#include <Mixture.h>

Inheritance diagram for lsst::meas::modelfit::Mixture:

Public Types
typedef MixtureComponent	Component

typedef MixtureUpdateRestriction	UpdateRestriction

typedef std::vector< Component >	ComponentList

typedef ComponentList::iterator	iterator

typedef ComponentList::const_iterator	const_iterator

Public Member Functions
iterator	begin ()
	Iterator and indexed access to components.

iterator	end ()

const_iterator	begin () const

const_iterator	end () const

Component &	operator[] (std::size_t i)

Component const &	operator[] (std::size_t i) const

std::size_t	size () const
	Return the number of components.

virtual int	getComponentCount () const
	Return the number of components.

std::shared_ptr< Mixture >	project (int dim) const
	Project the distribution onto the given dimensions (marginalize over all others)

std::shared_ptr< Mixture >	project (int dim1, int dim2) const
	Project the distribution onto the given dimensions (marginalize over all others)

int	getDimension () const
	Return the number of dimensions.

void	normalize ()
	Iterate over all components, rescaling their weights so they sum to one.

void	shift (int dim, Scalar offset)
	Shift the mixture in the given dimension, adding the given offset to all mu vectors.

std::size_t	clip (Scalar threshold=0.0)
	Iterate over all components, removing those with weight less than or equal to threshold.

Scalar	getDegreesOfFreedom () const
	Get the number of degrees of freedom in the component Student's T distributions (inf=Gaussian)

void	setDegreesOfFreedom (Scalar df=std::numeric_limits< Scalar >::infinity())
	Set the number of degrees of freedom in the component Student's T distributions (inf=Gaussian)

template<typename Derived >
Scalar	evaluate (Component const &component, Eigen::MatrixBase< Derived > const &x) const
	Evaluate the probability density at the given point for the given component distribution.

template<typename Derived >
Scalar	evaluate (Eigen::MatrixBase< Derived > const &x) const
	Evaluate the mixture distribution probability density function (PDF) at the given points.

void	evaluate (ndarray::Array< Scalar const, 2, 1 > const &x, ndarray::Array< Scalar, 1, 0 > const &p) const
	Evaluate the distribution probability density function (PDF) at the given points.

void	evaluateComponents (ndarray::Array< Scalar const, 2, 1 > const &x, ndarray::Array< Scalar, 2, 1 > const &p) const
	Evaluate the contributions of each component to the full probability at the given points.

void	evaluateDerivatives (ndarray::Array< Scalar const, 1, 1 > const &x, ndarray::Array< Scalar, 1, 1 > const &gradient, ndarray::Array< Scalar, 2, 1 > const &hessian) const
	Evaluate the derivative of the distribution at the given point.

void	evaluateDerivatives (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &x, Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &gradient, Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > &hessian) const
	Evaluate the derivative of the distribution at the given point.

void	evaluateDerivatives (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &x, Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &gradient) const
	Evaluate the derivative of the distribution at the given point.

void	draw (afw::math::Random &rng, ndarray::Array< Scalar, 2, 1 > const &x) const
	Draw random variates from the distribution.

void	updateEM (ndarray::Array< Scalar const, 2, 1 > const &x, ndarray::Array< Scalar const, 1, 0 > const &w, Scalar tau1=0.0, Scalar tau2=0.5)
	Perform an Expectation-Maximization step, updating the component parameters to match the given weighted samples.

void	updateEM (ndarray::Array< Scalar const, 2, 1 > const &x, ndarray::Array< Scalar const, 1, 0 > const &w, UpdateRestriction const &restriction, Scalar tau1=0.0, Scalar tau2=0.5)
	Perform an Expectation-Maximization step, updating the component parameters to match the given weighted samples.

void	updateEM (ndarray::Array< Scalar const, 2, 1 > const &x, UpdateRestriction const &restriction, Scalar tau1=0.0, Scalar tau2=0.5)
	Perform an Expectation-Maximization step, updating the component parameters to match the given unweighted samples.

virtual std::shared_ptr< Mixture >	clone () const
	Polymorphic deep copy.

