itk::Math::detail Namespace Reference

Typedefs
template<typename TReal>
using	CholeskyRowMajor = Eigen::Matrix<TReal, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>
template<typename TReal>
using	CholeskyVector = Eigen::Matrix<TReal, Eigen::Dynamic, 1>

Functions
template<typename TReal>
void	DynamicSquareSVDEigen (const TReal *inData, unsigned int n, TReal *uData, TReal *wData, TReal *vData)
template<unsigned int VRows, unsigned int VColumns, typename TReal>
void	MatrixExponentialEigen (const TReal *inData, TReal *outData)
template<typename TReal>
void	MatrixExponentialEigen (const TReal *inData, TReal *outData, unsigned int n)
template<typename TVector, typename TReal>
unsigned int	NumericalRank (const TVector &W, TReal rcond)
template<typename TMatrix, typename TVector, typename TReal>
TMatrix	PseudoInverse (const TMatrix &U, const TVector &W, const TMatrix &V, TReal rcond)
template<typename TMatrix, typename TVector, typename TReal>
TMatrix	Recompose (const TMatrix &U, const TVector &W, const TMatrix &V, TReal rcond)
template<typename TReal>
void	RectangularSVDEigen (const TReal *inData, unsigned int rows, unsigned int cols, TReal *uData, TReal *wData, TReal *vData)
template<typename TReal>
TReal	ResolveRcond (TReal rcond, unsigned int n)
template<typename TMatrix, typename TVector, typename TReal>
TVector	SolveLinear (const TMatrix &U, const TVector &W, const TMatrix &V, const TVector &b, TReal rcond)
template<unsigned int VDim, typename TReal>
void	SquareSVDEigen (const TReal *inData, TReal *uData, TReal *wData, TReal *vData)

Variables
constexpr unsigned int	kFixedSVDMaxDim = 16
constexpr unsigned int	kJacobiMaxDim = 6

Detailed Description

Cholesky-based linear algebra for symmetric matrices, backed by Eigen.

Matrix exponential exp(A) for a real square matrix, backed by Eigen.

SolveSymmetricPositiveDefinite() uses Eigen's robust LDL^T (pivoted) factor, which handles near-semidefinite regularized Gram matrices. CholeskyLowerTriangle() returns the L of A = L L^T via Eigen's LLT.

Replacement for the removed vnl_matrix_exp. Delegates to Eigen's MatrixBase::exp(), which uses scaling-and-squaring with a degree-13 Pade approximant (Higham, SIAM J. Matrix Anal. Appl. 26(4), 2005) – more robust than the truncated Taylor series formerly provided by vnl_matrix_exp.

Typedef Documentation

CholeskyRowMajor

template<typename TReal>

using itk::Math::detail::CholeskyRowMajor = Eigen::Matrix<TReal, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>

Definition at line 44 of file itkCholeskySolve.h.

CholeskyVector

template<typename TReal>

using itk::Math::detail::CholeskyVector = Eigen::Matrix<TReal, Eigen::Dynamic, 1>

Definition at line 46 of file itkCholeskySolve.h.

Function Documentation

DynamicSquareSVDEigen()

template<typename TReal>

void itk::Math::detail::DynamicSquareSVDEigen	(	const TReal *	inData,
		unsigned int	n,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 214 of file itkMathSVD.h.

References kJacobiMaxDim.

Referenced by SquareSVDEigen(), and itk::Math::SVD().

MatrixExponentialEigen() [1/2]

template<unsigned int VRows, unsigned int VColumns, typename TReal>

void itk::Math::detail::MatrixExponentialEigen	(	const TReal *	inData,
		TReal *	outData )

Definition at line 45 of file itkMatrixExponential.h.

Referenced by itk::Math::MatrixExponential(), itk::Math::MatrixExponential(), and itk::Math::MatrixExponential().

MatrixExponentialEigen() [2/2]

template<typename TReal>

void itk::Math::detail::MatrixExponentialEigen	(	const TReal *	inData,
		TReal *	outData,
		unsigned int	n )

Definition at line 55 of file itkMatrixExponential.h.

NumericalRank()

template<typename TVector, typename TReal>

unsigned int itk::Math::detail::NumericalRank	(	const TVector &	W,
		TReal	rcond )

Definition at line 87 of file itkMathSVD.h.

References ResolveRcond().

PseudoInverse()

template<typename TMatrix, typename TVector, typename TReal>

TMatrix itk::Math::detail::PseudoInverse	(	const TMatrix &	U,
		const TVector &	W,
		const TMatrix &	V,
		TReal	rcond )

Definition at line 53 of file itkMathSVD.h.

References ResolveRcond().

Recompose()

template<typename TMatrix, typename TVector, typename TReal>

TMatrix itk::Math::detail::Recompose	(	const TMatrix &	U,
		const TVector &	W,
		const TMatrix &	V,
		TReal	rcond )

Definition at line 106 of file itkMathSVD.h.

References ResolveRcond().

RectangularSVDEigen()

template<typename TReal>

void itk::Math::detail::RectangularSVDEigen	(	const TReal *	inData,
		unsigned int	rows,
		unsigned int	cols,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 289 of file itkMathSVD.h.

Referenced by itk::Math::SVD().

ResolveRcond()

template<typename TReal>

TReal itk::Math::detail::ResolveRcond	(	TReal	rcond,
		unsigned int	n )

Definition at line 44 of file itkMathSVD.h.

Referenced by NumericalRank(), PseudoInverse(), Recompose(), and SolveLinear().

SolveLinear()

template<typename TMatrix, typename TVector, typename TReal>

TVector itk::Math::detail::SolveLinear	(	const TMatrix &	U,
		const TVector &	W,
		const TMatrix &	V,
		const TVector &	b,
		TReal	rcond )

Definition at line 73 of file itkMathSVD.h.

References ResolveRcond().

SquareSVDEigen()

template<unsigned int VDim, typename TReal>

void itk::Math::detail::SquareSVDEigen	(	const TReal *	inData,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 251 of file itkMathSVD.h.

References DynamicSquareSVDEigen(), and kFixedSVDMaxDim.

Referenced by itk::Math::SVD().

Variable Documentation

kFixedSVDMaxDim

unsigned int itk::Math::detail::kFixedSVDMaxDim = 16

constexpr

Definition at line 204 of file itkMathSVD.h.

Referenced by SquareSVDEigen().

kJacobiMaxDim

unsigned int itk::Math::detail::kJacobiMaxDim = 6

constexpr

Definition at line 208 of file itkMathSVD.h.

Referenced by DynamicSquareSVDEigen().