Class for symmetric spherical harmonics The index scheme of the coefficients/basis values is like in the Descoteaux paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging". More...

#include <WSymmetricSphericalHarmonic.h>

Collaboration diagram for WSymmetricSphericalHarmonic< T >:

Public Member Functions
	WSymmetricSphericalHarmonic ()
	Default constructor. More...

	WSymmetricSphericalHarmonic (const WValue< T > &SHCoefficients)
	Constructor. More...

virtual	~WSymmetricSphericalHarmonic ()
	Destructor. More...

T	getValue (T theta, T phi) const
	Return the value on the sphere. More...

T	getValue (const WUnitSphereCoordinates< T > &coordinates) const
	Return the value on the sphere. More...

const WValue< T > &	getCoefficients () const
	Returns the used coefficients (stored like in the mentioned 2007 Descoteaux paper). More...

WValue< T >	getCoefficientsSchultz () const
	Returns the coefficients for Schultz' SH base. More...

WValue< std::complex< T > >	getCoefficientsComplex () const
	Returns the coefficients for the complex base. More...

void	setFromComplex (WValue< std::complex< T > > const &coeffs, int order)
	Set the coeffs from complex base coeffs. More...

void	applyFunkRadonTransformation (WMatrix< T > const &frtMat)
	Applies the Funk-Radon-Transformation. More...

size_t	getOrder () const
	Return the order of the spherical harmonic. More...

T	calcGFA (std::vector< WUnitSphereCoordinates< T > > const &orientations) const
	Calculate the generalized fractional anisotropy for this ODF. More...

T	calcGFA (WMatrix< T > const &B) const
	Calculate the generalized fractional anisotropy for this ODF. More...

void	normalize ()
	Normalize this SH in place. More...

Static Public Member Functions
static WMatrix< T >	getSHFittingMatrix (const std::vector< WMatrixFixed< T, 3, 1 > > &orientations, int order, T lambda, bool withFRT)
	This calculates the transformation/fitting matrix T like in the 2007 Descoteaux paper. More...

static WMatrix< T >	getSHFittingMatrix (const std::vector< WUnitSphereCoordinates< T > > &orientations, int order, T lambda, bool withFRT)
	This calculates the transformation/fitting matrix T like in the 2007 Descoteaux paper. More...

static WMatrix< T >	getSHFittingMatrixForConstantSolidAngle (const std::vector< WMatrixFixed< T, 3, 1 > > &orientations, int order, T lambda)
	This calculates the transformation/fitting matrix T like in the 2010 Aganj paper. More...

static WMatrix< T >	getSHFittingMatrixForConstantSolidAngle (const std::vector< WUnitSphereCoordinates< T > > &orientations, int order, T lambda)
	This calculates the transformation/fitting matrix T like in the 2010 Aganj paper. More...

static WMatrix< T >	calcBaseMatrix (const std::vector< WUnitSphereCoordinates< T > > &orientations, int order)
	Calculates the base matrix B like in the dissertation of Descoteaux. More...

static WMatrix< std::complex< T > >	calcComplexBaseMatrix (std::vector< WUnitSphereCoordinates< T > > const &orientations, int order)
	Calculates the base matrix B for the complex spherical harmonics. More...

static WValue< T >	calcEigenvalues (size_t order)
	Calc eigenvalues for SH elements. More...

static WMatrix< T >	calcMatrixWithEigenvalues (size_t order)
	Calc matrix with the eigenvalues of the SH elements on its diagonal. More...

static WMatrix< T >	calcSmoothingMatrix (size_t order)
	This calcs the smoothing matrix L from the 2007 Descoteaux Paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging". More...

static WMatrix< T >	calcFRTMatrix (size_t order)
	Calculates the Funk-Radon-Transformation-Matrix P from the 2007 Descoteaux Paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging". More...

static WMatrix< T >	calcSHToTensorSymMatrix (std::size_t order)
	Calculates a matrix that converts spherical harmonics to symmetric tensors of equal or lower order. More...

static WMatrix< T >	calcSHToTensorSymMatrix (std::size_t order, const std::vector< WUnitSphereCoordinates< T > > &orientations)
	Calculates a matrix that converts spherical harmonics to symmetric tensors of equal or lower order. More...

Private Attributes
size_t	m_order
	order of the spherical harmonic More...

WValue< T >	m_SHCoefficients
	coefficients of the spherical harmonic More...

Detailed Description

template<typename T>
class WSymmetricSphericalHarmonic< T >

Class for symmetric spherical harmonics The index scheme of the coefficients/basis values is like in the Descoteaux paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging".

Definition at line 51 of file WSymmetricSphericalHarmonic.h.

Constructor & Destructor Documentation

◆ WSymmetricSphericalHarmonic() [1/2]

template<typename T >

WSymmetricSphericalHarmonic< T >::WSymmetricSphericalHarmonic

Default constructor.

Definition at line 287 of file WSymmetricSphericalHarmonic.h.

