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 
19 namespace 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.
26 namespace 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.
31 template<typename MatrixType, typename OtherMatrixType>
32 struct 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 */
93 template <typename Derived, typename OtherDerived>
94 typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
95 umeyama(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   typedef typename Derived::Index Index;
101 
102   EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
103   EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
104     YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
105 
106   enum { Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
107 
108   typedef Matrix<Scalar, Dimension, 1> VectorType;
109   typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
110   typedef typename internal::plain_matrix_type_row_major<Derived>::type RowMajorMatrixType;
111 
112   const Index m = src.rows(); // dimension
113   const Index n = src.cols(); // number of measurements
114 
115   // required for demeaning ...
116   const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);
117 
118   // computation of mean
119   const VectorType src_mean = src.rowwise().sum() * one_over_n;
120   const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
121 
122   // demeaning of src and dst points
123   const RowMajorMatrixType src_demean = src.colwise() - src_mean;
124   const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
125 
126   // Eq. (36)-(37)
127   const Scalar src_var = src_demean.rowwise().squaredNorm().sum() * one_over_n;
128 
129   // Eq. (38)
130   const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
131 
132   JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
133 
134   // Initialize the resulting transformation with an identity matrix...
135   TransformationMatrixType Rt = TransformationMatrixType::Identity(m+1,m+1);
136 
137   // Eq. (39)
138   VectorType S = VectorType::Ones(m);
139   if (sigma.determinant()<Scalar(0)) S(m-1) = Scalar(-1);
140 
141   // Eq. (40) and (43)
142   const VectorType& d = svd.singularValues();
143   Index rank = 0; for (Index i=0; i<m; ++i) if (!internal::isMuchSmallerThan(d.coeff(i),d.coeff(0))) ++rank;
144   if (rank == m-1) {
145     if ( svd.matrixU().determinant() * svd.matrixV().determinant() > Scalar(0) ) {
146       Rt.block(0,0,m,m).noalias() = svd.matrixU()*svd.matrixV().transpose();
147     } else {
148       const Scalar s = S(m-1); S(m-1) = Scalar(-1);
149       Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
150       S(m-1) = s;
151     }
152   } else {
153     Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
154   }
155 
156   if (with_scaling)
157   {
158     // Eq. (42)
159     const Scalar c = Scalar(1)/src_var * svd.singularValues().dot(S);
160 
161     // Eq. (41)
162     Rt.col(m).head(m) = dst_mean;
163     Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
164     Rt.block(0,0,m,m) *= c;
165   }
166   else
167   {
168     Rt.col(m).head(m) = dst_mean;
169     Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
170   }
171 
172   return Rt;
173 }
174 
175 } // end namespace Eigen
176 
177 #endif // EIGEN_UMEYAMA_H
178