Program Listing for File rxso3.hpp

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#pragma once

#include "so3.hpp"

namespace Sophus {
template <class Scalar_, int Options = 0>
class RxSO3;
using RxSO3d = RxSO3<double>;
using RxSO3f = RxSO3<float>;
}  // namespace Sophus

namespace Eigen {
namespace internal {

template <class Scalar_, int Options_>
struct traits<Sophus::RxSO3<Scalar_, Options_>> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using QuaternionType = Eigen::Quaternion<Scalar, Options>;
};

template <class Scalar_, int Options_>
struct traits<Map<Sophus::RxSO3<Scalar_>, Options_>>
    : traits<Sophus::RxSO3<Scalar_, Options_>> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using QuaternionType = Map<Eigen::Quaternion<Scalar>, Options>;
};

template <class Scalar_, int Options_>
struct traits<Map<Sophus::RxSO3<Scalar_> const, Options_>>
    : traits<Sophus::RxSO3<Scalar_, Options_> const> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using QuaternionType = Map<Eigen::Quaternion<Scalar> const, Options>;
};
}  // namespace internal
}  // namespace Eigen

namespace Sophus {

template <class Derived>
class RxSO3Base {
 public:
  static constexpr int Options = Eigen::internal::traits<Derived>::Options;
  using Scalar = typename Eigen::internal::traits<Derived>::Scalar;
  using QuaternionType =
      typename Eigen::internal::traits<Derived>::QuaternionType;
  using QuaternionTemporaryType = Eigen::Quaternion<Scalar, Options>;

  static int constexpr DoF = 4;
  static int constexpr num_parameters = 4;
  static int constexpr N = 3;
  static int constexpr Dim = 3;
  using Transformation = Matrix<Scalar, N, N>;
  using Point = Vector3<Scalar>;
  using HomogeneousPoint = Vector4<Scalar>;
  using Line = ParametrizedLine3<Scalar>;
  using Hyperplane = Hyperplane3<Scalar>;
  using Tangent = Vector<Scalar, DoF>;
  using Adjoint = Matrix<Scalar, DoF, DoF>;

  struct TangentAndTheta {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    Tangent tangent;
    Scalar theta;
  };

  template <typename OtherDerived>
  using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<
      Scalar, typename OtherDerived::Scalar>::ReturnType;

  template <typename OtherDerived>
  using RxSO3Product = RxSO3<ReturnScalar<OtherDerived>>;

  template <typename PointDerived>
  using PointProduct = Vector3<ReturnScalar<PointDerived>>;

  template <typename HPointDerived>
  using HomogeneousPointProduct = Vector4<ReturnScalar<HPointDerived>>;

  SOPHUS_FUNC Adjoint Adj() const {
    Adjoint res;
    res.setIdentity();
    res.template topLeftCorner<3, 3>() = rotationMatrix();
    return res;
  }

  template <class NewScalarType>
  SOPHUS_FUNC RxSO3<NewScalarType> cast() const {
    return RxSO3<NewScalarType>(quaternion().template cast<NewScalarType>());
  }

  SOPHUS_FUNC Scalar* data() { return quaternion_nonconst().coeffs().data(); }

  SOPHUS_FUNC Scalar const* data() const {
    return quaternion().coeffs().data();
  }

  SOPHUS_FUNC RxSO3<Scalar> inverse() const {
    return RxSO3<Scalar>(quaternion().inverse());
  }

  SOPHUS_FUNC Tangent log() const { return logAndTheta().tangent; }

  SOPHUS_FUNC TangentAndTheta logAndTheta() const {
    using std::log;

    Scalar scale = quaternion().squaredNorm();
    TangentAndTheta result;
    result.tangent[3] = log(scale);
    auto omega_and_theta = SO3<Scalar>(quaternion()).logAndTheta();
    result.tangent.template head<3>() = omega_and_theta.tangent;
    result.theta = omega_and_theta.theta;
    return result;
  }
  SOPHUS_FUNC Transformation matrix() const {
    Transformation sR;

