A Multi-dimensional Gaussian functional. More...

#include <GaussianND.h>

Inheritance diagram for casacore::GaussianND< T >:

Public Member Functions
	GaussianND ()
	Makes a Gaussian using the indicated height, mean, variance & covariance. More...

	GaussianND (uInt ndim)

	GaussianND (uInt ndim, const T &height)

	GaussianND (uInt ndim, const T &height, const Vector< T > &mean)

	GaussianND (uInt ndim, const T &height, const Vector< T > &mean, const Vector< T > &variance)

	GaussianND (uInt ndim, const T &height, const Vector< T > &mean, const Matrix< T > &covar)

	GaussianND (const GaussianND &other)
	Copy constructor (deep copy) More...

GaussianND< T > &	operator= (const GaussianND< T > &other)
	Copy assignment (deep copy) More...

virtual	~GaussianND ()
	Destructor. More...

virtual T	eval (typename Function< T >::FunctionArg x) const
	Evaluate the Gaussian at `x`. More...

virtual Function< T > *	clone () const
	Return a copy of this object from the heap. More...

Public Member Functions inherited from casacore::GaussianNDParam< T >
	GaussianNDParam ()
	Constructs a Gaussian using the indicated height, mean, variance & covariance. More...

	GaussianNDParam (uInt ndim)

	GaussianNDParam (uInt ndim, const T &height)

	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean)

	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean, const Vector< T > &variance)

	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean, const Matrix< T > &covar)

	GaussianNDParam (const GaussianNDParam &other)
	Copy constructor (deep copy) More...

template<class W >
	GaussianNDParam (const GaussianNDParam< W > &other)

GaussianNDParam< T > &	operator= (const GaussianNDParam< T > &other)
	Copy assignment (deep copy) More...

virtual	~GaussianNDParam ()
	Destructor. More...

virtual const String &	name () const
	Give name of function. More...

virtual uInt	ndim () const
	Variable dimensionality. More...

T	height () const
	Get or set the peak height of the Gaussian. More...

void	setHeight (const T &height)

T	flux () const
	The analytical integrated area underneath the Gaussian. More...

void	setFlux (const T &flux)

Vector< T >	mean () const
	The center ordinate of the Gaussian. More...

void	setMean (const Vector< T > &mean)

Vector< T >	variance () const
	The FWHM of the Gaussian is `sqrt(8variancelog(2))`. More...

void	setVariance (const Vector< T > &variance)

Matrix< T >	covariance () const
	The covariance Matrix defines the correlations between all the axes. More...

void	setCovariance (const Matrix< T > &covar)

Public Member Functions inherited from casacore::Function< T >
	Function ()
	Constructors. More...

	Function (const uInt n)

	Function (const Vector< T > &in)

	Function (const FunctionParam< T > &other)

	Function (const Function< W, X > &other)

virtual	~Function ()
	Destructor. More...

uInt	nparameters () const
	Returns the number of parameters. More...

virtual U	eval (FunctionArg x) const =0
	Evaluate the function object. More...

T &	operator[] (const uInt n)
	Manipulate the nth parameter (0-based) with no index check. More...

const T &	operator[] (const uInt n) const

virtual U	operator() () const
	Evaluate this function object at `x`or at `x, y`. More...

virtual U	operator() (const ArgType &x) const

virtual U	operator() (const Vector< ArgType > &x) const

virtual U	operator() (FunctionArg x) const

virtual U	operator() (const ArgType &x, const ArgType &y) const

virtual U	operator() (const ArgType &x, const ArgType &y, const ArgType &z) const

Bool &	mask (const uInt n)
	Manipulate the mask associated with the nth parameter (e.g. More...

const Bool &	mask (const uInt n) const

const FunctionParam< T > &	parameters () const
	Return the parameter interface. More...

FunctionParam< T > &	parameters ()

const Vector< ArgType > &	argp () const
	Get `arg_p` and `parset_p`. More...

Bool	parsetp () const

void	lockParam ()
	Compiler cannot always find the correct 'const' version of parameter access. More...

void	unlockParam ()

virtual void	setMode (const RecordInterface &mode)
	get/set the function mode. More...

virtual void	getMode (RecordInterface &mode) const

virtual Bool	hasMode () const
	return True if the implementing function supports a mode. More...

ostream &	print (ostream &os) const
	Print the function (i.e. More...

virtual Function< typename FunctionTraits< T >::DiffType > *	cloneAD () const

virtual Function< typename FunctionTraits< T >::BaseType > *	cloneNonAD () const

Public Member Functions inherited from casacore::Functional< FunctionTraits< T >::ArgType, U >
virtual	~Functional ()
	Destructor. More...

virtual U	operator() (const FunctionTraits< T >::ArgType &x) const =0
	Map a Domain `x` into a Range `y` value. More...

