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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