Matrix4d MinEnFilter::iterate(MatrixXd& XY, VectorXd& Depth, MatrixXd& Flow) {
cout << "Iterate: " << iteration << endl;
iteration ++;
LieAlgebra grad, KE;
MatrixXd Hessian, KP;
// get and check dimensions
nPoints = XY.rows();
if (nPoints!= Depth.rows()) {
cerr << "Dimensions of XY do not coincide with Dimensions of Depth." << endl;
}
if (nPoints!= Flow.rows()) {
cerr << "Dimensions of XY do not coincide with Dimensions of Flow." << endl;
}
MatrixXd XYZ_inhom = inhomCoordinates(XY, false); // inhomogenous Coordinatestes
MatrixXd Flow_inhom = inhomCoordinates(Flow, true); // inhomogenous (relative) flow
MatrixXd Disps_inhom = XYZ_inhom + Flow_inhom; // inhomogenous Disparities
MatrixXd temp(nPoints, 4);
temp << XYZ_inhom, Depth.cwiseInverse();
MatrixXd Gk = (Depth*MatrixXd::Ones(1,4)).cwiseProduct(temp);
// compute Gradient of Hamiltonian
grad = computeGradient(currG , Gk, Disps_inhom);
// compute Hessian of Hamiltonian
Hessian = computeHessian(Gk, Disps_inhom);
// Numerical integration of state and second order information
for (size_t i = 0; i < numSteps; i++ ){
KE = integrateGImplicit(grad, Gk, Disps_inhom);
currG = currG * KE.exp_g();
KP = integratePExplicit(grad ,Hessian);
currP = currP + delta * KP;
}
return currG.E;
}
template<typename PointSource, typename PointTarget> double
pcl::NormalDistributionsTransform<PointSource, PointTarget>::computeStepLengthMT (const Eigen::Matrix<double, 6, 1> &x, Eigen::Matrix<double, 6, 1> &step_dir, double step_init, double step_max,
double step_min, double &score, Eigen::Matrix<double, 6, 1> &score_gradient, Eigen::Matrix<double, 6, 6> &hessian,
PointCloudSource &trans_cloud)
{
// Set the value of phi(0), Equation 1.3 [More, Thuente 1994]
double phi_0 = -score;
// Set the value of phi'(0), Equation 1.3 [More, Thuente 1994]
double d_phi_0 = -(score_gradient.dot (step_dir));
Eigen::Matrix<double, 6, 1> x_t;
if (d_phi_0 >= 0)
{
// Not a decent direction
if (d_phi_0 == 0)
return 0;
else
{
// Reverse step direction and calculate optimal step.
d_phi_0 *= -1;
step_dir *= -1;
}
}
// The Search Algorithm for T(mu) [More, Thuente 1994]
int max_step_iterations = 10;
int step_iterations = 0;
// Sufficient decreace constant, Equation 1.1 [More, Thuete 1994]
double mu = 1.e-4;
// Curvature condition constant, Equation 1.2 [More, Thuete 1994]
double nu = 0.9;
// Initial endpoints of Interval I,
double a_l = 0, a_u = 0;
// Auxiliary function psi is used until I is determined ot be a closed interval, Equation 2.1 [More, Thuente 1994]
double f_l = auxilaryFunction_PsiMT (a_l, phi_0, phi_0, d_phi_0, mu);
double g_l = auxilaryFunction_dPsiMT (d_phi_0, d_phi_0, mu);
double f_u = auxilaryFunction_PsiMT (a_u, phi_0, phi_0, d_phi_0, mu);
double g_u = auxilaryFunction_dPsiMT (d_phi_0, d_phi_0, mu);
// Check used to allow More-Thuente step length calculation to be skipped by making step_min == step_max
bool interval_converged = (step_max - step_min) > 0, open_interval = true;
double a_t = step_init;
a_t = std::min (a_t, step_max);
a_t = std::max (a_t, step_min);
x_t = x + step_dir * a_t;
final_transformation_ = (Eigen::Translation<float, 3>(static_cast<float> (x_t (0)), static_cast<float> (x_t (1)), static_cast<float> (x_t (2))) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (3)), Eigen::Vector3f::UnitX ()) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (4)), Eigen::Vector3f::UnitY ()) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (5)), Eigen::Vector3f::UnitZ ())).matrix ();
// New transformed point cloud
transformPointCloud (*input_, trans_cloud, final_transformation_);
// Updates score, gradient and hessian. Hessian calculation is unessisary but testing showed that most step calculations use the
// initial step suggestion and recalculation the reusable portions of the hessian would intail more computation time.
