Function Documentation

◆ noll_j()

int mx::sigproc::noll_j	(	unsigned	n,
		int	m
	)

Get the Noll index j corresponding to Zernike coefficients n,m.

Calculates the value j for(n,m) following Noll (1976) [16] See also: http://en.wikipedia.org/wiki/Zernike_polynomials

Return values

>=	0 on success
-1	on error (n-m odd)

Parameters

[in]	n	n the radial index of the Zernike polynomial
[in]	m	m the azimuthal index of the Zernnike polynomial.

◆ noll_nm()

int mx::sigproc::noll_nm	(	int &	n,
		int &	m,
		int	j
	)

Get the Zernike coefficients n,m corrresponding the Noll index j.

Calculates the values of (n,m) for an index j following Noll (1976) [16] See also: http://en.wikipedia.org/wiki/Zernike_polynomials

If j is odd, this returns m <= 0.

Return values

0	on success
-1	on error (j < 1)

Test:: Scenario: testing noll_nm [test doc]

Parameters

[out]	n	n the radial index of the Zernike polynomial
[out]	m	m the azimuthal index of the Zernnike polynomial. m < 0 if j odd.
[in]	j	j the Noll index, j > 0.

Definition at line 35 of file zernike.cpp.

References MXE_INVALIDARG, mxError, and mx::math::func::sign().

Referenced by SCENARIO(), mx::AO::analysis::zernikeTemporalPSD< _realT, aosysT >::singleLayerPSD(), mx::sigproc::zernike(), mx::sigproc::zernike(), mx::sigproc::zernikeQNorm(), and mx::sigproc::zernikeQNorm().

◆ nZernRadOrd()

int mx::sigproc::nZernRadOrd ( unsigned n )

Get the number of Zernikes up to and including a radial order.

Calculates the total number of Zernike polynomials through radial order n. See Noll (1976) [16] See also: http://en.wikipedia.org/wiki/Zernike_polynomials

Return values

the	number of
-1	on error (n-m odd)

Parameters

n	[n] the radial order

◆ zernike() [1/7]

template<typename arrayT , typename calcRealT >

int mx::sigproc::zernike	(	arrayT &	arr,
		int	j,
		typename arrayT::Scalar	rad = `-1`
	)

Fill in an Eigen-like array with a Zernike polynomial.

The geometric center of the array, 0.5*(arr.rows()-1), 0.5*(arr.cols()-1), is used as the center. Sets any pixel which is at rad <= r < rad+0.5 pixels to rho = 1, to be consistent with mx::circularPupil

Parameters

[out]	arr	is the allocated array with an Eigen-like interface. The rows() and cols() members are used to size the polynomial.
[in]	j	is the Noll index of the polynomial
[in]	rad	[optional] is the desired radius. If rad <= 0, then the maximum radius based on dimensions of m is used.

Template Parameters

arrayT	is an Eigen-like array of real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Definition at line 483 of file zernike.hpp.

◆ zernike() [2/7]

template<typename arrayT , typename calcRealT >

int mx::sigproc::zernike	(	arrayT &	arr,
		int	j,
		typename arrayT::Scalar	xcen,
		typename arrayT::Scalar	ycen,
		typename arrayT::Scalar	rad = `-1`
	)

Fill in an Eigen-like array with a Zernike polynomial.

Sets any pixel which is at rad <= r <= rad+0.5 pixels to rho = 1, to be consistent with mx::circularPupil

Parameters

[out]	arr	is the allocated array with an Eigen-like interface. The rows() and cols() members are used to size the polynomial.
[in]	j	is the Noll index of the polynomial
[in]	xcen	is the x coordinate of the desired center of the polynomial, in pixels
[in]	ycen	is the y coordinate of the desired center of the polynomial, in pixels
[in]	rad	[optional] is the desired radius. If rad <= 0, then the maximum radius based on dimensions of m is used.

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Definition at line 438 of file zernike.hpp.

References mx::sigproc::noll_nm().

◆ zernike() [3/7]

template<typename arrayT , typename calcRealT >

int mx::sigproc::zernike	(	arrayT &	arr,
		int	n,
		int	m,
		typename arrayT::Scalar	rad = `-1`
	)

Fill in an Eigen-like array with a Zernike polynomial.

