The Fourier Basis

#define	MX_FOURIER_BASIC   0
	Signifies the basic sine/cosine Fourier basis.
#define	MX_FOURIER_MODIFIED   1
	Signifies the modified Fourier basis.
template<class typeN>
int	mx::sigproc::makeFourierModeC (typeN im, typename typeN::Scalar m, typename typeN::Scalar n)
	Populate a single cosine mode.
template<class typeN>
int	mx::sigproc::makeFourierModeS (typeN im, typename typeN::Scalar m, typename typeN::Scalar n)
	Populate a single sine mode.
template<class typeN>
int	mx::sigproc::makeFourierMode (typeN im, typename typeN::Scalar m, typename typeN::Scalar n, int p)
	Populate a single basic Fourier mode.
template<class typeN>
int	mx::sigproc::makeModifiedFourierMode (typeN im, typename typeN::Scalar m, typename typeN::Scalar n, int p, typename typeN::Scalar ang=0)
	Populate a single modified Fourier mode.
template<class typeN>
int	mx::sigproc::makeModifiedFourierModeP (typeN im, typename typeN::Scalar m, typename typeN::Scalar n, typename typeN::Scalar ang=0)
	Populate a single modified Fourier +1 mode.
template<class typeN>
int	mx::sigproc::makeModifiedFourierModeM (typeN im, typename typeN::Scalar m, typename typeN::Scalar n, typename typeN::Scalar ang=0)
	Populate a single modified Fourier -1 mode.
bool	mx::sigproc::comp_fourierModeDef_PandV (const fourierModeDef &spf1, const fourierModeDef &spf2)
	Sorting functor for modes according to the Poyner and Veran (2005) standard.
int	mx::sigproc::makeFourierModeFreqs_PandV (std::vector< fourierModeDef > &spf, int N)
	Generate a Poyneer and Veran spatial frequency grid.
bool	mx::sigproc::comp_fourierModeDef (const fourierModeDef &spf1, const fourierModeDef &spf2)
	Sorting functor for modes according to the mx::AO standard.
int	mx::sigproc::makeFourierModeFreqs_Circ (std::vector< fourierModeDef > &spf, int N)
	Generate a circular spatial frequency grid.
int	mx::sigproc::fourierModeNumber (int m, int n, int p)
	Calculate the index of a Fourier mode given its (m,n,p) coordinates.
int	mx::sigproc::fourierModeCoordinates (int &m, int &n, int &p, int i)
	Calculate the (m,n,p) coordinates of a Fourier mode given its index.
int	mx::sigproc::makeFourierModeFreqs_Rect (std::vector< fourierModeDef > &spf, int N)
	Generate a rectangular spatial frequency grid.
template<typename cubeT>
int	mx::sigproc::fillFourierBasis (cubeT &cube, int dim, std::vector< fourierModeDef > spf, int basisType, typename cubeT::Scalar ang=0)
	Fill in a cube with a Fourier basis.
template<typename cubeT>
int	mx::sigproc::makeFourierBasis_PandV (cubeT &cube, int dim, int N, int basisType)
	Generate a rectangular modified Fourier basis with the Poyneer and Veran ordering.
template<typename cubeT>
int	mx::sigproc::makeFourierBasis_Circ (cubeT &cube, int dim, int N, int basisType)
	Generate a circular Fourier basis.
template<typename cubeT>
int	mx::sigproc::makeFourierBasis_Rect (cubeT &cube, int dim, int N, int basisType, typename cubeT::Scalar ang=0)
	Generate a rectangular Fourier basis.

Macro Definition Documentation

MX_FOURIER_BASIC

#define MX_FOURIER_BASIC   0

Signifies the basic sine/cosine Fourier basis.

Todo

need to handle references and r-values so this works directly with Eigen arrays.

Definition at line 55 of file fourierModes.hpp.

MX_FOURIER_MODIFIED

#define MX_FOURIER_MODIFIED   1

Signifies the modified Fourier basis.

Definition at line 60 of file fourierModes.hpp.

Referenced by mx::sigproc::fillFourierBasis(), and mx::AO::makeModfBasis().

Function Documentation

comp_fourierModeDef()

bool mx::sigproc::comp_fourierModeDef	(	const fourierModeDef &	spf1,
		const fourierModeDef &	spf2 )

Sorting functor for modes according to the mx::AO standard.