score = computeDerivatives (score_gradient, hessian, trans_cloud, x_t, true);
// Calculate phi(alpha_t)
double phi_t = -score;
// Calculate phi'(alpha_t)
double d_phi_t = -(score_gradient.dot (step_dir));
// Calculate psi(alpha_t)
double psi_t = auxilaryFunction_PsiMT (a_t, phi_t, phi_0, d_phi_0, mu);
// Calculate psi'(alpha_t)
double d_psi_t = auxilaryFunction_dPsiMT (d_phi_t, d_phi_0, mu);
// Iterate until max number of iterations, interval convergance or a value satisfies the sufficient decrease, Equation 1.1, and curvature condition, Equation 1.2 [More, Thuente 1994]
while (!interval_converged && step_iterations < max_step_iterations && !(psi_t <= 0 /*Sufficient Decrease*/ && d_phi_t <= -nu * d_phi_0 /*Curvature Condition*/))
{
// Use auxilary function if interval I is not closed
if (open_interval)
{
a_t = trialValueSelectionMT (a_l, f_l, g_l,
a_u, f_u, g_u,
a_t, psi_t, d_psi_t);
}
else
{
a_t = trialValueSelectionMT (a_l, f_l, g_l,
a_u, f_u, g_u,
a_t, phi_t, d_phi_t);
}
a_t = std::min (a_t, step_max);
a_t = std::max (a_t, step_min);
x_t = x + step_dir * a_t;
final_transformation_ = (Eigen::Translation<float, 3> (static_cast<float> (x_t (0)), static_cast<float> (x_t (1)), static_cast<float> (x_t (2))) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (3)), Eigen::Vector3f::UnitX ()) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (4)), Eigen::Vector3f::UnitY ()) *
Eigen::AngleAxis<float> (static_cast<float> (x_t (5)), Eigen::Vector3f::UnitZ ())).matrix ();
// New transformed point cloud
// Done on final cloud to prevent wasted computation
transformPointCloud (*input_, trans_cloud, final_transformation_);
// Updates score, gradient. Values stored to prevent wasted computation.
score = computeDerivatives (score_gradient, hessian, trans_cloud, x_t, false);
// Calculate phi(alpha_t+)
phi_t = -score;
// Calculate phi'(alpha_t+)
d_phi_t = -(score_gradient.dot (step_dir));
// Calculate psi(alpha_t+)
psi_t = auxilaryFunction_PsiMT (a_t, phi_t, phi_0, d_phi_0, mu);
// Calculate psi'(alpha_t+)
d_psi_t = auxilaryFunction_dPsiMT (d_phi_t, d_phi_0, mu);
// Check if I is now a closed interval
if (open_interval && (psi_t <= 0 && d_psi_t >= 0))
{
open_interval = false;
// Converts f_l and g_l from psi to phi
f_l = f_l + phi_0 - mu * d_phi_0 * a_l;
g_l = g_l + mu * d_phi_0;
// Converts f_u and g_u from psi to phi
f_u = f_u + phi_0 - mu * d_phi_0 * a_u;
g_u = g_u + mu * d_phi_0;
}
if (open_interval)
{
// Update interval end points using Updating Algorithm [More, Thuente 1994]
interval_converged = updateIntervalMT (a_l, f_l, g_l,
a_u, f_u, g_u,
a_t, psi_t, d_psi_t);
}
else
{
// Update interval end points using Modified Updating Algorithm [More, Thuente 1994]
interval_converged = updateIntervalMT (a_l, f_l, g_l,
a_u, f_u, g_u,
a_t, phi_t, d_phi_t);
}
step_iterations++;
}
// If inner loop was run then hessian needs to be calculated.
// Hessian is unnessisary for step length determination but gradients are required
// so derivative and transform data is stored for the next iteration.
if (step_iterations)
computeHessian (hessian, trans_cloud, x_t);
return (a_t);
}
