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- // 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;
- };
- }
- #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 residual 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;
- 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 = RealScalar(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 ( svd.matrixU().determinant() * svd.matrixV().determinant() < 0 )
- S(m-1) = -1;
- // Eq. (40) and (43)
- Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
- if (with_scaling)
- {
- // Eq. (42)
- const Scalar c = Scalar(1)/src_var * svd.singularValues().dot(S);
- // Eq. (41)
- Rt.col(m).head(m) = dst_mean;
- Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
- Rt.block(0,0,m,m) *= c;
- }
- else
- {
- Rt.col(m).head(m) = dst_mean;
- Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
- }
- return Rt;
- }
- } // end namespace Eigen
- #endif // EIGEN_UMEYAMA_H
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