◆ WSymmetricSphericalHarmonic() [2/2]

template<typename T >

WSymmetricSphericalHarmonic< T >::WSymmetricSphericalHarmonic ( const WValue< T > & SHCoefficients )

explicit

Constructor.

Parameters

SHCoefficients the initial coefficients (stored like in the mentioned Descoteaux paper).

Definition at line 294 of file WSymmetricSphericalHarmonic.h.

References WSymmetricSphericalHarmonic< T >::m_order, and WSymmetricSphericalHarmonic< T >::m_SHCoefficients.

◆ ~WSymmetricSphericalHarmonic()

template<typename T >

WSymmetricSphericalHarmonic< T >::~WSymmetricSphericalHarmonic

virtual

Destructor.

Definition at line 304 of file WSymmetricSphericalHarmonic.h.

Member Function Documentation

◆ applyFunkRadonTransformation()

template<typename T >

void WSymmetricSphericalHarmonic< T >::applyFunkRadonTransformation ( WMatrix< T > const & frtMat )

Applies the Funk-Radon-Transformation.

This is faster than matrix multiplication. ( O(n) instead of O(n²) )

Parameters

frtMat the frt matrix as calculated by calcFRTMatrix()

Definition at line 546 of file WSymmetricSphericalHarmonic.h.

References WMatrix< T >::getNbCols(), and WMatrix< T >::getNbRows().

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◆ calcBaseMatrix()

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcBaseMatrix	(	const std::vector< WUnitSphereCoordinates< T > > &	orientations,
		int	order
	)

static

Calculates the base matrix B like in the dissertation of Descoteaux.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create

Returns: The base Matrix B

Definition at line 656 of file WSymmetricSphericalHarmonic.h.

Referenced by WMCalculateGFA::moduleMain().

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◆ calcComplexBaseMatrix()

template<typename T >

WMatrix< std::complex< T > > WSymmetricSphericalHarmonic< T >::calcComplexBaseMatrix	(	std::vector< WUnitSphereCoordinates< T > > const &	orientations,
		int	order
	)

static

Calculates the base matrix B for the complex spherical harmonics.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create

Returns: The base Matrix B

Definition at line 692 of file WSymmetricSphericalHarmonic.h.

◆ calcEigenvalues()

template<typename T >

WValue< T > WSymmetricSphericalHarmonic< T >::calcEigenvalues ( size_t order )

static

Calc eigenvalues for SH elements.

Parameters

order The order of the spherical harmonic

Returns: The eigenvalues of the coefficients

Definition at line 722 of file WSymmetricSphericalHarmonic.h.

◆ calcFRTMatrix()

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcFRTMatrix ( size_t order )

static

Calculates the Funk-Radon-Transformation-Matrix P from the 2007 Descoteaux Paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging".

Parameters

order The order of the spherical harmonic

Returns: The Funk-Radon-Matrix P

Definition at line 763 of file WSymmetricSphericalHarmonic.h.

◆ calcGFA() [1/2]

template<typename T >

T WSymmetricSphericalHarmonic< T >::calcGFA ( std::vector< WUnitSphereCoordinates< T > > const & orientations ) const

Calculate the generalized fractional anisotropy for this ODF.

See: David S. Tuch, "Q-Ball Imaging", Magn. Reson. Med. 52, 2004, 1358-1372

Note: this only makes sense if this is an ODF, meaning funk-radon-transform was applied etc.

Parameters

orientations A vector of unit sphere coordinates.

Returns: The generalized fractional anisotropy.

Definition at line 450 of file WSymmetricSphericalHarmonic.h.

Referenced by WMCalculateGFA::perVoxelGFAFunc().

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◆ calcGFA() [2/2]

template<typename T >

T WSymmetricSphericalHarmonic< T >::calcGFA ( WMatrix< T > const & B ) const

Calculate the generalized fractional anisotropy for this ODF.

This version of the function uses precomputed base functions (because calculating the base function values is rather expensive). Use this version if you want to compute the GFA for multiple ODFs with the same base functions. The base function Matrix can be computed using

See also: calcBMatrix().

See: David S. Tuch, "Q-Ball Imaging", Magn. Reson. Med. 52, 2004, 1358-1372

Note: this only makes sense if this is an ODF, meaning funk-radon-transform was applied etc.

Parameters

B	The matrix of SH base functions.

Returns: The generalized fractional anisotropy.

Definition at line 494 of file WSymmetricSphericalHarmonic.h.

References WMatrix< T >::getNbCols(), and WMatrix< T >::getNbRows().

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◆ calcMatrixWithEigenvalues()

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcMatrixWithEigenvalues ( size_t order )

static

Calc matrix with the eigenvalues of the SH elements on its diagonal.

Parameters

order The order of the spherical harmonic

Returns: The matrix with the eigenvalues of the coefficients

Definition at line 739 of file WSymmetricSphericalHarmonic.h.

References WValue< T >::size().

