[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
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| 2 | // for linear algebra.
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| 3 | //
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| 4 | // Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
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| 5 | //
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| 6 | // This Source Code Form is subject to the terms of the Mozilla
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| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 9 |
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| 10 | #ifndef EIGEN_UMEYAMA_H
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| 11 | #define EIGEN_UMEYAMA_H
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| 12 |
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| 13 | // This file requires the user to include
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| 14 | // * Eigen/Core
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| 15 | // * Eigen/LU
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| 16 | // * Eigen/SVD
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| 17 | // * Eigen/Array
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| 18 |
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| 19 | namespace Eigen {
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| 20 |
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| 21 | #ifndef EIGEN_PARSED_BY_DOXYGEN
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| 22 |
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| 23 | // These helpers are required since it allows to use mixed types as parameters
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| 24 | // for the Umeyama. The problem with mixed parameters is that the return type
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| 25 | // cannot trivially be deduced when float and double types are mixed.
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| 26 | namespace internal {
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| 27 |
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| 28 | // Compile time return type deduction for different MatrixBase types.
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| 29 | // Different means here different alignment and parameters but the same underlying
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| 30 | // real scalar type.
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| 31 | template<typename MatrixType, typename OtherMatrixType>
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| 32 | struct umeyama_transform_matrix_type
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| 33 | {
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| 34 | enum {
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| 35 | MinRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),
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| 36 |
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| 37 | // When possible we want to choose some small fixed size value since the result
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| 38 | // is likely to fit on the stack. So here, EIGEN_SIZE_MIN_PREFER_DYNAMIC is not what we want.
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| 39 | HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic ? Dynamic : int(MinRowsAtCompileTime)+1
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| 40 | };
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| 41 |
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| 42 | typedef Matrix<typename traits<MatrixType>::Scalar,
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| 43 | HomogeneousDimension,
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| 44 | HomogeneousDimension,
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| 45 | AutoAlign | (traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor),
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| 46 | HomogeneousDimension,
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| 47 | HomogeneousDimension
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| 48 | > type;
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| 49 | };
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| 50 |
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| 51 | }
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| 52 |
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| 53 | #endif
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| 54 |
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| 55 | /**
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| 56 | * \geometry_module \ingroup Geometry_Module
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| 57 | *
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| 58 | * \brief Returns the transformation between two point sets.
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| 59 | *
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| 60 | * The algorithm is based on:
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| 61 | * "Least-squares estimation of transformation parameters between two point patterns",
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| 62 | * Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
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| 63 | *
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| 64 | * It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such that
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| 65 | * \f{align*}
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| 66 | * \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2
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| 67 | * \f}
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| 68 | * is minimized.
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| 69 | *
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| 70 | * The algorithm is based on the analysis of the covariance matrix
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| 71 | * \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
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| 72 | * of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where
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| 73 | * \f$d\f$ is corresponding to the dimension (which is typically small).
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| 74 | * The analysis is involving the SVD having a complexity of \f$O(d^3)\f$
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| 75 | * though the actual computational effort lies in the covariance
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| 76 | * matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when
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| 77 | * the input point sets have dimension \f$d \times m\f$.
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| 78 | *
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| 79 | * Currently the method is working only for floating point matrices.
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| 80 | *
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| 81 | * \todo Should the return type of umeyama() become a Transform?
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| 82 | *
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| 83 | * \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right) \f$.
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| 84 | * \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n \right) \f$.
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| 85 | * \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is passed.
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| 86 | * \return The homogeneous transformation
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| 87 | * \f{align*}
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| 88 | * T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix}
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| 89 | * \f}
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| 90 | * minimizing the resudiual above. This transformation is always returned as an
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| 91 | * Eigen::Matrix.
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| 92 | */
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| 93 | template <typename Derived, typename OtherDerived>
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| 94 | typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
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| 95 | umeyama(const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true)
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| 96 | {
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| 97 | typedef typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type TransformationMatrixType;
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| 98 | typedef typename internal::traits<TransformationMatrixType>::Scalar Scalar;
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| 99 | typedef typename NumTraits<Scalar>::Real RealScalar;
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| 100 | typedef typename Derived::Index Index;
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| 101 |
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| 102 | EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
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| 103 | EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
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| 104 | YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
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| 105 |
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| 106 | enum { Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
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| 107 |
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| 108 | typedef Matrix<Scalar, Dimension, 1> VectorType;
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| 109 | typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
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| 110 | typedef typename internal::plain_matrix_type_row_major<Derived>::type RowMajorMatrixType;
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| 111 |
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| 112 | const Index m = src.rows(); // dimension
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| 113 | const Index n = src.cols(); // number of measurements
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| 114 |
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| 115 | // required for demeaning ...
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| 116 | const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);
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| 117 |
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| 118 | // computation of mean
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| 119 | const VectorType src_mean = src.rowwise().sum() * one_over_n;
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| 120 | const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
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| 121 |
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| 122 | // demeaning of src and dst points
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| 123 | const RowMajorMatrixType src_demean = src.colwise() - src_mean;
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| 124 | const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
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| 125 |
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| 126 | // Eq. (36)-(37)
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| 127 | const Scalar src_var = src_demean.rowwise().squaredNorm().sum() * one_over_n;
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| 128 |
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| 129 | // Eq. (38)
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| 130 | const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
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| 131 |
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| 132 | JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
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| 133 |
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| 134 | // Initialize the resulting transformation with an identity matrix...
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| 135 | TransformationMatrixType Rt = TransformationMatrixType::Identity(m+1,m+1);
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| 136 |
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| 137 | // Eq. (39)
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| 138 | VectorType S = VectorType::Ones(m);
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| 139 | if (sigma.determinant()<Scalar(0)) S(m-1) = Scalar(-1);
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| 140 |
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| 141 | // Eq. (40) and (43)
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| 142 | const VectorType& d = svd.singularValues();
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| 143 | Index rank = 0; for (Index i=0; i<m; ++i) if (!internal::isMuchSmallerThan(d.coeff(i),d.coeff(0))) ++rank;
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| 144 | if (rank == m-1) {
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| 145 | if ( svd.matrixU().determinant() * svd.matrixV().determinant() > Scalar(0) ) {
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| 146 | Rt.block(0,0,m,m).noalias() = svd.matrixU()*svd.matrixV().transpose();
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| 147 | } else {
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| 148 | const Scalar s = S(m-1); S(m-1) = Scalar(-1);
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| 149 | Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
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| 150 | S(m-1) = s;
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| 151 | }
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| 152 | } else {
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| 153 | Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
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| 154 | }
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| 155 |
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| 156 | if (with_scaling)
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| 157 | {
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| 158 | // Eq. (42)
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| 159 | const Scalar c = Scalar(1)/src_var * svd.singularValues().dot(S);
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| 160 |
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| 161 | // Eq. (41)
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| 162 | Rt.col(m).head(m) = dst_mean;
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| 163 | Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
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| 164 | Rt.block(0,0,m,m) *= c;
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| 165 | }
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| 166 | else
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| 167 | {
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| 168 | Rt.col(m).head(m) = dst_mean;
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| 169 | Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
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| 170 | }
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| 171 |
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| 172 | return Rt;
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| 173 | }
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| 174 |
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| 175 | } // end namespace Eigen
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| 176 |
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| 177 | #endif // EIGEN_UMEYAMA_H
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