	Mixture (int dim, ComponentList &components, Scalar df=std::numeric_limits< Scalar >::infinity())
	Construct a mixture model.

virtual bool	isPersistable () const noexcept override
	Return true if this particular object can be persisted using afw::table::io.

void	writeFits (std::string const &fileName, std::string const &mode="w") const
	Write the object to a regular FITS file.

void	writeFits (fits::MemFileManager &manager, std::string const &mode="w") const
	Write the object to a FITS image in memory.

void	writeFits (fits::Fits &fitsfile) const
	Write the object to an already-open FITS object.

Static Public Member Functions
static std::shared_ptr< Mixture >	readFits (fits::Fits &fitsfile)
	Read an object from an already open FITS object.

static std::shared_ptr< Mixture >	readFits (std::string const &fileName, int hdu=fits::DEFAULT_HDU)
	Read an object from a regular FITS file.

static std::shared_ptr< Mixture >	readFits (fits::MemFileManager &manager, int hdu=fits::DEFAULT_HDU)
	Read an object from a FITS file in memory.

static std::shared_ptr< Mixture >	dynamicCast (std::shared_ptr< Persistable > const &ptr)
	Dynamically cast a shared_ptr.

Protected Types
using	OutputArchiveHandle = io::OutputArchiveHandle

Protected Member Functions
std::string	getPythonModule () const override
	Return the fully-qualified Python module that should be imported to guarantee that its factory is registered.

std::string	getPersistenceName () const override
	Return the unique name used to persist this object and look up its factory.

void	write (OutputArchiveHandle &handle) const override
	Write the object to one or more catalogs.

Friends
std::ostream &	operator<< (std::ostream &os, Mixture const &self)

Detailed Description

Definition at line 128 of file Mixture.h.

Member Typedef Documentation

◆ Component

typedef MixtureComponent lsst::meas::modelfit::Mixture::Component

Definition at line 131 of file Mixture.h.

◆ ComponentList

typedef std::vector<Component> lsst::meas::modelfit::Mixture::ComponentList

Definition at line 133 of file Mixture.h.

◆ const_iterator

typedef ComponentList::const_iterator lsst::meas::modelfit::Mixture::const_iterator

Definition at line 135 of file Mixture.h.

◆ iterator

typedef ComponentList::iterator lsst::meas::modelfit::Mixture::iterator

Definition at line 134 of file Mixture.h.

◆ OutputArchiveHandle

using lsst::afw::table::io::Persistable::OutputArchiveHandle = io::OutputArchiveHandle

protectedinherited

Definition at line 108 of file Persistable.h.

◆ UpdateRestriction

typedef MixtureUpdateRestriction lsst::meas::modelfit::Mixture::UpdateRestriction

Definition at line 132 of file Mixture.h.

Constructor & Destructor Documentation

◆ Mixture()

lsst::meas::modelfit::Mixture::Mixture	(	int	dim,
		ComponentList &	components,
		Scalar	df = std::numeric_limits< Scalar >::infinity() )

explicit

Construct a mixture model.

Parameters

[in]	dim	Dimensionality of the distribution
[in]	df	Number of degrees of freedom for component Student's T distributions (inf=Gaussian)
[in]	components	List of components; will be emptied on return.

The components will be automatically normalized after construction.

Member Function Documentation

◆ begin() [1/2]

iterator lsst::meas::modelfit::Mixture::begin ( )

inline

Iterator and indexed access to components.

While mutable iterators and accessors are provided, any modifications to the component weights should be followed by a call to normalize(), as other member functions will not work properly if the mixture is not normalized.

Definition at line 146 of file Mixture.h.

146{ return _components.begin(); }

std::vector::begin

T begin(T... args)

◆ begin() [2/2]

const_iterator lsst::meas::modelfit::Mixture::begin ( ) const

inline

Definition at line 149 of file Mixture.h.

149{ return _components.begin(); }

◆ clip()

std::size_t lsst::meas::modelfit::Mixture::clip ( Scalar threshold = 0.0 )

Iterate over all components, removing those with weight less than or equal to threshold.

The weights will be normalized if any are removed.

Returns: the number of components removed.

◆ clone()

virtual std::shared_ptr< Mixture > lsst::meas::modelfit::Mixture::clone ( ) const

virtual

Polymorphic deep copy.