    Scalar const vx_sq = quaternion().vec().x() * quaternion().vec().x();
    Scalar const vy_sq = quaternion().vec().y() * quaternion().vec().y();
    Scalar const vz_sq = quaternion().vec().z() * quaternion().vec().z();
    Scalar const w_sq = quaternion().w() * quaternion().w();
    Scalar const two_vx = Scalar(2) * quaternion().vec().x();
    Scalar const two_vy = Scalar(2) * quaternion().vec().y();
    Scalar const two_vz = Scalar(2) * quaternion().vec().z();
    Scalar const two_vx_vy = two_vx * quaternion().vec().y();
    Scalar const two_vx_vz = two_vx * quaternion().vec().z();
    Scalar const two_vx_w = two_vx * quaternion().w();
    Scalar const two_vy_vz = two_vy * quaternion().vec().z();
    Scalar const two_vy_w = two_vy * quaternion().w();
    Scalar const two_vz_w = two_vz * quaternion().w();

    sR(0, 0) = vx_sq - vy_sq - vz_sq + w_sq;
    sR(1, 0) = two_vx_vy + two_vz_w;
    sR(2, 0) = two_vx_vz - two_vy_w;

    sR(0, 1) = two_vx_vy - two_vz_w;
    sR(1, 1) = -vx_sq + vy_sq - vz_sq + w_sq;
    sR(2, 1) = two_vx_w + two_vy_vz;

    sR(0, 2) = two_vx_vz + two_vy_w;
    sR(1, 2) = -two_vx_w + two_vy_vz;
    sR(2, 2) = -vx_sq - vy_sq + vz_sq + w_sq;
    return sR;
  }

  template <class OtherDerived>
  SOPHUS_FUNC RxSO3Base<Derived>& operator=(
      RxSO3Base<OtherDerived> const& other) {
    quaternion_nonconst() = other.quaternion();
    return *this;
  }

  template <typename OtherDerived>
  SOPHUS_FUNC RxSO3Product<OtherDerived> operator*(
      RxSO3Base<OtherDerived> const& other) const {
    using std::sqrt;
    using ResultT = ReturnScalar<OtherDerived>;
    using QuaternionProductType =
        typename RxSO3Product<OtherDerived>::QuaternionType;

    QuaternionProductType result_quaternion(
        Sophus::SO3<double>::QuaternionProduct<QuaternionProductType>(
            quaternion(), other.quaternion()));

    ResultT scale = result_quaternion.squaredNorm();
    if (scale < Constants<ResultT>::epsilon()) {
      SOPHUS_ENSURE(scale > ResultT(0), "Scale must be greater zero.");
      result_quaternion.normalize();
      result_quaternion.coeffs() *= sqrt(Constants<ResultT>::epsilonPlus());
    }
    if (scale > ResultT(1.) / Constants<ResultT>::epsilon()) {
      result_quaternion.normalize();
      result_quaternion.coeffs() /= sqrt(Constants<ResultT>::epsilonPlus());
    }
    return RxSO3Product<OtherDerived>(result_quaternion);
  }

  template <typename PointDerived,
            typename = typename std::enable_if<
                IsFixedSizeVector<PointDerived, 3>::value>::type>
  SOPHUS_FUNC PointProduct<PointDerived> operator*(
      Eigen::MatrixBase<PointDerived> const& p) const {
    // Follows http:///eigen.tuxfamily.org/bz/show_bug.cgi?id=459
    Scalar scale = quaternion().squaredNorm();
    PointProduct<PointDerived> two_vec_cross_p = quaternion().vec().cross(p);
    two_vec_cross_p += two_vec_cross_p;
    return scale * p + (quaternion().w() * two_vec_cross_p +
                        quaternion().vec().cross(two_vec_cross_p));
  }

  template <typename HPointDerived,
            typename = typename std::enable_if<
                IsFixedSizeVector<HPointDerived, 4>::value>::type>
  SOPHUS_FUNC HomogeneousPointProduct<HPointDerived> operator*(
      Eigen::MatrixBase<HPointDerived> const& p) const {
    const auto rsp = *this * p.template head<3>();
    return HomogeneousPointProduct<HPointDerived>(rsp(0), rsp(1), rsp(2), p(3));
  }

  SOPHUS_FUNC Line operator*(Line const& l) const {
    return Line((*this) * l.origin(),
                (*this) * l.direction() / quaternion().squaredNorm());
  }

  SOPHUS_FUNC Hyperplane operator*(Hyperplane const& p) const {
    const auto this_scale = scale();
    return Hyperplane((*this) * p.normal() / this_scale,
                      this_scale * p.offset());
  }

  template <typename OtherDerived,
            typename = typename std::enable_if<
                std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type>
  SOPHUS_FUNC RxSO3Base<Derived>& operator*=(
      RxSO3Base<OtherDerived> const& other) {
    *static_cast<Derived*>(this) = *this * other;
    return *this;
  }

  SOPHUS_FUNC Sophus::Vector<Scalar, num_parameters> params() const {
    return quaternion().coeffs();
  }