Given a spatial frequency vector \( \vec{k} = k_u \hat{u} + k_v \hat{v} \), sorts first by \( | \vec{k} | \), then by the angle from the u axis. Results in mode numbers increasing radially and counter-clockwise from the \( u \) axis.

Parameters

[in]	spf1	is an object of type fourierModeDef to compare with spf2
[in]	spf2	is an object of type fourierModeDef to compare with spf1

Returns

the result of (spf1 < spf2) according to the above rules.

Definition at line 409 of file fourierModes.hpp.

References mx::sigproc::fourierModeDef::m, mx::sigproc::fourierModeDef::n, mx::sigproc::fourierModeDef::p, and mx::math::two_pi().

Referenced by makeFourierModeFreqs_Circ().

comp_fourierModeDef_PandV()

bool mx::sigproc::comp_fourierModeDef_PandV	(	const fourierModeDef &	spf1,
		const fourierModeDef &	spf2 )

Sorting functor for modes according to the Poyner and Veran (2005) standard.

Follows the conventions of Poyneer & Veran (2005) [26]

Parameters

[in]	spf1	is an object of type fourierModeDef to compare with spf2
[in]	spf2	is an object of type fourierModeDef to compare with spf1

Returns

the result of (spf1 < spf2) according to the above rules.

Definition at line 314 of file fourierModes.hpp.

References mx::sigproc::fourierModeDef::m, mx::sigproc::fourierModeDef::n, and mx::sigproc::fourierModeDef::p.

Referenced by makeFourierModeFreqs_PandV().

fillFourierBasis()

template<typename cubeT>

int mx::sigproc::fillFourierBasis	(	cubeT &	cube,
		int	dim,
		std::vector< fourierModeDef >	spf,
		int	basisType,
		typename cubeT::Scalar	ang = 0 )

Fill in a cube with a Fourier basis.

Fills the cube with either the basic or modified Fourier basis for the modes specified.

Template Parameters

cubeT

is an eigen-like cube, e.g. mx::improc::eigenCube.

Parameters

[out]	cube	will be allocated to hold and will be filled with the modes
[in]	dim	is the linear size of the maps
[in]	spf	is a vector of mode definitions to use for each mode
[in]	basisType	is either MX_FOURIER_MODIFIED or MX_FOURIER_BASIC
[in]	ang

Definition at line 622 of file fourierModes.hpp.

References makeFourierMode(), makeModifiedFourierMode(), and MX_FOURIER_MODIFIED.

Referenced by makeFourierBasis_Circ(), makeFourierBasis_PandV(), and makeFourierBasis_Rect().

fourierModeCoordinates()

int mx::sigproc::fourierModeCoordinates	(	int &	m,
		int &	n,
		int &	p,
		int	i )

Calculate the (m,n,p) coordinates of a Fourier mode given its index.

The index increases counter-clockwise from the m axis in squares, with p==-1 being even and p==1 being odd.

Returns

0 on success, a negative number otherwise

Return values

-1

if i < 0

Parameters

[out]	m	is the m spatial frequency coordinate
[out]	n	is the n spatial frequency coordinage
[out]	p	is the parity, +/-1, for sine or cosine or specifiying the modified Fourier mode.
[in]	i	is the mode index.

Definition at line 542 of file fourierModes.hpp.

References mx::invalidarg, and mx::internal::mxlib_error_report().

Referenced by makeFourierModeFreqs_Rect().

fourierModeNumber()

int mx::sigproc::fourierModeNumber	(	int	m,
		int	n,
		int	p )

Calculate the index of a Fourier mode given its (m,n,p) coordinates.

The index increases counter-clockwise from the m axis in squares, with p==-1 being even and p==1 being odd.

Parameters

[in]	m	is the m spatial frequency coordinate
[in]	n	is the n spatial frequency coordinage
[in]	p	is the parity, +/-1, for sine or cosine or specifiying the modified Fourier mode.

Returns

the index of the Fourier mode on success

-1 on an error.

Definition at line 498 of file fourierModes.hpp.

References mx::invalidarg, and mx::internal::mxlib_error_report().

makeFourierBasis_Circ()

template<typename cubeT>

int mx::sigproc::makeFourierBasis_Circ	(	cubeT &	cube,
		int	dim,
		int	N,
		int	basisType )

Generate a circular Fourier basis.

Fills the cube with modes generated by makeFourierMode or makeModifiedFourierMode using the mode grid generated by makeFourierModeFreqs_Circ.

