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      1 // This file is part of Eigen, a lightweight C++ template library
      2 // for linear algebra.
      3 //
      4 // Copyright (C) 2009 Hauke Heibel <hauke.heibel (at) gmail.com>
      5 //
      6 // This Source Code Form is subject to the terms of the Mozilla
      7 // Public License v. 2.0. If a copy of the MPL was not distributed
      8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
      9 
     10 #ifndef EIGEN_UMEYAMA_H
     11 #define EIGEN_UMEYAMA_H
     12 
     13 // This file requires the user to include
     14 // * Eigen/Core
     15 // * Eigen/LU
     16 // * Eigen/SVD
     17 // * Eigen/Array
     18 
     19 namespace Eigen {
     20 
     21 #ifndef EIGEN_PARSED_BY_DOXYGEN
     22 
     23 // These helpers are required since it allows to use mixed types as parameters
     24 // for the Umeyama. The problem with mixed parameters is that the return type
     25 // cannot trivially be deduced when float and double types are mixed.
     26 namespace internal {
     27 
     28 // Compile time return type deduction for different MatrixBase types.
     29 // Different means here different alignment and parameters but the same underlying
     30 // real scalar type.
     31 template<typename MatrixType, typename OtherMatrixType>
     32 struct umeyama_transform_matrix_type
     33 {
     34   enum {
     35     MinRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),
     36 
     37     // When possible we want to choose some small fixed size value since the result
     38     // is likely to fit on the stack. So here, EIGEN_SIZE_MIN_PREFER_DYNAMIC is not what we want.
     39     HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic ? Dynamic : int(MinRowsAtCompileTime)+1
     40   };
     41 
     42   typedef Matrix<typename traits<MatrixType>::Scalar,
     43     HomogeneousDimension,
     44     HomogeneousDimension,
     45     AutoAlign | (traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor),
     46     HomogeneousDimension,
     47     HomogeneousDimension
     48   > type;
     49 };
     50 
     51 }
     52 
     53 #endif
     54 
     55 /**
     56 * \geometry_module \ingroup Geometry_Module
     57 *
     58 * \brief Returns the transformation between two point sets.
     59 *
     60 * The algorithm is based on:
     61 * "Least-squares estimation of transformation parameters between two point patterns",
     62 * Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
     63 *
     64 * It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such that
     65 * \f{align*}
     66 *   \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2
     67 * \f}
     68 * is minimized.
     69 *
     70 * The algorithm is based on the analysis of the covariance matrix
     71 * \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
     72 * of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where
     73 * \f$d\f$ is corresponding to the dimension (which is typically small).
     74 * The analysis is involving the SVD having a complexity of \f$O(d^3)\f$
     75 * though the actual computational effort lies in the covariance
     76 * matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when
     77 * the input point sets have dimension \f$d \times m\f$.
     78 *
     79 * Currently the method is working only for floating point matrices.
     80 *
     81 * \todo Should the return type of umeyama() become a Transform?
     82 *
     83 * \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right) \f$.
     84 * \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n \right) \f$.
     85 * \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is passed.
     86 * \return The homogeneous transformation
     87 * \f{align*}
     88 *   T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix}
     89 * \f}
     90 * minimizing the resudiual above. This transformation is always returned as an
     91 * Eigen::Matrix.
     92 */
     93 template <typename Derived, typename OtherDerived>
     94 typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
     95 umeyama(const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true)
     96 {
     97   typedef typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type TransformationMatrixType;
     98   typedef typename internal::traits<TransformationMatrixType>::Scalar Scalar;
     99   typedef typename NumTraits<Scalar>::Real RealScalar;
    100 
    101   EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
    102   EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
    103     YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
    104 
    105   enum { Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
    106 
    107   typedef Matrix<Scalar, Dimension, 1> VectorType;
    108   typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
    109   typedef typename internal::plain_matrix_type_row_major<Derived>::type RowMajorMatrixType;
    110 
    111   const Index m = src.rows(); // dimension
    112   const Index n = src.cols(); // number of measurements
    113 
    114   // required for demeaning ...
    115   const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);
    116 
    117   // computation of mean
    118   const VectorType src_mean = src.rowwise().sum() * one_over_n;
    119   const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
    120 
    121   // demeaning of src and dst points
    122   const RowMajorMatrixType src_demean = src.colwise() - src_mean;
    123   const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
    124 
    125   // Eq. (36)-(37)
    126   const Scalar src_var = src_demean.rowwise().squaredNorm().sum() * one_over_n;
    127 
    128   // Eq. (38)
    129   const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
    130 
    131   JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
    132 
    133   // Initialize the resulting transformation with an identity matrix...
    134   TransformationMatrixType Rt = TransformationMatrixType::Identity(m+1,m+1);
    135 
    136   // Eq. (39)
    137   VectorType S = VectorType::Ones(m);
    138 
    139   if  ( svd.matrixU().determinant() * svd.matrixV().determinant() < 0 )
    140     S(m-1) = -1;
    141 
    142   // Eq. (40) and (43)
    143   Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
    144 
    145   if (with_scaling)
    146   {
    147     // Eq. (42)
    148     const Scalar c = Scalar(1)/src_var * svd.singularValues().dot(S);
    149 
    150     // Eq. (41)
    151     Rt.col(m).head(m) = dst_mean;
    152     Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
    153     Rt.block(0,0,m,m) *= c;
    154   }
    155   else
    156   {
    157     Rt.col(m).head(m) = dst_mean;
    158     Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
    159   }
    160 
    161   return Rt;
    162 }
    163 
    164 } // end namespace Eigen
    165 
    166 #endif // EIGEN_UMEYAMA_H
    167