The geometric center of the array, 0.5*(arr.rows()-1), 0.5*(arr.cols()-1), is used as the center. Sets any pixel which is at rad <= r < rad+0.5 pixels to rho = 1, to be consistent with mx::circularPupil

Parameters

[out]	arr	is the allocated array with an Eigen-like interface. The rows() and cols() members are used to size the polynomial.
[in]	j	is the Noll index of the polynomial
[in]	rad	[optional] is the desired radius. If rad <= 0, then the maximum radius based on dimensions of m is used.

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Definition at line 463 of file zernike.hpp.

◆ zernike() [4/7]

template<typename arrayT , typename calcRealT , int overscan = 2>

int mx::sigproc::zernike	(	arrayT &	arr,
		int	n,
		int	m,
		typename arrayT::Scalar	xcen,
		typename arrayT::Scalar	ycen,
		typename arrayT::Scalar	rad = `-1`
	)

Fill in an Eigen-like array with a Zernike polynomial.

Sets any pixel which is at rad <= r < rad+0.5 pixels to rho = 1, to be consistent with mx::circularPupil

Parameters

[out]	arr	is the allocated array with an Eigen-like interface. The rows() and cols() members are used to size the polynomial.
[in]	n	is the radial index of the polynomial
[in]	m	is the azimuthal index of the polynomial
[in]	xcen	is the x coordinate of the desired center of the polynomial, in pixels
[in]	ycen	is the y coordinate of the desired center of the polynomial, in pixels
[in]	rad	[optional] is the desired radius. If rad <= 0, then the maximum radius based on dimensions of m is used.

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Definition at line 372 of file zernike.hpp.

References mx::sigproc::zernike(), and mx::sigproc::zernikeRCoeffs().

◆ zernike() [5/7]

template<typename realT , typename calcRealT >

realT mx::sigproc::zernike	(	realT	rho,
		realT	phi,
		int	j
	)

Calculate the value of a Zernike radial polynomial at a given radius and angle.

Return values

-9999	indicates a possible error
R	the value of the Zernike radial polynomial otherwise

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Parameters

[in]	rho	the radial coordinate, \( 0 \le \rho \le 1 \).
[in]	phi	the azimuthal angle (in radians)
[in]	j	the Noll index of the Zernike polynomial.

Definition at line 335 of file zernike.hpp.

References mx::sigproc::noll_nm().

◆ zernike() [6/7]

template<typename realT , typename calcRealT >

realT mx::sigproc::zernike	(	realT	rho,
		realT	phi,
		int	n,
		int	m
	)

Calculate the value of a Zernike radial polynomial at a given radius and angle.

Return values

-9999	indicates a possible error
R	the value of the Zernike radial polynomial otherwise

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Parameters

[in]	rho	the radial coordinate, \( 0 \le \rho \le 1 \).
[in]	phi	the azimuthal angle (in radians)
[in]	n	the radial index of the Zernike polynomial.
[in]	m	the azimuthal index of the Zernike polynomial.

Definition at line 301 of file zernike.hpp.

◆ zernike() [7/7]

template<typename realT , typename calcRealT >

realT mx::sigproc::zernike	(	realT	rho,
		realT	phi,
		int	n,
		int	m,
		std::vector< calcRealT > &	c
	)

Calculate the value of a Zernike radial polynomial at a given radius and angle.

Parameters

[in]	rho	is the radial coordinate, \( 0 \le \rho \le 1 \).
[in]	phi	is the azimuthal angle (in radians)
[in]	n	is the radial index of the Zernike polynomial.
[in]	m	is the azimuthal index of the Zernike polynomial.
[in]	c	is contains the radial polynomial coeeficients, and must be of length \( 0.5(n-m)+1\).

Return values

-9999	indicates a possible error
R	the value of the Zernike radial polynomial otherwise

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Definition at line 254 of file zernike.hpp.

References mx::math::six_fifths().

Referenced by mx::sigproc::zernike().

◆ zernikeBasis()

template<typename cubeT , typename calcRealT >

int mx::sigproc::zernikeBasis	(	cubeT &	cube,
		typename cubeT::Scalar	rad = `-1`,
		int	minj = `2`
	)

Fill in an Eigencube-like array with Zernike polynomials in Noll order.

The cube is pre-allocated to set the image size and the number of modes.