◆ draw()

void lsst::meas::modelfit::Mixture::draw	(	afw::math::Random &	rng,
		ndarray::Array< Scalar, 2, 1 > const &	x ) const

Draw random variates from the distribution.

Parameters

[in,out]	rng	random number generator
[out]	x	array of points, shape=(numSamples, dim)

◆ dynamicCast()

template std::shared_ptr< meas::extensions::psfex::PsfexPsf > lsst::afw::table::io::PersistableFacade< Mixture >::dynamicCast ( std::shared_ptr< Persistable > const & ptr )

staticinherited

Dynamically cast a shared_ptr.

Dynamically cast a shared pointer and raise on failure.

You must provide an explicit template instantiation in the .cc file for each class that inherits from PersistableFacade. Designed to work around RTTI issues on macOS with hidden symbols;

Exceptions

lsst::pex::exceptions::LogicError if the cast fails

param[in] ptr The pointer to be cast.

Returns: The cast pointer.

Exceptions

lsst::pex::exceptions::TypeError If the dynamic cast fails.

Definition at line 218 of file Persistable.cc.

◆ end() [1/2]

iterator lsst::meas::modelfit::Mixture::end ( )

inline

Definition at line 147 of file Mixture.h.

147{ return _components.end(); }

std::vector::end

T end(T... args)

◆ end() [2/2]

const_iterator lsst::meas::modelfit::Mixture::end ( ) const

inline

Definition at line 150 of file Mixture.h.

150{ return _components.end(); }

◆ evaluate() [1/3]

template<typename Derived >

Scalar lsst::meas::modelfit::Mixture::evaluate	(	Component const &	component,
		Eigen::MatrixBase< Derived > const &	x ) const

inline

Evaluate the probability density at the given point for the given component distribution.

This evaluates the probability of a single component, including the current weight of that component.

Definition at line 198 of file Mixture.h.

                                                                                           {
        Scalar z = _computeZ(component, x);
        return component.weight * _evaluate(z) / component._sqrtDet;
    }

◆ evaluate() [2/3]

template<typename Derived >

Scalar lsst::meas::modelfit::Mixture::evaluate ( Eigen::MatrixBase< Derived > const & x ) const

inline

Evaluate the mixture distribution probability density function (PDF) at the given points.

Parameters

[in] x point to evaluate, as an Eigen expression, shape=(dim,)

Definition at line 209 of file Mixture.h.

                                                              {
        Scalar p = 0.0;
        for (const_iterator i = begin(); i != end(); ++i) {
            p += evaluate(*i, x);
        }
        return p;
    }

◆ evaluate() [3/3]

void lsst::meas::modelfit::Mixture::evaluate	(	ndarray::Array< Scalar const, 2, 1 > const &	x,
		ndarray::Array< Scalar, 1, 0 > const &	p ) const

Evaluate the distribution probability density function (PDF) at the given points.

Parameters

[in]	x	array of points, shape=(numSamples, dim)
[out]	p	array of probability values, shape=(numSamples,)

◆ evaluateComponents()

void lsst::meas::modelfit::Mixture::evaluateComponents	(	ndarray::Array< Scalar const, 2, 1 > const &	x,
		ndarray::Array< Scalar, 2, 1 > const &	p ) const

Evaluate the contributions of each component to the full probability at the given points.

Parameters

[in]	x	points to evaluate at, with number of columns equal to the number of dimensions
[in]	p	array to fill, with number of columns equal to the number of components

◆ evaluateDerivatives() [1/3]

void lsst::meas::modelfit::Mixture::evaluateDerivatives	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &	gradient ) const

Evaluate the derivative of the distribution at the given point.

Parameters

[in]	x	point to evaluate the derivative, with size equal to the number of dimensions
[in]	gradient	1st derivative array to fill

◆ evaluateDerivatives() [2/3]

void lsst::meas::modelfit::Mixture::evaluateDerivatives	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > &	gradient,
		Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > &	hessian ) const

Evaluate the derivative of the distribution at the given point.