Public Member Functions inherited from casacore::Functional< Vector< FunctionTraits< T >::ArgType >, U >
virtual	~Functional ()
	Destructor. More...

virtual U	operator() (const Vector< FunctionTraits< T >::ArgType > &x) const =0
	Map a Domain `x` into a Range `y` value. More...

Additional Inherited Members
Public Types inherited from casacore::GaussianNDParam< T >
enum	{ HEIGHT, CENTER }

Public Types inherited from casacore::Function< T >
typedef FunctionTraits< T > ::ArgType	ArgType

typedef const ArgType *	FunctionArg

Protected Member Functions inherited from casacore::GaussianNDParam< T >
void	repack (Matrix< T > &covar) const
	Functions to convert between internal Vector of parameters and the Covariance Matrix. More...

void	unpack (const Matrix< T > &covar)

Protected Attributes inherited from casacore::GaussianNDParam< T >
uInt	itsDim
	dimensionality More...

T	itsFlux2Hgt
	factor to convert from flux to height More...

Protected Attributes inherited from casacore::Function< T >
FunctionParam< T >	param_p
	The parameters and masks. More...

Vector< ArgType >	arg_p
	Aid for non-contiguous argument storage. More...

Bool	parset_p
	Indicate parameter written. More...

Bool	locked_p
	Indicate that parameters are expected to be locked from changing. More...

Detailed Description

template<class T>
class casacore::GaussianND< T >

A Multi-dimensional Gaussian functional.

Intended use:

Public interface

Review Status

Reviewed By:: UNKNOWN
Date Reviewed:: before2004/08/25
Test programs:: tGaussianND
Demo programs:: dGaussianND

Prerequisite

Synopsis

A GaussianND is used to calculate Gaussian functions of any dimension. A Gaussian1D class exists which is more appropriate for one dimensional Gaussian functions, and a Gaussian2D class exists for two dimensional functions.

A statistical description of the multi-dimensional Gaussian is used (see Kendall & Stuart "The Advanced Theory of Statistics"). A Gaussian is defined in terms of its height, mean (which is the location of the peak value), variance, (a measure of the width of the Gaussian), and covariance which skews the distribution with respect to the Axes.

In the general description the variance and covariance are specified using a covariance matrix. This is defined as (for a 4 dimensional Gaussian):

V = |     s1*s1 r12*s1*s2 r13*s1*s3 r14*s1*s4 | 
| r12*s1*s2     s2*s2 r23*s2*s3 r24*s2*s4 |
| r13*s1*s3 r23*s2*s3     s3*s3 r34*s3*s4 |
| r14*s1*s4 r24*s2*s4 r34*s3*s4     s4*s4 |

where s1 (sigma1) is the standard deviation of the Gaussian with respect to the first axis, and r12 (rho12) is the correlation between the the first and second axis. The correlation MUST be between -1 and 1, and this class checks this as well as ensuring that the diagonal is positive.

Warning: It is possible to have symmetric matrices that are of the above described form (ie; symmetric with -1 <= rho(ij) <=1) that do not generate a Gaussian function; This is because the Matrix is NOT positive definite (The limits on rho(ij) are upper limits); This class does check that the covariance Matrix is positive definite and will throw an exception (AipsError) if it is not;

The covariance Matrix can be specified by only its upper or lower triangular regions (ie. with zeros in the other triangle), otherwise it MUST be symmetric.

The Gaussian that is constructed from this covariance Matrix (V), along with mean (u) and height (h) is:

f(x) = h*exp(-1/2 * (x-u) * V^(-1) * (x-u))

where x, and u are vectors whose length is the dimensionality of the Gaussian and V^(-1) is the inverse of the covariance Matrix defined above. For a two dimensional Gaussian with zero mean this expression reduces to:

f(x) = h*exp(-1/(2*(1-r12^2))*(x1^2/s1^2 - 2*r12*x1*x2/(s1*s2) + x2^2/s2^2))

The amplitude of the Gaussian can be defined in two ways, either using the peak height (as is done in the constructors, and the setHeight function) or using the setFlux function. The flux in this context is the analytic integral of the Gaussian over all dimensions. Using the setFlux function does not modify the shape of the Gaussian just its height.

All the parameters of the Gaussian except its dimensionality can be modified using the set/get functions.

