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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2022 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: vitus@google.com (Mike Vitus)
// jodebo_beck@gmx.de (Johannes Beck)
#ifndef CERES_PUBLIC_INTERNAL_SPHERE_MANIFOLD_HELPERS_H_
#define CERES_PUBLIC_INTERNAL_SPHERE_MANIFOLD_HELPERS_H_
#include "ceres/internal/householder_vector.h"
// This module contains functions to compute the SphereManifold plus and minus
// operator and their Jacobians.
//
// As the parameters to these functions are shared between them, they are
// described here: The following variable names are used:
// Plus(x, delta) = x + delta = x_plus_delta,
// Minus(y, x) = y - x = y_minus_x.
//
// The remaining ones are v and beta which describe the Householder
// transformation of x, and norm_delta which is the norm of delta.
//
// The types of x, y, x_plus_delta and y_minus_x need to be equivalent to
// Eigen::Matrix<double, AmbientSpaceDimension, 1> and the type of delta needs
// to be equivalent to Eigen::Matrix<double, TangentSpaceDimension, 1>.
//
// The type of Jacobian plus needs to be equivalent to Eigen::Matrix<double,
// AmbientSpaceDimension, TangentSpaceDimension, Eigen::RowMajor> and for
// Jacobian minus Eigen::Matrix<double, TangentSpaceDimension,
// AmbientSpaceDimension, Eigen::RowMajor>.
//
// For all vector / matrix inputs and outputs, template parameters are
// used in order to allow also Eigen::Ref and Eigen block expressions to
// be passed to the function.
namespace ceres {
namespace internal {
template <typename VT, typename XT, typename DeltaT, typename XPlusDeltaT>
inline void ComputeSphereManifoldPlus(const VT& v,
double beta,
const XT& x,
const DeltaT& delta,
double norm_delta,
XPlusDeltaT* x_plus_delta) {
constexpr int AmbientDim = VT::RowsAtCompileTime;
// Map the delta from the minimum representation to the over parameterized
// homogeneous vector. See B.2 p.25 equation (106) - (107) for more details.
const double norm_delta_div_2 = 0.5 * norm_delta;
const double sin_delta_by_delta =
std::sin(norm_delta_div_2) / norm_delta_div_2;
Eigen::Matrix<double, AmbientDim, 1> y(v.size());
y << 0.5 * sin_delta_by_delta * delta, std::cos(norm_delta_div_2);
// Apply the delta update to remain on the sphere.
*x_plus_delta = x.norm() * ApplyHouseholderVector(y, v, beta);
}
template <typename VT, typename JacobianT>
inline void ComputeSphereManifoldPlusJacobian(const VT& x,
JacobianT* jacobian) {
constexpr int AmbientSpaceDim = VT::RowsAtCompileTime;
using AmbientVector = Eigen::Matrix<double, AmbientSpaceDim, 1>;
const int ambient_size = x.size();
const int tangent_size = x.size() - 1;
AmbientVector v(ambient_size);
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.
ComputeHouseholderVector<VT, double, AmbientSpaceDim>(x, &v, &beta);
// The Jacobian is equal to J = 0.5 * H.leftCols(size_ - 1) where H is the
// Householder matrix (H = I - beta * v * v').
for (int i = 0; i < tangent_size; ++i) {
(*jacobian).col(i) = -0.5 * beta * v(i) * v;
(*jacobian)(i, i) += 0.5;
}
(*jacobian) *= x.norm();
}
template <typename VT, typename XT, typename YT, typename YMinusXT>
inline void ComputeSphereManifoldMinus(
const VT& v, double beta, const XT& x, const YT& y, YMinusXT* y_minus_x) {
constexpr int AmbientSpaceDim = VT::RowsAtCompileTime;
constexpr int TangentSpaceDim =
AmbientSpaceDim == Eigen::Dynamic ? Eigen::Dynamic : AmbientSpaceDim - 1;
using AmbientVector = Eigen::Matrix<double, AmbientSpaceDim, 1>;
const int tanget_size = v.size() - 1;
const AmbientVector hy = ApplyHouseholderVector(y, v, beta) / x.norm();
// Calculate y - x. See B.2 p.25 equation (108).
double y_last = hy[tanget_size];
double hy_norm = hy.template head<TangentSpaceDim>(tanget_size).norm();
if (hy_norm == 0.0) {
y_minus_x->setZero();
} else {
*y_minus_x = 2.0 * std::atan2(hy_norm, y_last) / hy_norm *
hy.template head<TangentSpaceDim>(tanget_size);
}
}
template <typename VT, typename JacobianT>
inline void ComputeSphereManifoldMinusJacobian(const VT& x,
JacobianT* jacobian) {
constexpr int AmbientSpaceDim = VT::RowsAtCompileTime;
using AmbientVector = Eigen::Matrix<double, AmbientSpaceDim, 1>;
const int ambient_size = x.size();
const int tangent_size = x.size() - 1;
AmbientVector v(ambient_size);
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.
ComputeHouseholderVector<VT, double, AmbientSpaceDim>(x, &v, &beta);
// The Jacobian is equal to J = 2.0 * H.leftCols(size_ - 1) where H is the
// Householder matrix (H = I - beta * v * v').
for (int i = 0; i < tangent_size; ++i) {
(*jacobian).row(i) = -2.0 * beta * v(i) * v;
(*jacobian)(i, i) += 2.0;
}
(*jacobian) /= x.norm();
}
} // namespace internal
} // namespace ceres
#endif
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