Geometry_Module

Topics
	Global aligned box typedefs

Classes
class	AlignedBox< _Scalar, _AmbientDim >
	An axis aligned box. More...

class	AngleAxis< _Scalar >
	Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. More...

class	Homogeneous< MatrixType, _Direction >
	Expression of one (or a set of) homogeneous vector(s) More...

class	Hyperplane< _Scalar, _AmbientDim, _Options >
	A hyperplane. More...

class	Map< const Quaternion< _Scalar >, _Options >
	Quaternion expression mapping a constant memory buffer. More...

class	Map< Quaternion< _Scalar >, _Options >
	Expression of a quaternion from a memory buffer. More...

class	ParametrizedLine< _Scalar, _AmbientDim, _Options >
	A parametrized line. More...

class	Quaternion< _Scalar, _Options >
	The quaternion class used to represent 3D orientations and rotations. More...

class	QuaternionBase< Derived >
	Base class for quaternion expressions. More...

class	Rotation2D< _Scalar >
	Represents a rotation/orientation in a 2 dimensional space. More...

class	Scaling
	Represents a generic uniform scaling transformation. More...

class	Transform< _Scalar, _Dim, _Mode, _Options >
	Represents an homogeneous transformation in a N dimensional space. More...

class	Translation< _Scalar, _Dim >
	Represents a translation transformation. More...

Typedefs
typedef AngleAxis< double >	AngleAxisd

typedef AngleAxis< float >	AngleAxisf

typedef Quaternion< double >	Quaterniond

typedef Quaternion< float >	Quaternionf

typedef Map< Quaternion< double >, Aligned >	QuaternionMapAlignedd

typedef Map< Quaternion< float >, Aligned >	QuaternionMapAlignedf

typedef Map< Quaternion< double >, 0 >	QuaternionMapd

typedef Map< Quaternion< float >, 0 >	QuaternionMapf

typedef Rotation2D< double >	Rotation2Dd

typedef Rotation2D< float >	Rotation2Df

Functions
Matrix< Scalar, 3, 1 >	eulerAngles (Index a0, Index a1, Index a2) const

template<typename Derived, typename OtherDerived>
internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type	umeyama (const MatrixBase< Derived > &src, const MatrixBase< OtherDerived > &dst, bool with_scaling=true)
	Returns the transformation between two point sets.

Detailed Description

Typedef Documentation

◆ AngleAxisd

typedef AngleAxis<double> AngleAxisd

double precision angle-axis type

◆ AngleAxisf

typedef AngleAxis<float> AngleAxisf

single precision angle-axis type

◆ Quaterniond

typedef Quaternion<double> Quaterniond

double precision quaternion type

◆ Quaternionf

typedef Quaternion<float> Quaternionf

single precision quaternion type

◆ QuaternionMapAlignedd

typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd

Map a 16-bits aligned array of double precision scalars as a quaternion

◆ QuaternionMapAlignedf

typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf

Map a 16-bits aligned array of double precision scalars as a quaternion

◆ QuaternionMapd

typedef Map<Quaternion<double>, 0> QuaternionMapd

Map an unaligned array of double precision scalar as a quaternion

◆ QuaternionMapf

typedef Map<Quaternion<float>, 0> QuaternionMapf

Map an unaligned array of single precision scalar as a quaternion

◆ Rotation2Dd

typedef Rotation2D<double> Rotation2Dd

double precision 2D rotation type

◆ Rotation2Df

typedef Rotation2D<float> Rotation2Df

single precision 2D rotation type

Function Documentation

◆ eulerAngles()

template<typename Derived>

Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > eulerAngles	(	Index	a0,
		Index	a1,
		Index	a2 ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry>

Returns: the Euler-angles of the rotation matrix *this using the convention defined by the triplet (a0,a1,a2)

Each of the three parameters a0,a1,a2 represents the respective rotation axis as an integer in {0,1,2}. For instance, in:

Vector3f ea = mat.eulerAngles(2, 0, 2);

"2" represents the z axis and "0" the x axis, etc. The returned angles are such that we have the following equality:

mat == AngleAxisf(ea[0], Vector3f::UnitZ())
     * AngleAxisf(ea[1], Vector3f::UnitX())
     * AngleAxisf(ea[2], Vector3f::UnitZ()); 

This corresponds to the right-multiply conventions (with right hand side frames).

◆ umeyama()

template<typename Derived, typename OtherDerived>

internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type umeyama	(	const MatrixBase< Derived > &	src,
		const MatrixBase< OtherDerived > &	dst,
		bool	with_scaling = true )

Returns the transformation between two point sets.

This is defined in the Geometry module.

#include <Eigen/Geometry>

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 $c, \mathbf{R},$ and $\mathbf{t}$ such that

$\begin{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2 \end{align*}$

is minimized.

The algorithm is based on the analysis of the covariance matrix $\Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d}$ of the input point sets $\mathbf{x}$ and $\mathbf{y}$ where $d$ is corresponding to the dimension (which is typically small). The analysis is involving the SVD having a complexity of $O(d^3)$ though the actual computational effort lies in the covariance matrix computation which has an asymptotic lower bound of $O(dm)$ when the input point sets have dimension $d \times m$ .

Currently the method is working only for floating point matrices.

Parameters

src	Source points $\mathbf{x} = \left( x_1, \hdots, x_n \right)$ .
dst	Destination points $\mathbf{y} = \left( y_1, \hdots, y_n \right)$ .
with_scaling	Sets when `false` is passed.

Returns: The homogeneous transformation
$\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}$
minimizing the resudiual above. This transformation is always returned as an Eigen::Matrix.

References MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::asDiagonal(), DenseBase< Derived >::colwise(), ComputeFullU, ComputeFullV, MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::determinant(), JacobiSVD< _MatrixType, QRPreconditioner >::matrixU(), JacobiSVD< _MatrixType, QRPreconditioner >::matrixV(), DenseBase< Derived >::rowwise(), JacobiSVD< _MatrixType, QRPreconditioner >::singularValues(), VectorwiseOp< ExpressionType, Direction >::squaredNorm(), VectorwiseOp< ExpressionType, Direction >::sum(), and MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::transpose().

Topics

Classes

Typedefs

Functions