveQuadraticEquation.\n";
    std::vector<real> tmpRoots;
    solveQuadraticEquation (a, b, c, tmpRoots);
    for (const auto& tmpRoot: tmpRoots)
      if (!isNearlyZero (tmpRoot))
        roots.push_back (tmpRoot);
    return;
  }
 
  double a2 = a*a,
         a3 = a2*a,
         b2 = b*b,
         b3 = b2*b,
         alpha = ( (3.0*a*c-b2)/ (3.0*a2)),
         beta  = (2*b3/ (27.0*a3)) - ( (b*c)/ (3.0*a2)) + (d/a),
         alpha2 = alpha*alpha,
         alpha3 = alpha2*alpha,
         beta2 = beta*beta;
 
  // Value for resubstitution:
  double resubValue = b/ (3*a);
 
  //std::cout << "Trying to solve y^3 + "<<alpha<<"y + "<<beta<<"\n";
 
  double discriminant = (alpha3/27.0) + 0.25*beta2;
 
  //std::cout << "Discriminant is "<<discriminant<<"\n";
 
  if (isNearlyZero (discriminant))
  {
    if (!isNearlyZero (alpha) || !isNearlyZero (beta))
    {
      roots.push_back ( (-3.0*beta)/ (2.0*alpha) - resubValue);
      roots.push_back ( (3.0*beta)/alpha - resubValue);
    }
    else
    {
      roots.push_back (-resubValue);
    }
  }
  else if (discriminant > 0)
  {
    double sqrtDiscriminant = sqrt (discriminant);
    double d1 = -0.5*beta + sqrtDiscriminant,
           d2 = -0.5*beta - sqrtDiscriminant;
    if (d1 < 0)
      d1 = -pow (-d1, 1.0/3.0);
    else
      d1 = pow (d1, 1.0/3.0);
 
    if (d2 < 0)
      d2 = -pow (-d2, 1.0/3.0);
    else
      d2 = pow (d2, 1.0/3.0);
 
    //std::cout << PVAR (d1)<<", "<<PVAR (d2)<<"\n";
    roots.push_back (d1 + d2 - resubValue);
  }
  else
  {
    double tmp1 = sqrt (- (4.0/3.0)*alpha),
           tmp2 = std::acos (-sqrt (-27.0/alpha3)*0.5*beta)/3.0;
    roots.push_back (tmp1*std::cos (tmp2) - resubValue);
    roots.push_back (-tmp1*std::cos (tmp2 + M_PI/3.0) - resubValue);
    roots.push_back (-tmp1*std::cos (tmp2 - M_PI/3.0) - resubValue);
  }
 
#if 0
  std::cout << __PRETTY_FUNCTION__ << ": Found "<<roots.size ()<<" roots.\n";
  for (unsigned int i=0; i<roots.size (); i++)
  {
    real x=roots[i], x2=x*x, x3=x2*x;
    real result = a*x3 + b*x2 + c*x + d;
    if (std::abs (result) > 1e-4)
    {
      std::cout << "Something went wrong:\n";
      //roots.clear ();
    }
    std::cout << "Root "<<i<<" = "<<roots[i]<<". ("<<a<<"x^3 + "<<b<<"x^2 + "<<c<<"x + "<<d<<" = "<<result<<")\n";
  }
  std::cout << "\n\n";
#endif
}
 
 
template<typename real>
inline void
PolynomialCalculationsT<real>::solveQuarticEquation (real a, real b, real c, real d, real e,
                                                     std::vector<real>& roots) const
{
  //std::cout << "Trying to solve "<<a<<"x^4 + "<<b<<"x^3 + "<<c<<"x^2 + "<<d<<"x + "<<e<<" = 0\n";
 
  if (isNearlyZero (a))
  {
    //std::cout << "Highest order element is 0 => Calling solveCubicEquation.\n";
    solveCubicEquation (b, c, d, e, roots);
    return;
  }
 
  if (isNearlyZero (e))
  {
    roots.push_back (0.0);
    //std::cout << "Constant element is 0 => Adding root 0 and calling solveCubicEquation.\n";
    std::vector<real> tmpRoots;
    solveCubicEquation (a, b, c, d, tmpRoots);
    for (const auto& tmpRoot: tmpRoots)
      if (!isNearlyZero (tmpRoot))
        roots.push_back (tmpRoot);
    return;
  }
 
  double a2 = a*a,
         a3 = a2*a,
         a4 = a2*a2,
         b2 = b*b,
         b3 = b2*b,
         b4 = b2*b2,
         alpha = ( (-3.0*b2)/ (8.0*a2)) + (c/a),
         beta  = (b3/ (8.0*a3)) - ( (b*c)/ (2.0*a2)) + (d/a),
         gamma = ( (-3.0*b4)/ (256.0*a4)) + ( (c*b2)/ (16.0*a3)) - ( (b*d)/ (4.0*a2)) + (e/a),
         alpha2 = alpha*alpha;
 
