Eigen/src/Geometry/Umeyama.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0

#ifndef EIGEN_UMEYAMA_H
#define EIGEN_UMEYAMA_H

// This file requires the user to include
// * Eigen/Core
// * Eigen/LU
// * Eigen/SVD
// * Eigen/Array

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

// These helpers are required since they allow the use of mixed types as parameters
// for the Umeyama. The problem with mixed parameters is that the return type
// cannot trivially be deduced when float and double types are mixed.
namespace internal {

// Compile time return type deduction for different MatrixBase types.
// Different means here different alignment and parameters but the same underlying
// real scalar type.
template <typename MatrixType, typename OtherMatrixType>
struct umeyama_transform_matrix_type {
  enum {
    MinRowsAtCompileTime =
        internal::min_size_prefer_dynamic(MatrixType::RowsAtCompileTime, OtherMatrixType::RowsAtCompileTime),

    // When possible we want to choose some small fixed size value since the result
    // is likely to fit on the stack. So here, min_size_prefer_dynamic is not what we want.
    HomogeneousDimension = int(MinRowsAtCompileTime) == Dynamic ? Dynamic : int(MinRowsAtCompileTime) + 1
  };

  using type = Matrix<typename traits<MatrixType>::Scalar, HomogeneousDimension, HomogeneousDimension,
                      AutoAlign | (traits<MatrixType>::Flags & RowMajorBit ? RowMajor : ColMajor), HomogeneousDimension,
                      HomogeneousDimension>;
};

}  // namespace internal

/**
 * \geometry_module \ingroup Geometry_Module
 *
 * \brief Returns the transformation between two point sets.
 *
 * The algorithm is based on:
 * "Least-squares estimation of transformation parameters between two point patterns",
 * Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
 *
 * It estimates parameters \f$ c, \mathbf{R}, \f$ and \f$ \mathbf{t} \f$ such that
 * \f{align*}
 *   \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2
 * \f}
 * is minimized.
 *
 * The algorithm is based on the analysis of the covariance matrix
 * \f$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \f$
 * of the input point sets \f$ \mathbf{x} \f$ and \f$ \mathbf{y} \f$ where
 * \f$d\f$ is corresponding to the dimension (which is typically small).
 * The analysis involves the SVD having a complexity of \f$O(d^3)\f$
 * though the actual computational effort lies in the covariance
 * matrix computation which has an asymptotic lower bound of \f$O(dm)\f$ when
 * the input point sets have dimension \f$d \times m\f$.
 *
 * Currently the method is working only for floating point matrices.
 *
 * \todo Should the return type of umeyama() become a Transform?
 *
 * \param src Source points \f$ \mathbf{x} = \left( x_1, \hdots, x_n \right) \f$.
 * \param dst Destination points \f$ \mathbf{y} = \left( y_1, \hdots, y_n \right) \f$.
 * \param with_scaling Sets \f$ c=1 \f$ when <code>false</code> is passed.
 * \return The homogeneous transformation
 * \f{align*}
 *   T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix}
 * \f}
 * minimizing the residual above. This transformation is always returned as an
 * Eigen::Matrix.
 */
template <typename Derived, typename OtherDerived>
typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type umeyama(
    const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, bool with_scaling = true) {
  using TransformationMatrixType = typename internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type;
  using Scalar = typename internal::traits<TransformationMatrixType>::Scalar;
  using RealScalar = typename NumTraits<Scalar>::Real;

  EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL)
  EIGEN_STATIC_ASSERT(
      (std::is_same<Scalar, typename internal::traits<OtherDerived>::Scalar>::value),
      YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)

  enum { Dimension = internal::min_size_prefer_dynamic(Derived::RowsAtCompileTime, OtherDerived::RowsAtCompileTime) };

  using VectorType = Matrix<Scalar, Dimension, 1>;
  using MatrixType = Matrix<Scalar, Dimension, Dimension>;
  using RowMajorMatrixType = typename internal::plain_matrix_type_row_major<Derived>::type;

  const Index m = src.rows();  // dimension
  const Index n = src.cols();  // number of measurements

  // required for demeaning ...
  const RealScalar one_over_n = RealScalar(1) / static_cast<RealScalar>(n);

  // computation of mean
  const VectorType src_mean = src.rowwise().sum() * one_over_n;
  const VectorType dst_mean = dst.rowwise().sum() * one_over_n;

  // demeaning of src and dst points
  const RowMajorMatrixType src_demean = src.colwise() - src_mean;
  const RowMajorMatrixType dst_demean = dst.colwise() - dst_mean;

  // Eq. (38)
  const MatrixType sigma = one_over_n * dst_demean * src_demean.transpose();

  JacobiSVD<MatrixType, ComputeFullU | ComputeFullV> svd(sigma);

  // Initialize the resulting transformation with an identity matrix...
  TransformationMatrixType Rt = TransformationMatrixType::Identity(m + 1, m + 1);

  // Eq. (39)
  VectorType S = VectorType::Ones(m);

  if (svd.matrixU().determinant() * svd.matrixV().determinant() < 0) {
    Index tmp = m - 1;
    S(tmp) = -1;
  }

  // Eq. (40) and (43)
  Rt.block(0, 0, m, m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();

  if (with_scaling) {
    // Eq. (36)-(37)
    const Scalar src_var = src_demean.rowwise().squaredNorm().sum() * one_over_n;

    if (src_var <= Scalar(0)) {
      // Degenerate: source points have zero variance (all nearly identical).
      // Scaling is undefined; return the best-fit pure translation.
      Rt.col(m).head(m) = dst_mean - src_mean;
      return Rt;
    }

    // Eq. (42)
    const Scalar c = Scalar(1) / src_var * svd.singularValues().dot(S);

    // Eq. (41)
    Rt.col(m).head(m) = dst_mean;
    Rt.col(m).head(m).noalias() -= c * Rt.topLeftCorner(m, m) * src_mean;
    Rt.block(0, 0, m, m) *= c;
  } else {
    Rt.col(m).head(m) = dst_mean;
    Rt.col(m).head(m).noalias() -= Rt.topLeftCorner(m, m) * src_mean;
  }

  return Rt;
}

}  // end namespace Eigen

#endif  // EIGEN_UMEYAMA_H