text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #ifndef EIGEN_UMEYAMA_H
  #define EIGEN_UMEYAMA_H
  
  // This file requires the user to include
  // * Eigen/Core
  // * Eigen/LU
  // * Eigen/SVD
  // * Eigen/Array
  
  namespace Eigen {
  
  #ifndef EIGEN_PARSED_BY_DOXYGEN
  
  // These helpers are required since it allows to use mixed types as parameters
  // for the Umeyama. The problem with mixed parameters is that the return type
  // cannot trivially be deduced when float and double types are mixed.
  namespace internal {
  
  // Compile time return type deduction for different MatrixBase types.
  // Different means here different alignment and parameters but the same
  // underlying real scalar type.
  template <typename MatrixType, typename OtherMatrixType>
  struct umeyama_transform_matrix_type {
      enum {
          MinRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(
              MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),
  
          // When possible we want to choose some small fixed size value since the
          // result is likely to fit on the stack. So here,
          // EIGEN_SIZE_MIN_PREFER_DYNAMIC is not what we want.
          HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic
                                     ? Dynamic
                                     : int(MinRowsAtCompileTime) + 1
      };
  
      typedef Matrix<typename traits<MatrixType>::Scalar, HomogeneousDimension,
                     HomogeneousDimension,
                     AutoAlign |
                         (traits<MatrixType>::Flags & RowMajorBit ? RowMajor
                                                                  : ColMajor),
                     HomogeneousDimension, HomogeneousDimension>
          type;
  };
  
  } // namespace internal
  
  #endif
  
  /**
   * \geometry_module \ingroup Geometry_Module
   *
   * \brief Returns the transformation between two point sets.
   *
   * The algorithm is based on:
   * "Least-squares estimation of transformation parameters between two point
   * patterns", Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
   *
   * It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such
   * that \f{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i +
   * \mathbf{t}) \vert\vert_2^2 \f} is minimized.
   *
   * The algorithm is based on the analysis of the covariance matrix
   * \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
   * of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where
   * \f$d\f$ is corresponding to the dimension (which is typically small).
   * The analysis is involving the SVD having a complexity of \f$O(d^3)\f$
   * though the actual computational effort lies in the covariance
   * matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when
   * the input point sets have dimension \f$d \times m\f$.
   *
   * Currently the method is working only for floating point matrices.
   *
   * \todo Should the return type of umeyama() become a Transform?
   *
   * \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right)
   * \f$. \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n
   * \right) \f$. \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is
   * passed. \return The homogeneous transformation \f{align*} T = \begin{bmatrix}
   * c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \f} minimizing the
   * resudiual above. This transformation is always returned as an Eigen::Matrix.
   */
  template <typename Derived, typename OtherDerived>
  typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
  umeyama(const MatrixBase<Derived> &src, const MatrixBase<OtherDerived> &dst,
          bool with_scaling = true) {
      typedef typename internal::umeyama_transform_matrix_type<
          Derived, OtherDerived>::type TransformationMatrixType;
      typedef typename internal::traits<TransformationMatrixType>::Scalar Scalar;
      typedef typename NumTraits<Scalar>::Real RealScalar;
      typedef typename Derived::Index Index;
  
      EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex,
                          NUMERIC_TYPE_MUST_BE_REAL)
      EIGEN_STATIC_ASSERT(
          (internal::is_same<
              Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
          YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
  
      enum {
          Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(
              Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime)
      };
  
      typedef Matrix<Scalar, Dimension, 1> VectorType;
      typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
      typedef typename internal::plain_matrix_type_row_major<Derived>::type
          RowMajorMatrixType;
  
      const Index m = src.rows(); // dimension
      const Index n = src.cols(); // number of measurements
  
      // required for demeaning ...
      const RealScalar one_over_n = 1 / static_cast<RealScalar>(n);
  
      // computation of mean
      const VectorType src_mean = src.rowwise().sum() * one_over_n;
      const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
  
      // demeaning of src and dst points
      const RowMajorMatrixType src_demean = src.colwise() - src_mean;
      const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
  
      // Eq. (36)-(37)
      const Scalar src_var =
          src_demean.rowwise().squaredNorm().sum() * one_over_n;
  
      // Eq. (38)
      const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
  
      JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
  
      // Initialize the resulting transformation with an identity matrix...
      TransformationMatrixType Rt =
          TransformationMatrixType::Identity(m + 1, m + 1);
  
      // Eq. (39)
      VectorType S = VectorType::Ones(m);
      if (sigma.determinant() < 0)
          S(m - 1) = -1;
  
      // Eq. (40) and (43)
      const VectorType &d = svd.singularValues();
      Index rank = 0;
      for (Index i = 0; i < m; ++i)
          if (!internal::isMuchSmallerThan(d.coeff(i), d.coeff(0)))
              ++rank;
      if (rank == m - 1) {
          if (svd.matrixU().determinant() * svd.matrixV().determinant() > 0) {
              Rt.block(0, 0, m, m).noalias() =
                  svd.matrixU() * svd.matrixV().transpose();
          } else {
              const Scalar s = S(m - 1);
              S(m - 1) = -1;
              Rt.block(0, 0, m, m).noalias() =
                  svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
              S(m - 1) = s;
          }
      } else {
          Rt.block(0, 0, m, m).noalias() =
              svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
      }
  
      // Eq. (42)
      const Scalar c = 1 / src_var * svd.singularValues().dot(S);
  
      // Eq. (41)
      // Note that we first assign dst_mean to the destination so that there no
      // need for a temporary.
      Rt.col(m).head(m) = dst_mean;
      Rt.col(m).head(m).noalias() -= c * Rt.topLeftCorner(m, m) * src_mean;
  
      if (with_scaling)
          Rt.block(0, 0, m, m) *= c;
  
      return Rt;
  }
  
  } // end namespace Eigen
  
  #endif // EIGEN_UMEYAMA_H
  