Returns: 0 on success; -1 on error

Template Parameters

cubeT	is an Eigencube-like array with real floating point type
calcRealT	is a real floating type used for internal calculations, should be at least double

Parameters

	cube	[in.out] the pre-allocated cube which will be filled with the Zernike basis
[in]	rad	[optional] the radius of the aperture. If -1 then the full image size is used.
[in]	minj	[optional] the minimum j value to include. The default is j=2, which skips piston (j=1).

Definition at line 501 of file zernike.hpp.

◆ zernikePAstig()

template<typename realT >

realT mx::sigproc::zernikePAstig ( const realT & kD )

Calculate the spatial power spectrum of Astigmatism.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 732 of file zernike.hpp.

References mx::math::func::jincN(), and mx::math::six_fifths().

◆ zernikePComa()

template<typename realT >

realT mx::sigproc::zernikePComa ( const realT & kD )

Calculate the spatial power spectrum of Coma.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 739 of file zernike.hpp.

References mx::math::func::jincN(), and mx::math::six_fifths().

◆ zernikePDefocus()

template<typename realT >

realT mx::sigproc::zernikePDefocus ( const realT & kD )

Calculate the spatial power spectrum of Defocus.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 725 of file zernike.hpp.

References mx::math::func::jincN(), and mx::math::six_fifths().

◆ zernikePPiston()

template<typename realT >

realT mx::sigproc::zernikePPiston ( const realT & kD )

Calculate the spatial power spectrum of Piston.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 711 of file zernike.hpp.

References mx::math::func::jinc(), and mx::math::six_fifths().

◆ zernikePTipTilt()

template<typename realT >

realT mx::sigproc::zernikePTipTilt ( const realT & kD )

Calculate the spatial power spectrum of Tip & Tilt.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 718 of file zernike.hpp.

References mx::math::func::jincN(), and mx::math::six_fifths().

◆ zernikePTrefoil()

template<typename realT >

realT mx::sigproc::zernikePTrefoil ( const realT & kD )

Calculate the spatial power spectrum of Trefoil.

Parameters

[in] kD Spatial frequency in diameter units, i.e. cycles per aperture.

Definition at line 746 of file zernike.hpp.

References mx::math::func::jincN(), and mx::math::six_fifths().

◆ zernikeQ()

template<typename realT >

std::complex< realT > mx::sigproc::zernikeQ	(	realT	k,
		realT	phi,
		int	n,
		int	m
	)

Calculate the square-normed Fourier transform of a Zernike polynomial at position (k,phi)

Implements Equation (8) of Noll (1976) [16].

Todo:: need a more robust jinc_n function for n > 1

Test:: Scenario: testing zernikeQNorm [test doc]

Returns: the value of |Q(k,phi)|^2

Template Parameters

realT is the floating point type used for arithmetic

Parameters

[in]	k	the radial coordinate of normalized spatial frequency. This is in the [16] convention of cycles-per-radius.
[in]	phi	the azimuthal coordinate of normalized spatial frequency
[in]	n	the Zernike polynomial n
[in]	m	the Zernike polynomial m

Definition at line 541 of file zernike.hpp.

References mx::math::func::bessel_j(), and mx::math::six_fifths().

◆ zernikeQNorm() [1/3]

template<typename arrayT >

int mx::sigproc::zernikeQNorm	(	arrayT &	arr,
		arrayT &	k,
		arrayT &	phi,
		int	j
	)

Fill in an Eigen-like array with the square-normed Fourier transform of a Zernike polynomial.

The array is filled in with the values of |Q(k,phi)|^2 according to Equation (8) of Noll (1976) [16].

Test:: Scenario: testing zernikeQNorm [test doc]

Returns: 0 on success; -1 on error

Template Parameters

arrayT is the Eigen-like array type. Arithmetic will be done in arrayT::Scalar.

Parameters

[out]	arr	the allocated array. The rows() and cols() members are used to size the transform.
[in]	k	the normalized spatial frequency magnitude at each pixel. This is in the [16] convention of cycles-per-radius.
[in]	phi	the spatial frequency angle at each pixel
[in]	j	the polynomial index in the Noll convention [16]

Definition at line 675 of file zernike.hpp.

References MXE_INVALIDARG, mxError, mx::sigproc::noll_nm(), and mx::sigproc::zernikeQNorm().