Parameters

[out]	cube	will be allocated to hold and will be filled with the modes
[in]	dim	is the linear size of the maps, each is
[in]	N	is the linear number of degrees of freedom. The number of modes is ((N+1)(N+1) - 1).
[in]	basisType	is either MX_FOURIER_MODIFIED or MX_FOURIER_BASIC

Template Parameters

cubeT

is an eigen-like cube, e.g. mx::improc::eigenCube.

Definition at line 689 of file fourierModes.hpp.

References fillFourierBasis(), and makeFourierModeFreqs_Circ().

makeFourierBasis_PandV()

template<typename cubeT>

int mx::sigproc::makeFourierBasis_PandV	(	cubeT &	cube,
		int	dim,
		int	N,
		int	basisType )

Generate a rectangular modified Fourier basis with the Poyneer and Veran ordering.

Fills the cube with modes generated by makeFourierMode or makeModifiedFourierMode using the mode grid generated by makeFourierModeFreqs_PandV. This follows the convention of Poyneer and Veran (2005) [26] for mode numbering.

Parameters

[out]	cube	will be allocated to hold and will be filled with the modes
[in]	dim	is the linear size of the maps, each is
[in]	N	is the linear number of degrees of freedom. The number of modes is ((N+1)(N+1) - 1).
[in]	basisType	is either MX_FOURIER_MODIFIED or MX_FOURIER_BASIC

Template Parameters

cubeT

is an eigen-like cube, e.g. mx::improc::eigenCube.

Definition at line 665 of file fourierModes.hpp.

References fillFourierBasis(), and makeFourierModeFreqs_PandV().

makeFourierBasis_Rect()

template<typename cubeT>

int mx::sigproc::makeFourierBasis_Rect	(	cubeT &	cube,
		int	dim,
		int	N,
		int	basisType,
		typename cubeT::Scalar	ang = 0 )

Generate a rectangular Fourier basis.

Fills the cube with modes generated by makeFourierMode or makeModifiedFourierMode using the mode grid generated by makeFourierModeFreqs_Rect.

Parameters

[out]	cube	will be allocated to hold and will be filled with the modes
[in]	dim	is the linear size of the maps, each is
[in]	N	is the linear number of degrees of freedom. The number of modes is ((N+1)(N+1) - 1).
[in]	basisType	is either MX_FOURIER_MODIFIED or MX_FOURIER_BASIC

Template Parameters

cubeT

is an eigen-like cube, e.g. mx::improc::eigenCube.

Definition at line 713 of file fourierModes.hpp.

References fillFourierBasis(), and makeFourierModeFreqs_Rect().

Referenced by mx::AO::makeModfBasis().

makeFourierMode()

template<class typeN>

int mx::sigproc::makeFourierMode	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n,
		int	p )

Populate a single basic Fourier mode.

Makes a 2D image of the basic sine or cosine mode. Calls makeFourierModeC and makeFourierModeS.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	specifies the spatial frequency in the v direction
[in]	p	species sine (-1) or cosine (+1)

Return values

0	on success

Template Parameters

typeN

is an Eigen-like reference type.

Definition at line 169 of file fourierModes.hpp.

References mx::invalidarg, makeFourierModeC(), makeFourierModeS(), and mx::internal::mxlib_error_report().

Referenced by fillFourierBasis().

makeFourierModeC()

template<class typeN>

int mx::sigproc::makeFourierModeC	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n )

Populate a single cosine mode.

Makes a 2D image of the cosine mode:

\[M_c(\vec{q}) = \cos\left( 2\pi\frac{m}{D}u + 2\pi\frac{n}{D}v \right) \]

where

\[\vec{q} = u \hat{u} + v \hat{v} \]

and \( D \) is taken to be the maximum of the number of columns and rows in the image.

Return values

0	on success

Template Parameters

typeN

is an Eigen-like reference type.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	n specifies the spatial frequency in the v direction

Definition at line 78 of file fourierModes.hpp.

References mx::math::two_pi().

Referenced by makeFourierMode().

makeFourierModeFreqs_Circ()

int mx::sigproc::makeFourierModeFreqs_Circ	(	std::vector< fourierModeDef > &	spf,
		int	N )

Generate a circular spatial frequency grid.

Todo

use push_back instead of resize

Parameters

spf
N	is the linear number of degrees of freedom. The number of modes is ((N+1)(N+1) - 1).

Definition at line 441 of file fourierModes.hpp.

References comp_fourierModeDef(), mx::internal::mxlib_error_report(), mx::math::pi(), and mx::sizeerr.

