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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
pcl::BivariatePolynomialT
This represents a bivariate polynomial and provides some functionality for it.
Definition bivariate_polynomial.h:54
pcl::BivariatePolynomialT::parameters
real * parameters
Definition bivariate_polynomial.h:118
pcl::BivariatePolynomialT::setDegree
void setDegree(int new_degree)
Initialize members to default values.
Definition bivariate_polynomial.hpp:76
pcl::BivariatePolynomialT::getNoOfParametersFromDegree
static unsigned int getNoOfParametersFromDegree(int n)
How many parameters has a bivariate polynomial of the given degree.
Definition bivariate_polynomial.h:114
pcl::PolynomialCalculationsT::solveCubicEquation
void solveCubicEquation(real a, real b, real c, real d, std::vector< real > &roots) const
Solves an equation of the form ax^3 + bx^2 + cx + d = 0 See https://en.wikipedia.org/wiki/Cubic_equat...
Definition polynomial_calculations.hpp:145
pcl::PolynomialCalculationsT::isNearlyZero
bool isNearlyZero(real d) const
check if std::abs(d)<zeroValue
Definition polynomial_calculations.h:107
pcl::PolynomialCalculationsT::bivariatePolynomialApproximation
BivariatePolynomialT< 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
Get the bivariate polynomial approximation for Z(X,Y) from the given sample points.
Definition polynomial_calculations.hpp:397
pcl::PolynomialCalculationsT::sqrtIsNearlyZero
bool sqrtIsNearlyZero(real d) const
check if sqrt(std::abs(d))<zeroValue
Definition polynomial_calculations.h:114
pcl::PolynomialCalculationsT::solveQuarticEquation
void solveQuarticEquation(real a, real b, real c, real d, real e, std::vector< real > &roots) const
Solves an equation of the form ax^4 + bx^3 + cx^2 +dx + e = 0 See https://en.wikipedia....
Definition polynomial_calculations.hpp:246
pcl::PolynomialCalculationsT::solveQuadraticEquation
void solveQuadraticEquation(real a, real b, real c, std::vector< real > &roots) const
Solves an equation of the form ax^2 + bx + c = 0 See https://en.wikipedia.org/wiki/Quadratic_equation...
Definition polynomial_calculations.hpp:90
pcl::PolynomialCalculationsT::solveLinearEquation
void solveLinearEquation(real a, real b, std::vector< real > &roots) const
Solves an equation of the form ax + b = 0.
Definition polynomial_calculations.hpp:58
pcl
Definition convolution.h:46
M_PI
#define M_PI
Definition pcl_macros.h:201
pcl::PolynomialCalculationsT::Parameters::setZeroValue
void setZeroValue(real new_zero_value)
Set zero_value.
Definition polynomial_calculations.hpp:49
pcl::PolynomialCalculationsT::Parameters::sqr_zero_value
real sqr_zero_value
sqr of the above
Definition polynomial_calculations.h:62
pcl::PolynomialCalculationsT::Parameters::zero_value
real zero_value
Every value below this is considered to be zero.
Definition polynomial_calculations.h:61