Utilities for working with the Gaussian function

Functions
template<typename floatT >
constexpr floatT	mx::math::func::twosqrt2log2 ()
	Constant to convert between the Gaussian width parameter and FWHM. More...

template<typename floatT >
floatT	mx::math::func::fwhm2sigma (floatT fw)
	Convert from FWHM to the Gaussian width parameter. More...

template<typename floatT >
floatT	mx::math::func::sigma2fwhm (floatT sig)
	Convert from Gaussian width parameter to FWHM. More...

template<typename realT >
realT	mx::math::func::gaussian (const realT x, const realT G0, const realT G, const realT x0, const realT sigma)
	Find value at position (x) of the 1D arbitrarily-centered symmetric Gaussian. More...

template<typename realT >
realT	mx::math::func::gaussian2D (const realT x, const realT y, const realT G0, const realT G, const realT x0, const realT y0, const realT sigma)
	Find value at position (x,y) of the 2D arbitrarily-centered symmetric Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D (realT *arr, size_t nx, size_t ny, const realT G0, const realT G, const realT x0, const realT y0, const realT sigma)
	Fill in an array with the 2D arbitrarily-centered symmetric Gaussian. More...

template<typename realT >
realT	mx::math::func::gaussian2D (const realT x, const realT y, const realT G0, const realT G, const realT x0, const realT y0, const realT a, const realT b, const realT c)
	Find value at position (x,y) of the 2D general elliptical Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D_gen2rot (realT &sigma_x, realT &sigma_y, realT &theta, const realT a, const realT b, const realT c)
	Convert from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) for the elliptical Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D_rot2gen (realT &a, realT &b, realT &c, const realT sigma_x, const realT sigma_y, const realT theta)
	Convert from ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) to (a,b,c) for the elliptical Gaussian. More...

template<typename realT >
realT	mx::math::func::gaussian2D_ang (const realT x, const realT y, const realT G0, const realT G, const realT x0, const realT y0, const realT sigma_x, const realT sigma_y, const realT theta)
	Find value at position (x,y) of the 2D rotated elliptical Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D (realT *arr, size_t nx, size_t ny, const realT G0, const realT G, const realT x0, const realT y0, const realT a, const realT b, const realT c)
	Fill in an array with the 2D general elliptical Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D_ang (realT *arr, size_t nx, size_t ny, const realT G0, const realT G, const realT x0, const realT y0, const realT sigma_x, const realT sigma_y, const realT theta)
	Fill in an array with the 2D general elliptical Gaussian. More...

template<typename realT >
void	mx::math::func::gaussian2D_jacobian (realT *j, const realT x, const realT y, const realT G0, const realT G, const realT x0, const realT y0, const realT a, const realT b, const realT c)
	Calculate the Jacobian at position (x,y) for the 2D general elliptical Gaussian. More...

Function Documentation

◆ fwhm2sigma()

template<typename floatT >

floatT mx::math::func::fwhm2sigma ( floatT fw )

Convert from FWHM to the Gaussian width parameter.

Performs the conversion:

\( \sigma = 2\sqrt{2\log2}FWHM \)

Returns: the converted value of the Gaussian width parameter

Parameters

[in] fw the full-width at half maximum

Definition at line 65 of file gaussian.hpp.

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

◆ gaussian()

template<typename realT >

realT mx::math::func::gaussian	(	const realT	x,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	sigma
	)

Find value at position (x) of the 1D arbitrarily-centered symmetric Gaussian.

Computes: \( G(x) = G_0 + G\exp[-(0.5/\sigma^2)((x-x_0)^2)]\)

Returns: the value of the 1D arbitrarily-centered symmetric Gaussian at (x)

Template Parameters

realT is the type to use for arithmetic

Parameters

[in]	x	is the x-position at which to evaluate the Gaussian
[in]	G0	is the constant to add to the Gaussian
[in]	G	is the scaling factor (peak height = G-G0)
[in]	x0	is the x-coordinate of the center
[in]	sigma	is the width of the Gaussian.

Definition at line 100 of file gaussian.hpp.

References mx::astro::constants::G(), and mx::astro::constants::sigma().

◆ gaussian2D() [1/4]

template<typename realT >

realT mx::math::func::gaussian2D	(	const realT	x,
		const realT	y,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	a,
		const realT	b,
		const realT	c
	)

Find value at position (x,y) of the 2D general elliptical Gaussian.