  SOPHUS_FUNC void setQuaternion(Eigen::Quaternion<Scalar> const& quat) {
    SOPHUS_ENSURE(quat.squaredNorm() > Constants<Scalar>::epsilon() *
                                           Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(
        quat.squaredNorm() < Scalar(1.) / (Constants<Scalar>::epsilon() *
                                           Constants<Scalar>::epsilon()),
        "Inverse scale factor must be greater-equal epsilon.");
    static_cast<Derived*>(this)->quaternion_nonconst() = quat;
  }

  SOPHUS_FUNC QuaternionType const& quaternion() const {
    return static_cast<Derived const*>(this)->quaternion();
  }

  SOPHUS_FUNC Transformation rotationMatrix() const {
    QuaternionTemporaryType norm_quad = quaternion();
    norm_quad.normalize();
    return norm_quad.toRotationMatrix();
  }

  SOPHUS_FUNC
  Scalar scale() const { return quaternion().squaredNorm(); }

  SOPHUS_FUNC void setRotationMatrix(Transformation const& R) {
    using std::sqrt;
    Scalar saved_scale = scale();
    quaternion_nonconst() = R;
    quaternion_nonconst().coeffs() *= sqrt(saved_scale);
  }

  SOPHUS_FUNC
  void setScale(Scalar const& scale) {
    using std::sqrt;
    quaternion_nonconst().normalize();
    quaternion_nonconst().coeffs() *= sqrt(scale);
  }

  SOPHUS_FUNC void setScaledRotationMatrix(Transformation const& sR) {
    Transformation squared_sR = sR * sR.transpose();
    Scalar squared_scale =
        Scalar(1. / 3.) *
        (squared_sR(0, 0) + squared_sR(1, 1) + squared_sR(2, 2));
    SOPHUS_ENSURE(squared_scale >= Constants<Scalar>::epsilon() *
                                       Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(squared_scale < Scalar(1.) / (Constants<Scalar>::epsilon() *
                                                Constants<Scalar>::epsilon()),
                  "Inverse scale factor must be greater-equal epsilon.");
    Scalar scale = sqrt(squared_scale);
    quaternion_nonconst() = sR / scale;
    quaternion_nonconst().coeffs() *= sqrt(scale);
  }

  SOPHUS_FUNC void setSO3(SO3<Scalar> const& so3) {
    using std::sqrt;
    Scalar saved_scale = scale();
    quaternion_nonconst() = so3.unit_quaternion();
    quaternion_nonconst().coeffs() *= sqrt(saved_scale);
  }

  SOPHUS_FUNC SO3<Scalar> so3() const { return SO3<Scalar>(quaternion()); }

  SOPHUS_FUNC Matrix<Scalar, num_parameters, DoF> Dx_this_mul_exp_x_at_0()
      const {
    Matrix<Scalar, num_parameters, DoF> J;
    Eigen::Quaternion<Scalar> const q = quaternion();
    J.col(3) = q.coeffs() * Scalar(0.5);
    Scalar const c0 = Scalar(0.5) * q.w();
    Scalar const c1 = Scalar(0.5) * q.z();
    Scalar const c2 = -c1;
    Scalar const c3 = Scalar(0.5) * q.y();
    Scalar const c4 = Scalar(0.5) * q.x();
    Scalar const c5 = -c4;
    Scalar const c6 = -c3;
    J(0, 0) = c0;
    J(0, 1) = c2;
    J(0, 2) = c3;
    J(1, 0) = c1;
    J(1, 1) = c0;
    J(1, 2) = c5;
    J(2, 0) = c6;
    J(2, 1) = c4;
    J(2, 2) = c0;
    J(3, 0) = c5;
    J(3, 1) = c6;
    J(3, 2) = c2;
    return J;
  }

  SOPHUS_FUNC Matrix<Scalar, DoF, num_parameters> Dx_log_this_inv_by_x_at_this()
      const {
    auto& q = quaternion();
    Matrix<Scalar, DoF, num_parameters> J;
    // clang-format off
    J << q.w(),  q.z(), -q.y(), -q.x(),
        -q.z(),  q.w(),  q.x(), -q.y(),
         q.y(), -q.x(),  q.w(), -q.z(),
         q.x(),  q.y(),  q.z(),  q.w();
    // clang-format on
    const Scalar scaler = Scalar(2.) / q.squaredNorm();
    return J * scaler;
  }

 private:
  SOPHUS_FUNC QuaternionType& quaternion_nonconst() {
    return static_cast<Derived*>(this)->quaternion_nonconst();
  }
};

template <class Scalar_, int Options>
class RxSO3 : public RxSO3Base<RxSO3<Scalar_, Options>> {
 public:
  using Base = RxSO3Base<RxSO3<Scalar_, Options>>;
  static int constexpr DoF = Base::DoF;
  static int constexpr num_parameters = Base::num_parameters;

