ar_basalt/thirdparty/basalt-headers/include/basalt/camera/stereographic_param.hpp

/**
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https://gitlab.com/VladyslavUsenko/basalt-headers.git

Copyright (c) 2019, Vladyslav Usenko and Nikolaus Demmel.
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@file
@brief Implementation of stereographic parametrization for unit vectors
*/

#pragma once

#include <Eigen/Dense>

namespace basalt {

/// @brief Stereographic projection for minimal representation of unit vectors
///
/// \image html stereographic.png
/// The projection is defined on the entire
/// sphere, except at one point: the projection point. Where it is defined, the
/// mapping is smooth and bijective. The illustrations shows 3 examples of unit
/// vectors parametrized with a point in the 2D plane. See
/// \ref project and \ref unproject functions for more details.
template <typename Scalar_ = double>
class StereographicParam {
 public:
  using Scalar = Scalar_;
  using Vec2 = Eigen::Matrix<Scalar, 2, 1>;
  using Vec4 = Eigen::Matrix<Scalar, 4, 1>;

  using Mat24 = Eigen::Matrix<Scalar, 2, 4>;
  using Mat42 = Eigen::Matrix<Scalar, 4, 2>;

  /// @brief Project the point and optionally compute Jacobian
  ///
  /// Projection function is defined as follows:
  /// \f{align}{
  ///    \pi(\mathbf{x}) &=
  ///    \begin{bmatrix}
  ///   {\frac{x}{d + z}}
  ///    \\ {\frac{y}{d + z}}
  ///    \\ \end{bmatrix},
  ///    \\ d &= \sqrt{x^2 + y^2 + z^2}.
  /// \f}
  /// @param[in] p3d point to project [x,y,z]
  /// @param[out] d_r_d_p if not nullptr computed Jacobian of projection
  /// with respect to p3d
  /// @return 2D projection of the point which parametrizes the corresponding
  /// direction vector
  static inline Vec2 project(const Vec4& p3d, Mat24* d_r_d_p = nullptr) {
    const Scalar sqrt = p3d.template head<3>().norm();
    const Scalar norm = p3d[2] + sqrt;
    const Scalar norm_inv = Scalar(1) / norm;

    Vec2 res(p3d[0] * norm_inv, p3d[1] * norm_inv);

    if (d_r_d_p) {
      Scalar norm_inv2 = norm_inv * norm_inv;
      Scalar tmp = -norm_inv2 / sqrt;

      (*d_r_d_p).setZero();

      (*d_r_d_p)(0, 0) = norm_inv + p3d[0] * p3d[0] * tmp;
      (*d_r_d_p)(1, 0) = p3d[0] * p3d[1] * tmp;

      (*d_r_d_p)(1, 1) = norm_inv + p3d[1] * p3d[1] * tmp;
      (*d_r_d_p)(0, 1) = p3d[0] * p3d[1] * tmp;

      (*d_r_d_p)(0, 2) = p3d[0] * norm * tmp;
      (*d_r_d_p)(1, 2) = p3d[1] * norm * tmp;

      (*d_r_d_p)(0, 3) = Scalar(0);
      (*d_r_d_p)(1, 3) = Scalar(0);
    }

    return res;
  }

  /// @brief Unproject the point and optionally compute Jacobian
  ///
  /// The unprojection function is computed as follows: \f{align}{
  ///    \pi^{-1}(\mathbf{u}) &=
  ///    \frac{2}{1 + x^2 + y^2}
  ///    \begin{bmatrix}
  ///    u \\ v \\ 1
  ///    \\ \end{bmatrix}-\begin{bmatrix}
  ///    0 \\ 0 \\ 1
  ///    \\ \end{bmatrix}.
  /// \f}
  ///
  /// @param[in] proj point to unproject [u,v]
  /// @param[out] d_r_d_p if not nullptr computed Jacobian of unprojection
  /// with respect to proj
  /// @return unprojected 3D unit vector
  static inline Vec4 unproject(const Vec2& proj, Mat42* d_r_d_p = nullptr) {
    const Scalar x2 = proj[0] * proj[0];
    const Scalar y2 = proj[1] * proj[1];
    const Scalar r2 = x2 + y2;

    const Scalar norm_inv = Scalar(2) / (Scalar(1) + r2);

    Vec4 res(proj[0] * norm_inv, proj[1] * norm_inv, norm_inv - Scalar(1),
             Scalar(0));

    if (d_r_d_p) {
      const Scalar norm_inv2 = norm_inv * norm_inv;
      const Scalar xy = proj[0] * proj[1];

      (*d_r_d_p)(0, 0) = (norm_inv - x2 * norm_inv2);
      (*d_r_d_p)(0, 1) = -xy * norm_inv2;

      (*d_r_d_p)(1, 0) = -xy * norm_inv2;
      (*d_r_d_p)(1, 1) = (norm_inv - y2 * norm_inv2);

      (*d_r_d_p)(2, 0) = -proj[0] * norm_inv2;
      (*d_r_d_p)(2, 1) = -proj[1] * norm_inv2;

      (*d_r_d_p)(3, 0) = Scalar(0);
      (*d_r_d_p)(3, 1) = Scalar(0);
    }

    return res;
  }
};

}  // namespace basalt