Referenced by makeFourierBasis_Circ().

makeFourierModeFreqs_PandV()

int mx::sigproc::makeFourierModeFreqs_PandV	(	std::vector< fourierModeDef > &	spf,
		int	N )

Generate a Poyneer and Veran spatial frequency grid.

Follows the convention of Poyneer and Veran (2005) [26]

Todo

use push_back instead of resize

Parameters

spf
N	is the linear number of degrees of freedom.

Definition at line 341 of file fourierModes.hpp.

References comp_fourierModeDef_PandV(), mx::internal::mxlib_error_report(), and mx::sizeerr.

Referenced by makeFourierBasis_PandV().

makeFourierModeFreqs_Rect()

int mx::sigproc::makeFourierModeFreqs_Rect	(	std::vector< fourierModeDef > &	spf,
		int	N )

Generate a rectangular spatial frequency grid.

Parameters

spf	is a vector of fourierModeDef structures, which will be resized and populated.
N	is the linear number of degrees of freedom. The number of modes is ((N+1)(N+1) - 1).

Definition at line 593 of file fourierModes.hpp.

References fourierModeCoordinates().

Referenced by mx::AO::analysis::fourierTemporalPSD< _realT, aosysT >::analyzePSDGrid(), mx::AO::analysis::fourierCovarMap(), mx::AO::analysis::fourierTemporalPSD< _realT, aosysT >::intensityPSD(), makeFourierBasis_Rect(), and mx::AO::analysis::fourierTemporalPSD< _realT, aosysT >::makePSDGrid().

makeFourierModeS()

template<class typeN>

int mx::sigproc::makeFourierModeS	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n )

Populate a single sine mode.

Makes a 2D image of the cosine mode:

\[M_s(\vec{q}) = \sin\left( 2\pi\frac{m}{D}u + 2\pi\frac{n}{D}v \right) \]

where

\[\vec{q} = u \hat{u} + v \hat{v} \]

and \( D \) is taken to be the maximum of the number of columns and rows in the image.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	specifies the spatial frequency in the v direction

Return values

0	on success

Template Parameters

typeN

is an Eigen-like reference type.

Definition at line 128 of file fourierModes.hpp.

References mx::math::two_pi().

Referenced by makeFourierMode().

makeModifiedFourierMode()

template<class typeN>

int mx::sigproc::makeModifiedFourierMode	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n,
		int	p,
		typename typeN::Scalar	ang = 0 )

Populate a single modified Fourier mode.

Makes a 2D image of the modified fourier mode:

\[ M_p(\vec{q}) = \cos\left( 2\pi \frac{m}{D}u + 2\pi\frac{n}{D}v \right) + p \sin\left( 2\pi\frac{m}{D}u + 2\pi\frac{n}{D}v \right) \]

where \( p = \pm 1 \), \( \vec{q} = u \hat{u} + v \hat{v} \), and \( D \) is taken to be the maximum of the number of columns in the image.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	specifies the spatial frequency in the v direction
[in]	p	is +/- 1 specifying which modified Fourier mode

Return values

0	on success

Template Parameters

typeN

is an Eigen-like reference type.

Definition at line 200 of file fourierModes.hpp.

References mx::math::dtor(), mx::invalidarg, mx::internal::mxlib_error_report(), and mx::math::two_pi().

Referenced by fillFourierBasis(), makeModifiedFourierModeM(), and makeModifiedFourierModeP().

makeModifiedFourierModeM()

template<class typeN>

int mx::sigproc::makeModifiedFourierModeM	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n,
		typename typeN::Scalar	ang = 0 )

Populate a single modified Fourier -1 mode.

See makeModifiedFourierMode.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	specifies the spatial frequency in the v direction

Template Parameters

typeN

is an Eigen-like reference type.

Definition at line 282 of file fourierModes.hpp.

References makeModifiedFourierMode().

makeModifiedFourierModeP()

template<class typeN>

int mx::sigproc::makeModifiedFourierModeP	(	typeN	im,
		typename typeN::Scalar	m,
		typename typeN::Scalar	n,
		typename typeN::Scalar	ang = 0 )

Populate a single modified Fourier +1 mode.

See makeModifiedFourierMode.

Parameters

[out]	im	is an Eigen-like image
[in]	m	specifies the spatial frequency in the u direction
[in]	n	specifies the spatial frequency in the v direction

Template Parameters

typeN

is an Eigen-like reference type.

Definition at line 264 of file fourierModes.hpp.

References makeModifiedFourierMode().