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FITS Embedded Function Proposal
Jonathan McDowell
1998 Feb 9
1. This document describes a proposal the storage of a function or functional in a FITS binary
table HDU. The HDU provides a recipe for returning a function or function value given the
values of its parameters. The proposal would have the status of an ASC internal convention.
2. The HDU represents a function­valued function, called the table function. The table func­
tion is a mapping from an n­dimensional space, whose axes are called the table function
parameters, to a space of functions, called the return function space. In other words, you
give the HDU an n­tuple of numbers and it gives you back a function, which we call the
return function. The return function space is not arbitrary; it is required to be the set of
functions which are restrictions of a single, specified, function (the model function) of m+p
parameters to an m­dimensional domain. We call the m parameters of the return function
the `independent variables' and the remaining p parameters of the model function the `model
function parameters'.
For example: Consider modelling an instrument point spread function P(x,y) as a circular
gaussian G(x,y,sigma,norm) whose sigma and norm depend further on the external parameters
energy and off­axis angle theta. Then we have:
ffl Model function G : R 4 ! R = Gauss(x; y; sigma; norm) (i.e. G is a function from 4D
space to 1D space).
ffl Independent variables: x,y
ffl Model function parameters: sigma, norm
ffl Return function P : R 2 ! R = P (x; y), the restriction of G(x,y,sigma,norm) to the 2­D
subdomain given by a particular choice of sigma and norm.
ffl Table function parameters: energy, theta
ffl Table function: T : R 2 ! F (R 2 ! R), (i.e. T is a function from 2D space to the space
of functions which map 2D space to 1D); T(energy,theta) = P(x,y) returns a function
of x and y for any given energy and theta.
3. The model function may be the constant function with the single parameter of its value,
in which case the table function is a simple map from an n­tuple to a function value. In
this case, if it is further true that the table function parameters are all tabulated on regular
constant­spaced intervals, the standard FITS IMAGE storage mechanism can and should be
used.
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4. The HDU shall be recognized as containing a function by the (mandatory) presence of the
keyword
HDUCLAS1 = 'FUNCTION'
The value of HDUCLASS may be 'ASC' and the value of HDUCLAS2 may indicate the
function type.
5. A good, but not compulsory, choice of EXTNAME is 'FUNCTION'.
6. The mandatory keyword FUNCNAME gives the name of the table function. This name will
be recognized by other software as the user name of this function, e.g. in the ASCFIT model
syntax.
7. The keyword MODEL gives the definition in ASC model syntax of the model function. The
special value 'None' for a constant function indicates that a function value is returned; al­
ternatively, MODEL may be omitted. Use of COMMENT keywords to explain the model
function definition is encouraged.
8. The mandatory keyword FNAXIS gives the number of table function parameters.
9. The mandatory indexed keywords FTYPEi (i = 1 to FNAXIS) give the name of the ith
table function parameter. This name must be identical to one of the TTYPE values. (We
key off the name rather than the column number so as to be more robust to changes by file
manipulation tools.)
10. The optional indexed keywords FNAXISi give the number of tabulated values for the ith
table function parameter. This only makes sense if the same number of values for FTYPE1
are tabulated at each value of FTYPE2, etc., in other words the grid of points at which the
function is tabulated is regular like
FTYPE1 ­­­­?
x...x.x..xx ---
........... ---
x...x.x..xx ---
........... ---
........... V
x...x.x..xx FTYPE2
x...x.x..xx
as opposed to
FTYPE1 ­­­­?
x...x.x..x. ---
........... ---
x.....x..xx ---
..x........ ---
........... V
....x.x...x FTYPE2
x...x.x..xx
2

