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