Computes:

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

where, for a counter-clockwise rotation \( \theta \), we have

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

In this version the parameters are specified directly as (a,b,c), in particular avoiding the trig function calls. This should be much more efficient, and so this version should be used inside fitting routines, etc. However note that the matrix {a b}{b c} must be positive-definite otherwise infinities can result from the argument of the exponent being positive.

The functions gaussian2D_gen2rot() and gaussian2D_rot2gen() provide conversions from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) and back. The function gaussian2D_ang() is a wrapper for this function, which instead accepts ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) as inputs.

Returns: the value of the 2D elliptical Gaussian at (x,y)

Template Parameters

realT is the type to use for arithmetic

Parameters

[in]	x	the x-position at which to evaluate the Gaussian
[in]	y	the y-positoin at which to evaluate the Gaussian
[in]	G0	the constant to add to the Gaussian
[in]	G	the scaling factor (peak = G-G0)
[in]	x0	the x-coordinate of the center
[in]	y0	the y-coordinate of the center
[in]	a	the first rotation and scaling factor
[in]	b	the second rotation and scaling factor
[in]	c	the third rotation and scaling factor

Definition at line 203 of file gaussian.hpp.

References mx::astro::constants::c(), and mx::astro::constants::G().

◆ gaussian2D() [2/4]

template<typename realT >

realT mx::math::func::gaussian2D	(	const realT	x,
		const realT	y,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	sigma
	)

Find value at position (x,y) of the 2D arbitrarily-centered symmetric Gaussian.

Computes:

\[ f(x,y) = G_0 + G \exp[-(0.5/\sigma^2)((x-x_0)^2+(y-y_0)^2)] \]

Returns: the value of the 2D arbitrarily-centered symmetric Gaussian at (x,y)

Template Parameters

realT is the type to use for arithmetic

Test:: Scenario: Verify direction and accuracy of various image shifts [test doc]

Parameters

[in]	x	the x-position at which to evaluate the Gaussian
[in]	y	the y-positoin at which to evaluate the Gaussian
[in]	G0	the constant to add to the Gaussian
[in]	G	the scaling factor (peak height is G-G0)
[in]	x0	the x-coordinate of the center
[in]	y0	the y-coordinate of the center
[in]	sigma	the width of the Gaussian.

Definition at line 127 of file gaussian.hpp.

References mx::astro::constants::G(), and mx::astro::constants::sigma().

◆ gaussian2D() [3/4]

template<typename realT >

void mx::math::func::gaussian2D	(	realT *	arr,
		size_t	nx,
		size_t	ny,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	a,
		const realT	b,
		const realT	c
	)

Fill in an array with the 2D general elliptical Gaussian.

At each pixel (x,y) of the array this computes:

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

where, for a counter-clockwise rotation \( \theta \), we have

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

In this version the parameters are specified directly as (a,b,c), in particular avoiding the trig function calls. This should be much more efficient, and so this version should be used inside fitting routines, etc.

The functions gaussian2D_gen2rot() and gaussian2D_rot2gen() provide conversions from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) and back. The function gaussian2D_ang() is a wrapper for this function, which instead accepts ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) as inputs.

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	arr	the native array to populate with the Gaussian function
[in]	nx	the x-size of the array, in pixels
[in]	ny	the y-size of the array, in pixels
[in]	G0	the constant to add to the Gaussian
[in]	G	the scaling factor (peak = G)
[in]	x0	the x-coordinate of the center, in pixels
[in]	y0	the y-coordinate of the center, in pixels
[in]	a	is the first rotation and scaling factor
[in]	b	is the second rotation and scaling factor
[in]	c	is the third rotation and scaling factor

Definition at line 420 of file gaussian.hpp.

References mx::astro::constants::c(), and mx::astro::constants::G().

Referenced by mx::math::func::gaussian2D(), and mx::math::func::gaussian2D_ang().

◆ gaussian2D() [4/4]

template<typename realT >

void mx::math::func::gaussian2D	(	realT *	arr,
		size_t	nx,
		size_t	ny,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	sigma
	)

Fill in an array with the 2D arbitrarily-centered symmetric Gaussian.

At each pixel (x,y) of the array this computes:

\( f(x,y) = G_0 + G\exp[-(0.5/\sigma^2)((x-x_0)^2+(y-y_0)^2)] \)

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	arr	is the allocated array to fill in
[in]	nx	is the size of the x dimension of the array
[in]	ny	is the size of the y dimension of the array
[in]	G0	is the constant to add to the Gaussian
[in]	G	is the scaling factor (peak height = G-G0)
[in]	x0	is the x-coordinate of the center
[in]	y0	is the y-coordinate of the center
[in]	sigma	is the third rotation and scaling factor

Definition at line 150 of file gaussian.hpp.