  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;
  using QuaternionMember = Eigen::Quaternion<Scalar, Options>;

  friend class RxSO3Base<RxSO3<Scalar_, Options>>;

  using Base::operator=;

  SOPHUS_FUNC RxSO3& operator=(RxSO3 const& other) = default;

  EIGEN_MAKE_ALIGNED_OPERATOR_NEW

  SOPHUS_FUNC RxSO3()
      : quaternion_(Scalar(1), Scalar(0), Scalar(0), Scalar(0)) {}

  SOPHUS_FUNC RxSO3(RxSO3 const& other) = default;

  template <class OtherDerived>
  SOPHUS_FUNC RxSO3(RxSO3Base<OtherDerived> const& other)
      : quaternion_(other.quaternion()) {}

  SOPHUS_FUNC explicit RxSO3(Transformation const& sR) {
    this->setScaledRotationMatrix(sR);
  }

  SOPHUS_FUNC RxSO3(Scalar const& scale, Transformation const& R)
      : quaternion_(R) {
    SOPHUS_ENSURE(scale >= Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(scale < Scalar(1.) / Constants<Scalar>::epsilon(),
                  "Inverse scale factor must be greater-equal epsilon.");
    using std::sqrt;
    quaternion_.coeffs() *= sqrt(scale);
  }

  SOPHUS_FUNC RxSO3(Scalar const& scale, SO3<Scalar> const& so3)
      : quaternion_(so3.unit_quaternion()) {
    SOPHUS_ENSURE(scale >= Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(scale < Scalar(1.) / Constants<Scalar>::epsilon(),
                  "Inverse scale factor must be greater-equal epsilon.");
    using std::sqrt;
    quaternion_.coeffs() *= sqrt(scale);
  }

  template <class D>
  SOPHUS_FUNC explicit RxSO3(Eigen::QuaternionBase<D> const& quat)
      : quaternion_(quat) {
    static_assert(std::is_same<typename D::Scalar, Scalar>::value,
                  "must be same Scalar type.");
    SOPHUS_ENSURE(quaternion_.squaredNorm() >= Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(
        quat.squaredNorm() < Scalar(1.) / Constants<Scalar>::epsilon(),
        "Inverse scale factor must be greater-equal epsilon.");
  }

  template <class D>
  SOPHUS_FUNC explicit RxSO3(Scalar const& scale,
                             Eigen::QuaternionBase<D> const& unit_quat)
      : RxSO3(scale, SO3<Scalar>(unit_quat)) {}

  SOPHUS_FUNC QuaternionMember const& quaternion() const { return quaternion_; }

  SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF>
  Dx_exp_x_at_0() {
    static Scalar const h(0.5);
    return h * Sophus::Matrix<Scalar, num_parameters, DoF>::Identity();
  }

  SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF> Dx_exp_x(
      const Tangent& a) {
    using std::exp;
    using std::sqrt;
    Sophus::Matrix<Scalar, num_parameters, DoF> J;
    Vector3<Scalar> const omega = a.template head<3>();
    Scalar const sigma = a[3];
    Eigen::Quaternion<Scalar> quat = SO3<Scalar>::exp(omega).unit_quaternion();
    Scalar const scale = sqrt(exp(sigma));
    Scalar const scale_half = scale * Scalar(0.5);

    J.template block<4, 3>(0, 0) = SO3<Scalar>::Dx_exp_x(omega) * scale;
    J.col(3) = quat.coeffs() * scale_half;
    return J;
  }

  SOPHUS_FUNC static Sophus::Matrix<Scalar, 3, DoF> Dx_exp_x_times_point_at_0(
      Point const& point) {
    Sophus::Matrix<Scalar, 3, DoF> j;
    j << Sophus::SO3<Scalar>::hat(-point), point;
    return j;
  }

  SOPHUS_FUNC static Transformation Dxi_exp_x_matrix_at_0(int i) {
    return generator(i);
  }
  SOPHUS_FUNC static RxSO3<Scalar> exp(Tangent const& a) {
    Scalar theta;
    return expAndTheta(a, &theta);
  }

