Umeyama.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
00005 //
00006 // This Source Code Form is subject to the terms of the Mozilla
00007 // Public License v. 2.0. If a copy of the MPL was not distributed
00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00009 
00010 #ifndef EIGEN_UMEYAMA_H
00011 #define EIGEN_UMEYAMA_H
00012 
00013 // This file requires the user to include 
00014 // * Eigen/Core
00015 // * Eigen/LU 
00016 // * Eigen/SVD
00017 // * Eigen/Array
00018 
00019 namespace Eigen { 
00020 
00021 #ifndef EIGEN_PARSED_BY_DOXYGEN
00022 
00023 // These helpers are required since it allows to use mixed types as parameters
00024 // for the Umeyama. The problem with mixed parameters is that the return type
00025 // cannot trivially be deduced when float and double types are mixed.
00026 namespace internal {
00027 
00028 // Compile time return type deduction for different MatrixBase types.
00029 // Different means here different alignment and parameters but the same underlying
00030 // real scalar type.
00031 template<typename MatrixType, typename OtherMatrixType>
00032 struct umeyama_transform_matrix_type
00033 {
00034   enum {
00035     MinRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),
00036 
00037     // When possible we want to choose some small fixed size value since the result
00038     // is likely to fit on the stack. So here, EIGEN_SIZE_MIN_PREFER_DYNAMIC is not what we want.
00039     HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic ? Dynamic : int(MinRowsAtCompileTime)+1
00040   };
00041 
00042   typedef Matrix<typename traits<MatrixType>::Scalar,
00043     HomogeneousDimension,
00044     HomogeneousDimension,
00045     AutoAlign | (traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor),
00046     HomogeneousDimension,
00047     HomogeneousDimension
00048   > type;
00049 };
00050 
00051 }
00052 
00053 #endif
00054 
00055 /**
00056 * \geometry_module \ingroup Geometry_Module
00057 *
00058 * \brief Returns the transformation between two point sets.
00059 *
00060 * The algorithm is based on:
00061 * "Least-squares estimation of transformation parameters between two point patterns",
00062 * Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
00063 *
00064 * It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such that
00065 * \f{align*}
00066 *   \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2
00067 * \f}
00068 * is minimized.
00069 *
00070 * The algorithm is based on the analysis of the covariance matrix
00071 * \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
00072 * of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where 
00073 * \f$d\f$ is corresponding to the dimension (which is typically small).
00074 * The analysis is involving the SVD having a complexity of \f$O(d^3)\f$
00075 * though the actual computational effort lies in the covariance
00076 * matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when 
00077 * the input point sets have dimension \f$d \times m\f$.
00078 *
00079 * Currently the method is working only for floating point matrices.
00080 *
00081 * \todo Should the return type of umeyama() become a Transform?
00082 *
00083 * \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right) \f$.
00084 * \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n \right) \f$.
00085 * \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is passed.
00086 * \return The homogeneous transformation 
00087 * \f{align*}
00088 *   T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix}
00089 * \f}
00090 * minimizing the resudiual above. This transformation is always returned as an 
00091 * Eigen::Matrix.
00092 */
00093 template <typename Derived, typename OtherDerived>
00094 typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type
00095 umeyama (const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true)
00096 {
00097   typedef typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type TransformationMatrixType;
00098   typedef typename internal::traits<TransformationMatrixType>::Scalar Scalar;
00099   typedef typename NumTraits<Scalar>::Real RealScalar;
00100   typedef typename Derived::Index Index;
00101 
00102   EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
00103   EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
00104     YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
00105 
00106   enum { Dimension = EIGEN_SIZE_MIN_PREFER_DYNAMIC(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };
00107 
00108   typedef Matrix<Scalar, Dimension, 1> VectorType;
00109   typedef Matrix<Scalar, Dimension, Dimension> MatrixType;
00110   typedef typename internal::plain_matrix_type_row_major<Derived>::type RowMajorMatrixType;
00111 
00112   const Index m = src.rows(); // dimension
00113   const Index n = src.cols(); // number of measurements
00114 
00115   // required for demeaning ...
00116   const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);
00117 
00118   // computation of mean
00119   const VectorType src_mean = src.rowwise().sum() * one_over_n;
00120   const VectorType dst_mean = dst.rowwise().sum() * one_over_n;
00121 
00122   // demeaning of src and dst points
00123   const RowMajorMatrixType src_demean = src.colwise() - src_mean;
00124   const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;
00125 
00126   // Eq. (36)-(37)
00127   const Scalar src_var = src_demean.rowwise().squaredNorm ().sum() * one_over_n;
00128 
00129   // Eq. (38)
00130   const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();
00131 
00132   JacobiSVD<MatrixType> svd(sigma, ComputeFullU | ComputeFullV);
00133 
00134   // Initialize the resulting transformation with an identity matrix...
00135   TransformationMatrixType Rt = TransformationMatrixType::Identity(m+1,m+1);
00136 
00137   // Eq. (39)
00138   VectorType S = VectorType::Ones(m);
00139   if (sigma.determinant()<Scalar(0)) S(m-1) = Scalar(-1);
00140 
00141   // Eq. (40) and (43)
00142   const VectorType& d = svd.singularValues ();
00143   Index rank = 0; for (Index i=0; i<m; ++i) if (!internal::isMuchSmallerThan(d.coeff(i),d.coeff(0))) ++rank;
00144   if (rank == m-1) {
00145     if ( svd.matrixU ().determinant() * svd.matrixV ().determinant() > Scalar(0) ) {
00146       Rt.block(0,0,m,m).noalias() = svd.matrixU ()*svd.matrixV ().transpose();
00147     } else {
00148       const Scalar s = S(m-1); S(m-1) = Scalar(-1);
00149       Rt.block(0,0,m,m).noalias() = svd.matrixU () * S.asDiagonal() * svd.matrixV ().transpose();
00150       S(m-1) = s;
00151     }
00152   } else {
00153     Rt.block(0,0,m,m).noalias() = svd.matrixU () * S.asDiagonal() * svd.matrixV ().transpose();
00154   }
00155 
00156   if (with_scaling)
00157   {
00158     // Eq. (42)
00159     const Scalar c = Scalar(1)/src_var * svd.singularValues ().dot(S);
00160 
00161     // Eq. (41)
00162     Rt.col(m).head(m) = dst_mean;
00163     Rt.col(m).head(m).noalias() -= c*Rt.topLeftCorner(m,m)*src_mean;
00164     Rt.block(0,0,m,m) *= c;
00165   }
00166   else
00167   {
00168     Rt.col(m).head(m) = dst_mean;
00169     Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m,m)*src_mean;
00170   }
00171 
00172   return Rt;
00173 }
00174 
00175 } // end namespace Eigen
00176 
00177 #endif // EIGEN_UMEYAMA_H