References mx::astro::constants::G(), mx::math::func::gaussian2D(), and mx::astro::constants::sigma().

◆ gaussian2D_ang() [1/2]

template<typename realT >

realT mx::math::func::gaussian2D_ang	(	const realT	x,
		const realT	y,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	sigma_x,
		const realT	sigma_y,
		const realT	theta
	)

Find value at position (x,y) of the 2D rotated elliptical Gaussian.

Computes: \( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

where, for a counter-clockwise rotation \( \theta \), we have

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

This is a convenience wrapper for the general elliptical Gaussian function gaussian2D(), where here (a,b,c) are first calculated from ( \(\sigma_x\), \(\sigma_y\), \(\theta\)). This will in general be slower due to the trig function calls, so the (a,b,c) version should be used most of the time.

The functions gaussian2D_gen2rot() and gaussian2D_rot2gen() provide conversions from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) and back. The function gaussian2D_rot() is a wrapper for this function, which instead accepts ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) as inputs.

Returns: the value of the 2D elliptical Gaussian at (x,y)

Template Parameters

realT is the type to use for arithmetic

Parameters

[in]	x	the x-position at which to evaluate the Gaussian
[in]	y	the y-positoin at which to evaluate the Gaussian
[in]	G0	the constant to add to the Gaussian
[in]	G	the scaling factor (peak height = G-G0)
[in]	x0	the x-coordinate of the center
[in]	y0	the y-coordinate of the center
[in]	sigma_x	the width in the rotated x direction
[in]	sigma_y	the width in the rotated y direction
[in]	theta	the counter-clockwise rotation angle

Definition at line 376 of file gaussian.hpp.

References mx::astro::constants::c(), mx::astro::constants::G(), mx::math::func::gaussian2D(), and mx::math::func::gaussian2D_rot2gen().

◆ gaussian2D_ang() [2/2]

template<typename realT >

void mx::math::func::gaussian2D_ang	(	realT *	arr,
		size_t	nx,
		size_t	ny,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	sigma_x,
		const realT	sigma_y,
		const realT	theta
	)

Fill in an array with the 2D general elliptical Gaussian.

At each pixel (x,y) of the array this computes:

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

where, for a counter-clockwise rotation \( \theta \), we have

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

This is a convenience wrapper for the general elliptical Gaussian function gaussian2D( realT *, size_t, size_t, const realT, const realT, const realT, const realT, const realT, const realT, const realT) , where here (a,b,c) are first calculated from ( \(\sigma_x\), \(\sigma_y\), \(\theta\)).

The functions gaussian2D_gen2rot() and gaussian2D_rot2gen() provide conversions from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) and back.

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	arr	the native array to populate with the Gaussian function
[in]	nx	the x-size of the array, in pixels
[in]	ny	the y-size of the array, in pixels
[in]	G0	the constant to add to the Gaussian
[in]	G	the scaling factor (peak = G)
[in]	x0	the x-coordinate of the center, in pixels
[in]	y0	the y-coordinate of the center, in pixels
[in]	sigma_x	the width in the rotated x direction, in pixels
[in]	sigma_y	the width in the rotated y direction, in pixels
[in]	theta	the counter-clockwise rotation angle, in radians

Definition at line 473 of file gaussian.hpp.

References mx::astro::constants::c(), mx::astro::constants::G(), mx::math::func::gaussian2D(), and mx::math::func::gaussian2D_rot2gen().

◆ gaussian2D_gen2rot()

template<typename realT >

void mx::math::func::gaussian2D_gen2rot	(	realT &	sigma_x,
		realT &	sigma_y,
		realT &	theta,
		const realT	a,
		const realT	b,
		const realT	c
	)

Convert from (a,b,c) to ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) for the elliptical Gaussian.

The general 2D elliptical Gaussian

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

can be expressed in terms of widths and a rotation angle using:

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

where \( \theta \) specifies a counter-clockwise rotation.

This function calculates ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) from inputs (a,b,c). It adopts the convention that the long axis of the ellipse is \(\sigma_x\), and \(\theta\) is chosen appropriately. Note that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	sigma_x	the width parameter in the rotated x-direction
[out]	sigma_y	the width parameter in the rotated y-direction
[out]	theta	the c.c.w rotation
[in]	a	the first rotation and scaling parameter
[in]	b	the second rotation and scaling parameter
[in]	c	the the third rotation and scaling parameter

Definition at line 245 of file gaussian.hpp.

References mx::astro::constants::c().