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◆ calcSHToTensorSymMatrix() [1/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcSHToTensorSymMatrix ( std::size_t order )

static

Calculates a matrix that converts spherical harmonics to symmetric tensors of equal or lower order.

Parameters

order The order of the symmetric tensor.

Returns: the conversion matrix

Definition at line 782 of file WSymmetricSphericalHarmonic.h.

Referenced by WMCalculateTensors::moduleMain().

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◆ calcSHToTensorSymMatrix() [2/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcSHToTensorSymMatrix	(	std::size_t	order,
		const std::vector< WUnitSphereCoordinates< T > > &	orientations
	)

static

Calculates a matrix that converts spherical harmonics to symmetric tensors of equal or lower order.

Parameters

order	The order of the symmetric tensor.
orientations	A vector of at least (orderTensor+1) * (orderTensor+2) / 2 orientations.

Returns: the conversion matrix

Definition at line 802 of file WSymmetricSphericalHarmonic.h.

◆ calcSmoothingMatrix()

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::calcSmoothingMatrix ( size_t order )

static

This calcs the smoothing matrix L from the 2007 Descoteaux Paper "Regularized, Fast, and Robust Analytical Q-Ball Imaging".

Parameters

order The order of the spherical harmonic

Returns: The smoothing matrix L

Definition at line 751 of file WSymmetricSphericalHarmonic.h.

References WValue< T >::size().

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◆ getCoefficients()

template<typename T >

const WValue< T > & WSymmetricSphericalHarmonic< T >::getCoefficients

Returns the used coefficients (stored like in the mentioned 2007 Descoteaux paper).

Returns: coefficient list

Definition at line 342 of file WSymmetricSphericalHarmonic.h.

◆ getCoefficientsComplex()

template<typename T >

WValue< std::complex< T > > WSymmetricSphericalHarmonic< T >::getCoefficientsComplex

Returns the coefficients for the complex base.

Returns: coefficiend list

Definition at line 372 of file WSymmetricSphericalHarmonic.h.

◆ getCoefficientsSchultz()

template<typename T >

WValue< T > WSymmetricSphericalHarmonic< T >::getCoefficientsSchultz

Returns the coefficients for Schultz' SH base.

Returns: coefficient list

Definition at line 348 of file WSymmetricSphericalHarmonic.h.

◆ getOrder()

template<typename T >

size_t WSymmetricSphericalHarmonic< T >::getOrder

Return the order of the spherical harmonic.

Returns: order of SH

Definition at line 558 of file WSymmetricSphericalHarmonic.h.

◆ getSHFittingMatrix() [1/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::getSHFittingMatrix	(	const std::vector< WMatrixFixed< T, 3, 1 > > &	orientations,
		int	order,
		T	lambda,
		bool	withFRT
	)

static

This calculates the transformation/fitting matrix T like in the 2007 Descoteaux paper.

The orientations are given as WMatrixFixed< T, 3, 1 >.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create
lambda	Regularization parameter for smoothing matrix
withFRT	include the Funk-Radon-Transformation?

Returns: Transformation matrix

Definition at line 564 of file WSymmetricSphericalHarmonic.h.

◆ getSHFittingMatrix() [2/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::getSHFittingMatrix	(	const std::vector< WUnitSphereCoordinates< T > > &	orientations,
		int	order,
		T	lambda,
		bool	withFRT
	)

static

This calculates the transformation/fitting matrix T like in the 2007 Descoteaux paper.

The orientations are given as WUnitSphereCoordinates< T >.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create
lambda	Regularization parameter for smoothing matrix
withFRT	include the Funk-Radon-Transformation?

Returns: Transformation matrix

Definition at line 579 of file WSymmetricSphericalHarmonic.h.

References WMatrix< T >::transposed().

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◆ getSHFittingMatrixForConstantSolidAngle() [1/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::getSHFittingMatrixForConstantSolidAngle	(	const std::vector< WMatrixFixed< T, 3, 1 > > &	orientations,
		int	order,
		T	lambda
	)

static

This calculates the transformation/fitting matrix T like in the 2010 Aganj paper.

The orientations are given as WUnitSphereCoordinates< T >.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create
lambda	Regularization parameter for smoothing matrix

Returns: Transformation matrix

Definition at line 603 of file WSymmetricSphericalHarmonic.h.

Referenced by WMHARDIToSphericalHarmonics::moduleMain().

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◆ getSHFittingMatrixForConstantSolidAngle() [2/2]

template<typename T >

WMatrix< T > WSymmetricSphericalHarmonic< T >::getSHFittingMatrixForConstantSolidAngle	(	const std::vector< WUnitSphereCoordinates< T > > &	orientations,
		int	order,
		T	lambda
	)

static

This calculates the transformation/fitting matrix T like in the 2010 Aganj paper.

The orientations are given as WUnitSphereCoordinates< T >.

Parameters

orientations	The vector with the used orientation on the unit sphere (usually the gradients of the HARDI)
order	The order of the spherical harmonics intended to create
lambda	Regularization parameter for smoothing matrix