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// Ceres Solver - A fast non-linear least squares minimizer
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// http://ceres-solver.org/
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// Author: jodebo_beck@gmx.de (Johannes Beck)
//
#ifndef CERES_PUBLIC_INTERNAL_LINE_PARAMETERIZATION_H_
#define CERES_PUBLIC_INTERNAL_LINE_PARAMETERIZATION_H_
#include "householder_vector.h"
namespace ceres {
template <int AmbientSpaceDimension>
bool LineParameterization<AmbientSpaceDimension>::Plus(
const double* x_ptr,
const double* delta_ptr,
double* x_plus_delta_ptr) const {
// We seek a box plus operator of the form
//
// [o*, d*] = Plus([o, d], [delta_o, delta_d])
//
// where o is the origin point, d is the direction vector, delta_o is
// the delta of the origin point and delta_d the delta of the direction and
// o* and d* is the updated origin point and direction.
//
// We separate the Plus operator into the origin point and directional part
// d* = Plus_d(d, delta_d)
// o* = Plus_o(o, d, delta_o)
//
// The direction update function Plus_d is the same as for the homogeneous
// vector parameterization:
//
// d* = H_{v(d)} [0.5 sinc(0.5 |delta_d|) delta_d, cos(0.5 |delta_d|)]^T
//
// where H is the householder matrix
// H_{v} = I - (2 / |v|^2) v v^T
// and
// v(d) = d - sign(d_n) |d| e_n.
//
// The origin point update function Plus_o is defined as
//
// o* = o + H_{v(d)} [0.5 delta_o, 0]^T.
static constexpr int kDim = AmbientSpaceDimension;
using AmbientVector = Eigen::Matrix<double, kDim, 1>;
using AmbientVectorRef = Eigen::Map<Eigen::Matrix<double, kDim, 1>>;
using ConstAmbientVectorRef =
Eigen::Map<const Eigen::Matrix<double, kDim, 1>>;
using ConstTangentVectorRef =
Eigen::Map<const Eigen::Matrix<double, kDim - 1, 1>>;
ConstAmbientVectorRef o(x_ptr);
ConstAmbientVectorRef d(x_ptr + kDim);
ConstTangentVectorRef delta_o(delta_ptr);
ConstTangentVectorRef delta_d(delta_ptr + kDim - 1);
AmbientVectorRef o_plus_delta(x_plus_delta_ptr);
AmbientVectorRef d_plus_delta(x_plus_delta_ptr + kDim);
const double norm_delta_d = delta_d.norm();
o_plus_delta = o;
// Shortcut for zero delta direction.
if (norm_delta_d == 0.0) {
d_plus_delta = d;
if (delta_o.isZero(0.0)) {
return true;
}
}
// Calculate the householder transformation which is needed for f_d and f_o.
AmbientVector v;
double beta;
// NOTE: The explicit template arguments are needed here because
// ComputeHouseholderVector is templated and some versions of MSVC
// have trouble deducing the type of v automatically.
internal::ComputeHouseholderVector<ConstAmbientVectorRef, double, kDim>(
d, &v, &beta);
if (norm_delta_d != 0.0) {
// Map the delta from the minimum representation to the over parameterized
// homogeneous vector. See section A6.9.2 on page 624 of Hartley & Zisserman
// (2nd Edition) for a detailed description. Note there is a typo on Page
// 625, line 4 so check the book errata.
const double norm_delta_div_2 = 0.5 * norm_delta_d;
const double sin_delta_by_delta =
std::sin(norm_delta_div_2) / norm_delta_div_2;
// Apply the delta update to remain on the unit sphere. See section A6.9.3
// on page 625 of Hartley & Zisserman (2nd Edition) for a detailed
// description.
AmbientVector y;
y.template head<kDim - 1>() = 0.5 * sin_delta_by_delta * delta_d;
y[kDim - 1] = std::cos(norm_delta_div_2);
d_plus_delta = d.norm() * (y - v * (beta * (v.transpose() * y)));
}
// The null space is in the direction of the line, so the tangent space is
// perpendicular to the line direction. This is achieved by using the
// householder matrix of the direction and allow only movements
// perpendicular to e_n.
//
// The factor of 0.5 is used to be consistent with the line direction
// update.
AmbientVector y;
y << 0.5 * delta_o, 0;
o_plus_delta += y - v * (beta * (v.transpose() * y));
return true;
}
template <int AmbientSpaceDimension>
bool LineParameterization<AmbientSpaceDimension>::ComputeJacobian(
const double* x_ptr, double* jacobian_ptr) const {
static constexpr int kDim = AmbientSpaceDimension;
using AmbientVector = Eigen::Matrix<double, kDim, 1>;
using ConstAmbientVectorRef =
Eigen::Map<const Eigen::Matrix<double, kDim, 1>>;
using MatrixRef = Eigen::Map<
Eigen::Matrix<double, 2 * kDim, 2 * (kDim - 1), Eigen::RowMajor>>;
ConstAmbientVectorRef d(x_ptr + kDim);
MatrixRef jacobian(jacobian_ptr);
// Clear the Jacobian as only half of the matrix is not zero.
jacobian.setZero();
AmbientVector v;
double beta;
// NOTE: The explicit template arguments are needed here because
// ComputeHouseholderVector is templated and some versions of MSVC
// have trouble deducing the type of v automatically.
internal::ComputeHouseholderVector<ConstAmbientVectorRef, double, kDim>(
d, &v, &beta);
// The Jacobian is equal to J = 0.5 * H.leftCols(kDim - 1) where H is
// the Householder matrix (H = I - beta * v * v') for the origin point. For
// the line direction part the Jacobian is scaled by the norm of the
// direction.
for (int i = 0; i < kDim - 1; ++i) {
jacobian.block(0, i, kDim, 1) = -0.5 * beta * v(i) * v;
jacobian.col(i)(i) += 0.5;
}
jacobian.template block<kDim, kDim - 1>(kDim, kDim - 1) =
jacobian.template block<kDim, kDim - 1>(0, 0) * d.norm();
return true;
}
} // namespace ceres
#endif // CERES_PUBLIC_INTERNAL_LINE_PARAMETERIZATION_H_