Parameters

[in]	x	point to evaluate the derivative, with size equal to the number of dimensions
[in]	gradient	1st derivative array to fill
[in]	hessian	2nd derivative array to fill

◆ evaluateDerivatives() [3/3]

void lsst::meas::modelfit::Mixture::evaluateDerivatives	(	ndarray::Array< Scalar const, 1, 1 > const &	x,
		ndarray::Array< Scalar, 1, 1 > const &	gradient,
		ndarray::Array< Scalar, 2, 1 > const &	hessian ) const

Evaluate the derivative of the distribution at the given point.

Parameters

[in]	x	point to evaluate the derivative, with size equal to the number of dimensions
[in]	gradient	1st derivative array to fill
[in]	hessian	2nd derivative array to fill

◆ getComponentCount()

virtual int lsst::meas::modelfit::Mixture::getComponentCount ( ) const

inlinevirtual

Return the number of components.

Definition at line 160 of file Mixture.h.

160{ return size(); }

lsst::meas::modelfit::Mixture::size

std::size_t size() const

Return the number of components.

Definition Mixture.h:157

◆ getDegreesOfFreedom()

Scalar lsst::meas::modelfit::Mixture::getDegreesOfFreedom ( ) const

inline

Get the number of degrees of freedom in the component Student's T distributions (inf=Gaussian)

Definition at line 187 of file Mixture.h.

187{ return _df; }

◆ getDimension()

int lsst::meas::modelfit::Mixture::getDimension ( ) const

inline

Return the number of dimensions.

Definition at line 169 of file Mixture.h.

169{ return _dim; }

◆ getPersistenceName()

std::string lsst::meas::modelfit::Mixture::getPersistenceName ( ) const

overrideprotectedvirtual

Return the unique name used to persist this object and look up its factory.

Must be less than ArchiveIndexSchema::MAX_NAME_LENGTH characters.

Reimplemented from lsst::afw::table::io::Persistable.

◆ getPythonModule()

std::string lsst::meas::modelfit::Mixture::getPythonModule ( ) const

inlineoverrideprotectedvirtual

Return the fully-qualified Python module that should be imported to guarantee that its factory is registered.

Must be less than ArchiveIndexSchema::MAX_MODULE_LENGTH characters.

Will be ignored if empty.

Reimplemented from lsst::afw::table::io::Persistable.

Definition at line 368 of file Mixture.h.

368{ return "lsst.meas.modelfit"; }

◆ isPersistable()

virtual bool lsst::meas::modelfit::Mixture::isPersistable ( ) const

inlineoverridevirtualnoexcept

Return true if this particular object can be persisted using afw::table::io.

Reimplemented from lsst::afw::table::io::Persistable.

Definition at line 364 of file Mixture.h.

364{ return true; }

◆ normalize()

void lsst::meas::modelfit::Mixture::normalize ( )

Iterate over all components, rescaling their weights so they sum to one.

◆ operator[]() [1/2]

Component & lsst::meas::modelfit::Mixture::operator[] ( std::size_t i )

inline

Definition at line 152 of file Mixture.h.

152{ return _components[i]; }

◆ operator[]() [2/2]

Component const & lsst::meas::modelfit::Mixture::operator[] ( std::size_t i ) const

inline

Definition at line 153 of file Mixture.h.

153{ return _components[i]; }

◆ project() [1/2]

std::shared_ptr< Mixture > lsst::meas::modelfit::Mixture::project ( int dim ) const

Project the distribution onto the given dimensions (marginalize over all others)

◆ project() [2/2]

std::shared_ptr< Mixture > lsst::meas::modelfit::Mixture::project	(	int	dim1,
		int	dim2 ) const

Project the distribution onto the given dimensions (marginalize over all others)

◆ readFits() [1/3]

static std::shared_ptr< Mixture > lsst::afw::table::io::PersistableFacade< Mixture >::readFits ( fits::Fits & fitsfile )

inlinestaticinherited

Read an object from an already open FITS object.

Parameters

[in] fitsfile FITS object to read from, already positioned at the desired HDU.

Definition at line 183 of file Persistable.h.

◆ readFits() [2/3]

static std::shared_ptr< Mixture > lsst::afw::table::io::PersistableFacade< Mixture >::readFits	(	fits::MemFileManager &	manager,
		int	hdu = fits::DEFAULT_HDU )

inlinestaticinherited

Read an object from a FITS file in memory.