◆ zernikeQNorm() [2/3]

template<typename realT >

realT mx::sigproc::zernikeQNorm	(	realT	k,
		realT	phi,
		int	j
	)

Calculate the square-normed Fourier transform of a Zernike polynomial at position (k,phi)

Implements Equation (8) of Noll (1976) [16].

Returns: the value of |Q(k,phi)|^2

Template Parameters

realT is the floating point type used for arithmetic

Parameters

[in]	k	the radial coordinate of normalized spatial frequency. This is in the [16] convention of cycles-per-radius.
[in]	phi	the azimuthal coordinate of normalized spatial frequency
[in]	j	the Zernike polynomial index j (Noll convention)

Definition at line 651 of file zernike.hpp.

References mx::sigproc::noll_nm(), and mx::sigproc::zernikeQNorm().

◆ zernikeQNorm() [3/3]

template<typename realT >

realT mx::sigproc::zernikeQNorm	(	realT	k,
		realT	phi,
		int	n,
		int	m
	)

Calculate the square-normed Fourier transform of a Zernike polynomial at position (k,phi)

Implements Equation (8) of Noll (1976) [16].

Todo:: need a more robust jinc_n function for n > 1

Test:: Scenario: testing zernikeQNorm [test doc]

Returns: the value of |Q(k,phi)|^2

Template Parameters

realT is the floating point type used for arithmetic

Parameters

[in]	k	the radial coordinate of normalized spatial frequency. This is in the [16] convention of cycles-per-radius.
[in]	phi	the azimuthal coordinate of normalized spatial frequency
[in]	n	the Zernike polynomial n
[in]	m	the Zernike polynomial m

Definition at line 595 of file zernike.hpp.

References mx::math::func::bessel_j(), and mx::math::six_fifths().

Referenced by mx::AO::analysis::F_zernike(), SCENARIO(), mx::sigproc::zernikeQNorm(), and mx::sigproc::zernikeQNorm().

◆ zernikeR() [1/2]

template<typename realT , typename calcRealT >

realT mx::sigproc::zernikeR	(	realT	rho,
		int	n,
		int	m
	)

Calculate the value of a Zernike radial polynomial at a given separation.

Return values

-9999	indicates a possible error
R	the value of the Zernike radial polynomial otherwise

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calculations, should be at least double

Parameters

[in]	rho	the radial coordinate, \( 0 \le \rho \le 1 \).
[in]	n	the radial index of the Zernike polynomial.
[in]	m	the azimuthal index of the Zernike polynomial.

Definition at line 208 of file zernike.hpp.

References mx::sigproc::zernikeRCoeffs().

◆ zernikeR() [2/2]

template<typename realT , typename calcRealT >

realT mx::sigproc::zernikeR	(	realT	rho,
		int	n,
		int	m,
		std::vector< calcRealT > &	c
	)

Calculate the value of a Zernike radial polynomial at a given separation.

Return values

-9999	indicates a possible error
R	the value of the Zernike radial polynomial otherwise

Template Parameters

realT	is a real floating type
calcRealT	is a real floating type used for internal calcs, should be at least double.

Parameters

[in]	rho	the radial coordinate, \( 0 \le \rho \le 1 \).
[in]	n	the radial index of the Zernike polynomial.
[in]	m	the azimuthal index of the Zernike polynomial.
[in]	c	contains the radial polynomial coeeficients, and must be of length \( 0.5(n-m)+1\).

Definition at line 157 of file zernike.hpp.

References MXE_INVALIDARG, and mxError.

◆ zernikeRCoeffs()

template<typename realT >

int mx::sigproc::zernikeRCoeffs	(	std::vector< realT > &	c,
		int	n,
		int	m
	)

Calculate the coefficients of a Zernike radial polynomial.

Return values

0	on success
-1	on error

Template Parameters

realT is a real floating type

Parameters

[out]	c	allocated to length \( 0.5(n-m)+1\) and filled with the coefficients.
[in]	n	the radial index of the Zernike polynomial.
[in]	m	the azimuthal index of the Zernike polynomial.

Definition at line 101 of file zernike.hpp.

References MXE_INVALIDARG, and mxError.

Referenced by mx::sigproc::zernike(), and mx::sigproc::zernikeR().