| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com> |
| // |
| // Eigen is free software; you can redistribute it and/or |
| // modify it under the terms of the GNU Lesser General Public |
| // License as published by the Free Software Foundation; either |
| // version 3 of the License, or (at your option) any later version. |
| // |
| // Alternatively, you can redistribute it and/or |
| // modify it under the terms of the GNU General Public License as |
| // published by the Free Software Foundation; either version 2 of |
| // the License, or (at your option) any later version. |
| // |
| // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY |
| // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS |
| // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the |
| // GNU General Public License for more details. |
| // |
| // You should have received a copy of the GNU Lesser General Public |
| // License and a copy of the GNU General Public License along with |
| // Eigen. If not, see <http://www.gnu.org/licenses/>. |
| |
| #ifndef EIGEN_UMEYAMA_H |
| #define EIGEN_UMEYAMA_H |
| |
| // This file requires the user to include |
| // * Eigen/Core |
| // * Eigen/LU |
| // * Eigen/SVD |
| // * Eigen/Array |
| |
| #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 |
| { |
| // 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 ei_umeyama_transform_matrix_type |
| { |
| enum { |
| MinRowsAtCompileTime = EIGEN_SIZE_MIN(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. Here EIGEN_ENUM_MIN is really what we want. |
| HomogeneousDimension = EIGEN_ENUM_MIN(MinRowsAtCompileTime+1, Dynamic) |
| }; |
| |
| typedef Matrix<typename ei_traits<MatrixType>::Scalar, |
| HomogeneousDimension, |
| HomogeneousDimension, |
| AutoAlign | (ei_traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor), |
| HomogeneousDimension, |
| HomogeneousDimension |
| > type; |
| }; |
| } |
| |
| #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 ei_umeyama_transform_matrix_type<Derived, OtherDerived>::type |
| umeyama(const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true) |
| { |
| typedef typename ei_umeyama_transform_matrix_type<Derived, OtherDerived>::type TransformationMatrixType; |
| typedef typename ei_traits<TransformationMatrixType>::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| |
| EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL) |
| EIGEN_STATIC_ASSERT((ei_is_same_type<Scalar, typename ei_traits<OtherDerived>::Scalar>::ret), |
| YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY) |
| |
| enum { Dimension = EIGEN_SIZE_MIN(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) }; |
| |
| typedef Matrix<Scalar, Dimension, 1> VectorType; |
| typedef Matrix<Scalar, Dimension, Dimension> MatrixType; |
| typedef typename ei_plain_matrix_type_row_major<Derived>::type RowMajorMatrixType; |
| |
| const int m = src.rows(); // dimension |
| const int 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(); |
| |
| SVD<MatrixType> svd(sigma); |
| |
| // 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(); |
| int rank = 0; for (int i=0; i<m; ++i) if (!ei_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; |
| } |
| |
| #endif // EIGEN_UMEYAMA_H |