Parameters

[in]	manager	Manager for the memory to read from.
[in]	hdu	HDU to read, where 0 is the primary. The special value of afw::fits::DEFAULT_HDU skips the primary HDU if it is empty.

Definition at line 205 of file Persistable.h.

◆ readFits() [3/3]

static std::shared_ptr< Mixture > lsst::afw::table::io::PersistableFacade< Mixture >::readFits	(	std::string const &	fileName,
		int	hdu = fits::DEFAULT_HDU )

inlinestaticinherited

Read an object from a regular FITS file.

Parameters

[in]	fileName	Name of the file to read.
[in]	hdu	HDU to read, where 0 is the primary. The special value of afw::fits::DEFAULT_HDU skips the primary HDU if it is empty.

Definition at line 194 of file Persistable.h.

◆ setDegreesOfFreedom()

void lsst::meas::modelfit::Mixture::setDegreesOfFreedom ( Scalar df = std::numeric_limits< Scalar >::infinity() )

Set the number of degrees of freedom in the component Student's T distributions (inf=Gaussian)

◆ shift()

void lsst::meas::modelfit::Mixture::shift	(	int	dim,
		Scalar	offset )

Shift the mixture in the given dimension, adding the given offset to all mu vectors.

◆ size()

std::size_t lsst::meas::modelfit::Mixture::size ( ) const

inline

Return the number of components.

Definition at line 157 of file Mixture.h.

157{ return _components.size(); }

std::vector::size

T size(T... args)

◆ updateEM() [1/3]

void lsst::meas::modelfit::Mixture::updateEM	(	ndarray::Array< Scalar const, 2, 1 > const &	x,
		ndarray::Array< Scalar const, 1, 0 > const &	w,
		Scalar	tau1 = 0.0,
		Scalar	tau2 = 0.5 )

Perform an Expectation-Maximization step, updating the component parameters to match the given weighted samples.

Parameters

[in]	x	array of variables, shape=(numSamples, dim)
[in]	w	array of weights, shape=(numSamples,)
[in]	tau1	damping parameter (see below)
[in]	tau2	damping parameter (see below)

The updates to the \(\sigma\) matrices are damped according to:

\[ \sigma_d = \alpha\sigma_1 + (1-\alpha)\sigma_0 \]

Where \(\sigma_0\) is the previous matrix, \(\sigma_1\) is the undamped update, and \(\sigma_d\) is the damped update. The parameter \(\alpha\) is set by the ratio of the determinants:

\[ r \equiv \frac{|\sigma_1|}{|\sigma_0|} \]

When \(r \ge \tau_1\), \(\alpha=1\); when \(r \lt \tau_1\), it is rolled off quadratically to \(\tau_2\).

◆ updateEM() [2/3]

void lsst::meas::modelfit::Mixture::updateEM	(	ndarray::Array< Scalar const, 2, 1 > const &	x,
		ndarray::Array< Scalar const, 1, 0 > const &	w,
		UpdateRestriction const &	restriction,
		Scalar	tau1 = 0.0,
		Scalar	tau2 = 0.5 )

Perform an Expectation-Maximization step, updating the component parameters to match the given weighted samples.

Parameters

[in]	x	array of variables, shape=(numSamples, dim)
[in]	w	array of weights, shape=(numSamples,)
[in]	restriction	Functor used to restrict the form of the updated mu and sigma
[in]	tau1	damping parameter (see Mixture::updateEM)
[in]	tau2	damping parameter (see Mixture::updateEM)

◆ updateEM() [3/3]

void lsst::meas::modelfit::Mixture::updateEM	(	ndarray::Array< Scalar const, 2, 1 > const &	x,
		UpdateRestriction const &	restriction,
		Scalar	tau1 = 0.0,
		Scalar	tau2 = 0.5 )

Perform an Expectation-Maximization step, updating the component parameters to match the given unweighted samples.

Parameters

[in]	x	array of variables, shape=(numSamples, dim)
[in]	restriction	Functor used to restrict the form of the updated mu and sigma
[in]	tau1	damping parameter (see Mixture::updateEM)
[in]	tau2	damping parameter (see Mixture::updateEM)

◆ write()

void lsst::meas::modelfit::Mixture::write ( OutputArchiveHandle & handle ) const

overrideprotectedvirtual

Write the object to one or more catalogs.