  // Value for resubstitution:
  double resubValue = b/ (4*a);
 
  //std::cout << "Trying to solve y^4 + "<<alpha<<"y^2 + "<<beta<<"y + "<<gamma<<"\n";
 
  if (isNearlyZero (beta))
  {  // y^4 + alpha*y^2 + gamma\n";
    //std::cout << "Using beta=0 condition\n";
    std::vector<real> tmpRoots;
    solveQuadraticEquation (1.0, alpha, gamma, tmpRoots);
    for (const auto& quadraticRoot: tmpRoots)
    {
      if (sqrtIsNearlyZero (quadraticRoot))
      {
        roots.push_back (-resubValue);
      }
      else if (quadraticRoot > 0.0)
      {
        double root1 = sqrt (quadraticRoot);
        roots.push_back (root1 - resubValue);
        roots.push_back (-root1 - resubValue);
      }
    }
  }
  else
  {
    //std::cout << "beta != 0\n";
    double alpha3 = alpha2*alpha,
           beta2 = beta*beta,
           p = (-alpha2/12.0)-gamma,
           q = (-alpha3/108.0)+ ( (alpha*gamma)/3.0)- (beta2/8.0),
           q2 = q*q,
           p3 = p*p*p,
           u = (0.5*q) + sqrt ( (0.25*q2)+ (p3/27.0));
    if (u > 0.0)
      u = pow (u, 1.0/3.0);
    else if (isNearlyZero (u))
      u = 0.0;
    else
      u = -pow (-u, 1.0/3.0);
 
    double y = (-5.0/6.0)*alpha - u;
    if (!isNearlyZero (u))
      y += p/ (3.0*u);
 
    double w = alpha + 2.0*y;
 
    if (w > 0)
    {
      w = sqrt (w);
    }
    else if (isNearlyZero (w))
    {
      w = 0;
    }
    else
    {
      //std::cout << "Found no roots\n";
      return;
    }
 
    double tmp1 = - (3.0*alpha + 2.0*y + 2.0* (beta/w)),
           tmp2 = - (3.0*alpha + 2.0*y - 2.0* (beta/w));
 
    if (tmp1 > 0)
    {
      tmp1 = sqrt (tmp1);
      double root1 = - (b/ (4.0*a)) + 0.5* (w+tmp1);
      double root2 = - (b/ (4.0*a)) + 0.5* (w-tmp1);
      roots.push_back (root1);
      roots.push_back (root2);
    }
    else if (isNearlyZero (tmp1))
    {
      double root1 = - (b/ (4.0*a)) + 0.5*w;
      roots.push_back (root1);
    }
 
   if (tmp2 > 0)
   {
      tmp2 = sqrt (tmp2);
      double root3 = - (b/ (4.0*a)) + 0.5* (-w+tmp2);
      double root4 = - (b/ (4.0*a)) + 0.5* (-w-tmp2);
      roots.push_back (root3);
      roots.push_back (root4);
    }
    else if (isNearlyZero (tmp2))
    {
      double root3 = - (b/ (4.0*a)) - 0.5*w;
      roots.push_back (root3);
    }
 
    //std::cout << "Test: " << alpha<<", "<<beta<<", "<<gamma<<", "<<p<<", "<<q<<", "<<u <<", "<<y<<", "<<w<<"\n";
  }
 
#if 0
  std::cout << __PRETTY_FUNCTION__ << ": Found "<<roots.size ()<<" roots.\n";
  for (unsigned int i=0; i<roots.size (); i++)
  {
    real x=roots[i], x2=x*x, x3=x2*x, x4=x2*x2;
    real result = a*x4 + b*x3 + c*x2 + d*x + e;
    if (std::abs (result) > 1e-4)
    {
      std::cout << "Something went wrong:\n";
      //roots.clear ();
    }
    std::cout << "Root "<<i<<" = "<<roots[i]
         << ". ("<<a<<"x^4 + "<<b<<"x^3 + "<<c<<"x^2 + "<<d<<"x + "<<e<<" = "<<result<<")\n";
  }
  std::cout << "\n\n";
#endif
}
 