  SOPHUS_FUNC static RxSO3<Scalar> expAndTheta(Tangent const& a,
                                               Scalar* theta) {
    SOPHUS_ENSURE(theta != nullptr, "must not be nullptr.");
    using std::exp;
    using std::max;
    using std::min;
    using std::sqrt;

    Vector3<Scalar> const omega = a.template head<3>();
    Scalar sigma = a[3];
    Scalar scale = exp(sigma);
    // Ensure that scale-factor constraint is always valid
    scale = max(scale, Constants<Scalar>::epsilonPlus());
    scale = min(scale, Scalar(1.) / Constants<Scalar>::epsilonPlus());
    Scalar sqrt_scale = sqrt(scale);
    Eigen::Quaternion<Scalar> quat =
        SO3<Scalar>::expAndTheta(omega, theta).unit_quaternion();
    quat.coeffs() *= sqrt_scale;
    return RxSO3<Scalar>(quat);
  }

  SOPHUS_FUNC static Transformation generator(int i) {
    SOPHUS_ENSURE(i >= 0 && i <= 3, "i should be in range [0,3].");
    Tangent e;
    e.setZero();
    e[i] = Scalar(1);
    return hat(e);
  }

  SOPHUS_FUNC static Transformation hat(Tangent const& a) {
    Transformation A;
    // clang-format off
    A <<  a(3), -a(2),  a(1),
          a(2),  a(3), -a(0),
         -a(1),  a(0),  a(3);
    // clang-format on
    return A;
  }

  SOPHUS_FUNC static Tangent lieBracket(Tangent const& a, Tangent const& b) {
    Vector3<Scalar> const omega1 = a.template head<3>();
    Vector3<Scalar> const omega2 = b.template head<3>();
    Vector4<Scalar> res;
    res.template head<3>() = omega1.cross(omega2);
    res[3] = Scalar(0);
    return res;
  }

  template <class UniformRandomBitGenerator>
  static RxSO3 sampleUniform(UniformRandomBitGenerator& generator) {
    std::uniform_real_distribution<Scalar> uniform(Scalar(-1), Scalar(1));
    using std::exp2;
    return RxSO3(exp2(uniform(generator)),
                 SO3<Scalar>::sampleUniform(generator));
  }

  SOPHUS_FUNC static Tangent vee(Transformation const& Omega) {
    using std::abs;
    return Tangent(Omega(2, 1), Omega(0, 2), Omega(1, 0), Omega(0, 0));
  }

 protected:
  SOPHUS_FUNC QuaternionMember& quaternion_nonconst() { return quaternion_; }

  QuaternionMember quaternion_;
};

}  // namespace Sophus

namespace Eigen {

template <class Scalar_, int Options>
class Map<Sophus::RxSO3<Scalar_>, Options>
    : public Sophus::RxSO3Base<Map<Sophus::RxSO3<Scalar_>, Options>> {
 public:
  using Base = Sophus::RxSO3Base<Map<Sophus::RxSO3<Scalar_>, Options>>;
  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;

  friend class Sophus::RxSO3Base<Map<Sophus::RxSO3<Scalar_>, Options>>;

  using Base::operator=;
  using Base::operator*=;
  using Base::operator*;

  SOPHUS_FUNC explicit Map(Scalar* coeffs) : quaternion_(coeffs) {}

  SOPHUS_FUNC
  Map<Eigen::Quaternion<Scalar>, Options> const& quaternion() const {
    return quaternion_;
  }

 protected:
  SOPHUS_FUNC Map<Eigen::Quaternion<Scalar>, Options>& quaternion_nonconst() {
    return quaternion_;
  }

  Map<Eigen::Quaternion<Scalar>, Options> quaternion_;
};

template <class Scalar_, int Options>
class Map<Sophus::RxSO3<Scalar_> const, Options>
    : public Sophus::RxSO3Base<Map<Sophus::RxSO3<Scalar_> const, Options>> {
 public:
  using Base = Sophus::RxSO3Base<Map<Sophus::RxSO3<Scalar_> const, Options>>;
  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;

  using Base::operator*=;
  using Base::operator*;

  SOPHUS_FUNC
  explicit Map(Scalar const* coeffs) : quaternion_(coeffs) {}

  SOPHUS_FUNC
  Map<Eigen::Quaternion<Scalar> const, Options> const& quaternion() const {
    return quaternion_;
  }

 protected:
  Map<Eigen::Quaternion<Scalar> const, Options> const quaternion_;
};
}  // namespace Eigen