◆ gaussian2D_jacobian()

template<typename realT >

void mx::math::func::gaussian2D_jacobian	(	realT *	j,
		const realT	x,
		const realT	y,
		const realT	G0,
		const realT	G,
		const realT	x0,
		const realT	y0,
		const realT	a,
		const realT	b,
		const realT	c
	)

Calculate the Jacobian at position (x,y) for the 2D general elliptical Gaussian.

Note: this has not been verified and may be incorrect.

Given:

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

we calculate

\( \frac{\partial G}{\partial G_0} = 1 \)

\( \frac{\partial G}{\partial G} = (G - G_0)/A \)

\( \frac{\partial G}{\partial x_0} = 0.5(G-G_0)( 2a(x-x_0) + b(y-y_0)) \)

\( \frac{\partial G}{\partial y_0} = 0.5(G-G_0) ( b(x-x_0) + 2c(y-y_0)) \)

\( \frac{\partial G}{\partial a} = -0.5(G-G_0) ( (x-x_0)^2 ) \)

\( \frac{\partial G}{\partial b} = -0.5(G-G_0) ( (x-x_0)(y-y_0) ) \)

\( \frac{\partial G}{\partial c} = -0.5(G-G_0) ( (y-y_0)^2 )\)

Parameters

j	is a 7 element vector which is populated with the derivatives
x	is the x-position at which to evaluate the Gaussian
y	is the y-positoin at which to evaluate the Gaussian
G0	is the constant to add to the Gaussian
G	is the scaling factor (peak = A)
x0	is the x-coordinate of the center
y0	is the y-coordinate of the center
a	is the first rotation and scaling factor
b	is the second rotation and scaling factor
c	is the third rotation and scaling factor

Template Parameters

realT is the type to use for arithmetic

Definition at line 535 of file gaussian.hpp.

References mx::astro::constants::c(), and mx::astro::constants::G().

◆ gaussian2D_rot2gen()

template<typename realT >

void mx::math::func::gaussian2D_rot2gen	(	realT &	a,
		realT &	b,
		realT &	c,
		const realT	sigma_x,
		const realT	sigma_y,
		const realT	theta
	)

Convert from ( \(\sigma_x\), \(\sigma_y\), \(\theta\)) to (a,b,c) for the elliptical Gaussian.

The general 2D elliptical Gaussian

\( f(x,y) = G_0 + G\exp [-0.5( a(x-x_0)^2 + b(x-x_0)(y-y_0) + c(y-y_0)^2 ]\)

can be expressed in terms of widths and a rotation angle using:

\( a = \frac{\cos^2 \theta}{\sigma_x^2} + \frac{\sin^2\theta}{\sigma_y^2}\)

\( b = \frac{\sin 2\theta}{2} \left(\frac{1}{\sigma_x^2} - \frac{1}{\sigma_y^2} \right)\)

\( c = \frac{\sin^2 \theta}{\sigma_x^2} + \frac{\cos^2\theta}{\sigma_y^2}\)

where \( \theta \) specifies a counter-clockwise rotation.

This function calculates (a,b,c) from inputs ( \(\sigma_x\), \(\sigma_y\), \(\theta\)).

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	a	the first rotation and scaling parameter
[out]	b	the second rotation and scaling parameter
[out]	c	the the third rotation and scaling parameter
[in]	sigma_x	the width parameter in the rotated x-direction
[in]	sigma_y	the width parameter in the rotated y-direction
[in]	theta	the c.c.w rotation

Definition at line 328 of file gaussian.hpp.

References mx::astro::constants::c().

Referenced by mx::math::func::gaussian2D_ang().

◆ sigma2fwhm()

template<typename floatT >

floatT mx::math::func::sigma2fwhm ( floatT sig )

Convert from Gaussian width parameter to FWHM.

Performs the conversion:

\( FWHM = 2\sqrt{2\log2}\sigma \)

Returns: the converted value of the full-width at half maximum

Parameters

[in] sig the Gaussian width parameter

Definition at line 80 of file gaussian.hpp.

Referenced by mx::improc::imCenterCircleSym< realT >::dumpResults(), and mx::math::fit::fitGaussian2D< fitterT >::fwhm().

◆ twosqrt2log2()

template<typename floatT >

constexpr floatT mx::math::func::twosqrt2log2 ( )

constexpr

Constant to convert between the Gaussian width parameter and FWHM.

Used for

\( FWHM = 2\sqrt{2\log2}\sigma \)

This was calculated in long double precision.

Definition at line 50 of file gaussian.hpp.