The parameter interface (see FunctionParam class), is used to provide an interface to the Fitting classes. There are always 4 parameter sets. The parameters are, in order:

height (1 term). No assumptions on what quantity the height represents, and it can be negative
mean (ndim terms).
variance (ndim terms). The variance is always positive, and an exception (AipsError) will be thrown if you try to set a negative value.
covariance (ndim*(ndim-1)/2 terms) The order is (assuming ndim=5) v12,v13,v14,v15,v23,v24,v25,v34,v35,v45. The restrictions described above for the covariance (ie. -1 < r12 < +1) are enforced.

Example

Construct a two dimensional Gaussian with mean=(0,1), variance=(.1,7) and height = 1;

uInt ndim = 2;
Float height = 1;
Vector<Float> mean(ndim); mean(0) = 0, mean(1) = 1;
Vector<Float> variance(ndim); variance(0) =.1, variance(1) = 7;
GaussianND<Float> g(ndim, height, mean, variance); 
Vector<Float> x(ndim); x = 0;
cout << "g("<< x <<") = " << g(x) <<endl; // g([0,0])=1*exp(-1/2*1/7);
x(1)++;
cout << "g("<< x <<") = " <<g(x) <<endl;  // g([0,1])= 1
cout << "Height: " << g.height() <<endl;    // Height: 1
cout << "Flux: " << g.flux() << endl;       // Flux: 2*Pi*Sqrt(.1*7)
cout << "Mean: " << g.mean() << endl;  // Mean: [0, -1]
cout << "Variance: " << g.variance() <<endl;  // Variance: [.1, 7]
cout << "Covariance: "<< g.covariance()<<endl;// Covariance: [.1, 0]
//             [0,  7]
g.setFlux(1);
cout << "g("<< x <<") = " <<g(x) <<endl;  //g([0,1])=1/(2*Pi*Sqrt(.7))
cout << "Height: " << g.height() <<endl;    // Height: 1/(2*Pi*Sqrt(.7))
cout << "Flux: " << g.flux() << endl;       // Flux: 1
cout << "Mean: " << g.mean() << endl;  // Mean: [0, -1]
cout << "Variance: " << g.variance() <<endl;  // Variance: [.1, 7]
cout << "Covariance: "<< g.covariance()<<endl;// Covariance: [.1, 0]
//             [0,  7]

Motivation

A Gaussian Functional was needed for modeling the sky with a series of components. It was later realised that it was too general and Gaussian2D was written.

Template Type Argument Requirements (T)

T should have standard numerical operators and exp() function. Current implementation only tested for real types.

To Do

Nothing I know off, apart from possible optimization

Definition at line 178 of file GaussianND.h.

Constructor & Destructor Documentation

template<class T>

casacore::GaussianND< T >::GaussianND ( )

inline

Makes a Gaussian using the indicated height, mean, variance & covariance.

ndim defaults to 2, mean defaults to 0, height to Pi^(-ndim/2) (the flux is unity) variance defaults to 1.0, covariance defaults to 0.0,

Definition at line 190 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND ( uInt ndim )

inlineexplicit

Definition at line 191 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND	(	uInt	ndim,
		const T &	height
	)

inline

Definition at line 193 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean
	)

inline

Definition at line 195 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean,
		const Vector< T > &	variance
	)

inline

Definition at line 197 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean,
		const Matrix< T > &	covar
	)

inline

Definition at line 200 of file GaussianND.h.

template<class T>

casacore::GaussianND< T >::GaussianND ( const GaussianND< T > & other )

inline

Copy constructor (deep copy)

Definition at line 207 of file GaussianND.h.

template<class T>

virtual casacore::GaussianND< T >::~GaussianND ( )

inlinevirtual

Destructor.

Definition at line 215 of file GaussianND.h.

Member Function Documentation

template<class T>

virtual Function<T>* casacore::GaussianND< T >::clone ( ) const

inlinevirtual

Return a copy of this object from the heap.

The caller is responsible for deleting this pointer.

Implements casacore::Function< T >.

Definition at line 227 of file GaussianND.h.

template<class T>

virtual T casacore::GaussianND< T >::eval ( typename Function< T >::FunctionArg x ) const

virtual

Evaluate the Gaussian at x.

template<class T>

GaussianND<T>& casacore::GaussianND< T >::operator= ( const GaussianND< T > & other )

inline

Copy assignment (deep copy)

Definition at line 211 of file GaussianND.h.

References casacore::GaussianNDParam< T >::operator=().

The documentation for this class was generated from the following file:

scimath/Functionals/GaussianND.h

Public Member Functions

Additional Inherited Members

Detailed Description

template<class T> class casacore::GaussianND< T >

Intended use:

Review Status

Prerequisite

Synopsis

Example

Motivation

Template Type Argument Requirements (T)

To Do

Constructor & Destructor Documentation

Member Function Documentation

template<class T>
class casacore::GaussianND< T >