 
template<typename real>
inline pcl::BivariatePolynomialT<real>
PolynomialCalculationsT<real>::bivariatePolynomialApproximation (
  std::vector<Eigen::Matrix<real, 3, 1>, Eigen::aligned_allocator<Eigen::Matrix<real, 3, 1> > >& samplePoints, unsigned int polynomial_degree, bool& error) const
{
  pcl::BivariatePolynomialT<real> ret;
  error = bivariatePolynomialApproximation (samplePoints, polynomial_degree, ret);
  return ret;
}
 
 
template<typename real>
inline bool
PolynomialCalculationsT<real>::bivariatePolynomialApproximation (
  std::vector<Eigen::Matrix<real, 3, 1>, Eigen::aligned_allocator<Eigen::Matrix<real, 3, 1> > >& samplePoints, unsigned int polynomial_degree,
  pcl::BivariatePolynomialT<real>& ret) const
{
  const auto parameters_size = BivariatePolynomialT<real>::getNoOfParametersFromDegree (polynomial_degree);
 
  // Too many parameters for this number of equations (points)?
  if (parameters_size > samplePoints.size ())
  {
    return false;
    // Reduce degree of polynomial
    //polynomial_degree = (unsigned int) (0.5f* (std::sqrt (8*samplePoints.size ()+1) - 3));
    //parameters_size = BivariatePolynomialT<real>::getNoOfParametersFromDegree (polynomial_degree);
    //std::cout << "Not enough points, so degree of polynomial was decreased to "<<polynomial_degree
    //     << " ("<<samplePoints.size ()<<" points => "<<parameters_size<<" parameters)\n";
  }
 
  ret.setDegree (polynomial_degree);
 
  // A is a symmetric matrix
  Eigen::Matrix<real, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor> A (parameters_size, parameters_size);
  A.setZero();
  Eigen::Matrix<real, Eigen::Dynamic, 1> b (parameters_size);
  b.setZero();
 
  {
    std::vector<real> C (parameters_size);
    for (const auto& point: samplePoints)
    {
      real currentX = point[0], currentY = point[1], currentZ = point[2];
 
      {
        auto CRevPtr = C.rbegin ();
        real tmpX = 1.0;
        for (unsigned int xDegree=0; xDegree<=polynomial_degree; ++xDegree)
        {
          real tmpY = 1.0;
          for (unsigned int yDegree=0; yDegree<=polynomial_degree-xDegree; ++yDegree, ++CRevPtr)
          {
            *CRevPtr = tmpX*tmpY;
            tmpY *= currentY;
          }
          tmpX *= currentX;
        }
      }
 
      for (std::size_t i=0; i<parameters_size; ++i)
      {
        b[i] += currentZ * C[i];
        // fill the upper right triangular matrix
        for (std::size_t j=i; j<parameters_size; ++j)
        {
          A (i, j) += C[i] * C[j];
        }
      }
    //A += DMatrix<real>::outProd (C);
    //b += currentZ * C;
    }
  }
 
  // The Eigen only solution is slow for small matrices. Maybe file a bug
  // A.traingularView<Eigen::StrictlyLower> = A.transpose();
  // copy upper-right elements to lower-left
  for (std::size_t i = 0; i < parameters_size; ++i)
  {
    for (std::size_t j = 0; j < i; ++j)
    {
        A (i, j) = A (j, i);
    }
  }
 
  Eigen::Matrix<real, Eigen::Dynamic, 1> parameters;
  //double choleskyStartTime=-get_time ();
  //parameters = A.choleskySolve (b);
  //std::cout << "Cholesky took "<< (choleskyStartTime+get_time ())*1000<<"ms.\n";
 
  //double invStartTime=-get_time ();
  parameters = A.inverse () * b;
  //std::cout << "Inverse took "<< (invStartTime+get_time ())*1000<<"ms.\n";
 
  //std::cout << PVARC (A)<<PVARC (b)<<PVARN (parameters);
 
  real inversionCheckResult = (A*parameters - b).norm ();
  if (inversionCheckResult > 1e-5)
  {
    //std::cout << "Inversion result: "<< inversionCheckResult<<" for matrix "<<A<<"\n";
    return false;
  }
 
  std::copy_n(parameters.data(), parameters_size, ret.parameters);
 
  //std::cout << "Resulting polynomial is "<<ret<<"\n";
 
  //Test of gradient: ret.calculateGradient ();
 
  return true;
}
 
} // namespace pcl