add vcglib
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at the mozilla.org home page
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#ifndef EIGEN_UMEYAMA_H
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#define EIGEN_UMEYAMA_H
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// This file requires the user to include
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// * Eigen/Core
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// * Eigen/LU
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// * Eigen/SVD
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// * Eigen/Array
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namespace Eigen {
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#ifndef EIGEN_PARSED_BY_DOXYGEN
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// These helpers are required since it allows to use mixed types as parameters
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// for the Umeyama. The problem with mixed parameters is that the return type
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// cannot trivially be deduced when float and double types are mixed.
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namespace internal {
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// Compile time return type deduction for different MatrixBase types.
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// Different means here different alignment and parameters but the same underlying
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// real scalar type.
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template<typename MatrixType, typename OtherMatrixType>
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struct umeyama_transform_matrix_type
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{
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enum {
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MinRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),
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// When possible we want to choose some small fixed size value since the result
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// is likely to fit on the stack. So here, EIGEN_SIZE_MIN_PREFER_DYNAMIC is not what we want.
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HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic ? Dynamic : int(MinRowsAtCompileTime)+1
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};
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typedef Matrix<typename traits<MatrixType>::Scalar,
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HomogeneousDimension,
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HomogeneousDimension,
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AutoAlign | (traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor),
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HomogeneousDimension,
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HomogeneousDimension
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> type;
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};
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}
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#endif
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/**
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* \geometry_module \ingroup Geometry_Module
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*
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* \brief Returns the transformation between two point sets.
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*
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* The algorithm is based on:
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* "Least-squares estimation of transformation parameters between two point patterns",
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* Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
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*
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* It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such that
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* \f{align*}
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* \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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* \f}
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* is minimized.
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*
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* The algorithm is based on the analysis of the covariance matrix
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* \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
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* of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where
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* \f$d\f$ is corresponding to the dimension (which is typically small).
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* The analysis is involving the SVD having a complexity of \f$O(d^3)\f$
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* though the actual computational effort lies in the covariance
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* matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when
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* the input point sets have dimension \f$d \times m\f$.
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*
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* Currently the method is working only for floating point matrices.
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*
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* \todo Should the return type of umeyama() become a Transform?
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*
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* \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right) \f$.
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* \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n \right) \f$.
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* \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is passed.
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* \return The homogeneous transformation
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* \f{align*}
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* T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix}
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* \f}
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* minimizing the residual above. This transformation is always returned as an
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* Eigen::Matrix.
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*/
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template <typename Derived, typename OtherDerived>
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typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
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umeyama(const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true)
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{
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typedef typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type TransformationMatrixType;
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typedef typename internal::traits<TransformationMatrixType>::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
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EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
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YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
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enum { Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
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typedef Matrix<Scalar, Dimension, 1> VectorType;
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typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
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typedef typename internal::plain_matrix_type_row_major<Derived>::type RowMajorMatrixType;
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const Index m = src.rows(); // dimension
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const Index n = src.cols(); // number of measurements
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// required for demeaning ...
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const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);
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// computation of mean
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const VectorType src_mean = src.rowwise().sum() * one_over_n;
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const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
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// demeaning of src and dst points
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const RowMajorMatrixType src_demean = src.colwise() - src_mean;
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const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
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// Eq. (36)-(37)
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const Scalar src_var = src_demean.rowwise().squaredNorm().sum() * one_over_n;
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// Eq. (38)
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const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
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JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
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// Initialize the resulting transformation with an identity matrix...
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TransformationMatrixType Rt = TransformationMatrixType::Identity(m+1,m+1);
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// Eq. (39)
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VectorType S = VectorType::Ones(m);
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if ( svd.matrixU().determinant() * svd.matrixV().determinant() < 0 )
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S(m-1) = -1;
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// Eq. (40) and (43)
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Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
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if (with_scaling)
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{
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// Eq. (42)
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const Scalar c = Scalar(1)/src_var * svd.singularValues().dot(S);
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// Eq. (41)
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Rt.col(m).head(m) = dst_mean;
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Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
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Rt.block(0,0,m,m) *= c;
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}
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else
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{
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Rt.col(m).head(m) = dst_mean;
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Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
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}
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return Rt;
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}
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} // end namespace Eigen
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#endif // EIGEN_UMEYAMA_H
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