The handle object passed to this function provides an interface for adding new catalogs and adding nested objects to the same archive (while checking for duplicates). See OutputArchiveHandle for more information.

Reimplemented from lsst::afw::table::io::Persistable.

◆ writeFits() [1/3]

void lsst::afw::table::io::Persistable::writeFits ( fits::Fits & fitsfile ) const

inherited

Write the object to an already-open FITS object.

Parameters

[in] fitsfile Open FITS object to write to.

Definition at line 18 of file Persistable.cc.

                                                    {
    OutputArchive archive;
    archive.put(this);
    archive.writeFits(fitsfile);
}

◆ writeFits() [2/3]

void lsst::afw::table::io::Persistable::writeFits	(	fits::MemFileManager &	manager,
		std::string const &	mode = "w" ) const

inherited

Write the object to a FITS image in memory.

Parameters

[in]	manager	Name of the file to write to.
[in]	mode	If "w", any existing file with the given name will be overwritten. If "a", new HDUs will be appended to an existing file.

Definition at line 29 of file Persistable.cc.

                                                                                    {
    fits::Fits fitsfile(manager, mode, fits::Fits::AUTO_CLOSE | fits::Fits::AUTO_CHECK);
    writeFits(fitsfile);
}

◆ writeFits() [3/3]

void lsst::afw::table::io::Persistable::writeFits	(	std::string const &	fileName,
		std::string const &	mode = "w" ) const

inherited

Write the object to a regular FITS file.

Parameters

[in]	fileName	Name of the file to write to.
[in]	mode	If "w", any existing file with the given name will be overwritten. If "a", new HDUs will be appended to an existing file.

Definition at line 24 of file Persistable.cc.

                                                                                  {
    fits::Fits fitsfile(fileName, mode, fits::Fits::AUTO_CLOSE | fits::Fits::AUTO_CHECK);
    writeFits(fitsfile);
}

Friends And Related Symbol Documentation

◆ operator<<

std::ostream & operator<<	(	std::ostream &	os,
		Mixture const &	self )

friend

Definition at line 359 of file Mixture.h.

                                                                          {
        self._stream(os);
        return os;
    }

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

/j/snowflake/release/lsstsw/stack/lsst-scipipe-9.0.0/Linux64/meas_modelfit/ga5288a1d22+2903d499ea/include/lsst/meas/modelfit/Mixture.h

Public Types

Public Member Functions

Static Public Member Functions

Protected Types

Protected Member Functions

Friends

Detailed Description

Member Typedef Documentation

◆ Component

◆ ComponentList

◆ const_iterator

◆ iterator

◆ OutputArchiveHandle

◆ UpdateRestriction

Constructor & Destructor Documentation

◆ Mixture()

Member Function Documentation

◆ begin() [1/2]

◆ begin() [2/2]

◆ clip()

◆ clone()

◆ draw()

◆ dynamicCast()

◆ end() [1/2]

◆ end() [2/2]

◆ evaluate() [1/3]

◆ evaluate() [2/3]

◆ evaluate() [3/3]

◆ evaluateComponents()

◆ evaluateDerivatives() [1/3]

◆ evaluateDerivatives() [2/3]

◆ evaluateDerivatives() [3/3]

◆ getComponentCount()

◆ getDegreesOfFreedom()

◆ getDimension()

◆ getPersistenceName()

◆ getPythonModule()

◆ isPersistable()

◆ normalize()

◆ operator[]() [1/2]

◆ operator[]() [2/2]

◆ project() [1/2]

◆ project() [2/2]

◆ readFits() [1/3]

◆ readFits() [2/3]

◆ readFits() [3/3]

◆ setDegreesOfFreedom()

◆ shift()

◆ size()

◆ updateEM() [1/3]

◆ updateEM() [2/3]

◆ updateEM() [3/3]

◆ write()

◆ writeFits() [1/3]

◆ writeFits() [2/3]

◆ writeFits() [3/3]

Friends And Related Symbol Documentation

◆ operator<<