The latter form is more difficult to interpolate in, but may be necessary in practice as many
measurements do not occur on fully sampled grids (e.g. a measurement at 6.5 keV may
not have been done for one off­axis angle). Hence, the presence of the FNAXISi keywords
indicates that for any tabulated value of the other parameters, you always get FNAXISi
values of parameter i and moreover they will be the same values. Further, the total number
of rows will be the product of all the FNAXISi.
11. The model parameters are given as table columns, using names consistent with the ASCFIT
model syntax. However, if a model parameter has a constant value for all the rows, it may
be stored as a keyword instead.
12. For the constant model, the column name giving the model value shall be the same as the
table function name (value of FUNCNAME).
13. The keyword NPARAM gives the number of parameters.
14. The indexed keywords PARAMi repeat the names of the model parameters, so that the
columns or keywords containing model parameters can be easily identified.
15. Other columns may be present in the table and do not affect the interpretation of the function.
For instance, if the table is used to store the results of a fitting process, one of the table
function parameters might be `iteration number' and an additional column might be the
goodness of fit parameter.
16. Note: ASCFIT distinguishes between a model's independent variables and its parameters. If
we want to use the present convention as providing a user­defined table model for ASCFIT,
we may need extra keywords to say which of the table function parameters are parameters in
the ASCFIT sense, and which are independent variables. For example, suppose in example
1 below we want to fit for the focal length given measurements of the effective area. We
then want Energy and Theta as independent variables and F as a parameter. We could add
keywords to the file to say this. However, next time we might want to fit for theta too; it may
be better to make ASCFIT able to choose on the fly which arguments to treat as parameters.
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EXAMPLE 1: Simple function. We tabulate the HRMA effective area as a function of energy,
off­axis angle, and assumed focal length: HRMA AREA(ENERGY,THETA,F).
XTENSION = 'BINTABLE'
BITPIX = 8
NAXIS = 2
NAXIS1 = 16
NAXIS2 = 12
PCOUNT = 0
GCOUNT = 1
TFIELDS = 4
TTYPE1 = 'ENERGY ' / Energy
TUNIT1 = 'keV '
TFORM1 = '1E '
TTYPE2 = 'THETA ' / Off­axis angle
TUNIT2 = 'arcmin '
TFORM2 = '1E '
TTYPE3 = 'F ' / Focal length
TUNIT3 = 'mm '
TFORM3 = '1E '
TTYPE4 = 'HRMA—AREA'
TUNIT4 = 'cm**2 '
TFORM4 = '1E '
EXTNAME = 'AREA—TABLE'
HDUCLASS = 'ASC '
HDUCLAS1 = 'FUNCTION'
HDUCLAS2 = 'TABLE'
COMMENT
COMMENT Mission and instrument are relevant when units include counts
COMMENT and pixels
COMMENT
MISSION = 'AXAF '
TELESCOP = 'AXAF '
FUNCNAME = 'HRMA—AREA' /
FNAXIS = 3 /
FTYPE1 = 'ENERGY' /
FTYPE2 = 'THETA' /
FTYPE3 = 'F ' /
FNAXIS1 = 2 /
FNAXIS2 = 3 /
FNAXIS3 = 2 /
MODEL = 'None' /
ENERGY THETA F HRMA—AREA
4

1.5 0.0 10065.0 4.8
6.4 0.0 10065.0 2.3
1.5 1.0 10065.0 4.6
6.4 1.0 10065.0 2.1
1.5 2.0 10065.0 4.0
6.4 2.0 10065.0 1.8
1.5 0.0 10258.0 5.8
6.4 0.0 10258.0 3.3
1.5 1.0 10258.0 5.6
6.4 1.0 10258.0 3.1
1.5 2.0 10258.0 5.0
6.4 2.0 10258.0 3.8
5

EXAMPLE 2: Functional case. We give the ACIS PSF, which is modelled as a functional of
summed gaussians. The table indicates the values of the gaussian parameters as a function of the
independent variables.
XTENSION = 'BINTABLE'
BITPIX = 8
NAXIS = 2
NAXIS1 = 32
NAXIS2 = 6
PCOUNT = 0
GCOUNT = 1
TFIELDS = 8
TTYPE1 = 'ENERGY ' / Energy
TUNIT1 = 'keV '
TFORM1 = '1E '
TTYPE2 = 'THETA ' / Off­axis angle
TUNIT2 = 'arcmin '
TFORM2 = '1E '
TTYPE3 = 'CORE.SIGMA'
TUNIT3 = 'pixel '
TFORM3 = '1E '
TTYPE4 = 'CORE.NORM'
TUNIT4 = 'count '
TFORM4 = '1E '
TTYPE5 = 'HALO.X0 ' / Halo center X0
TUNIT5 = 'pixel '
TFORM5 = '1E '
TTYPE6 = 'HALO.Y0 ' / Halo center Y0
TUNIT6 = 'pixel '
TFORM6 = '1E '
TTYPE7 = 'HALO.SIGMA'
TUNIT7 = 'pixel '
TFORM7 = '1E '
TTYPE8 = 'HALO.NORM'
TUNIT8 = 'count '
TFORM8 = '1E '
TTYPE9 = 'TRW—ID ' / Test used to derive this data point
TFORM9 = '20A ' / (Example of an extraneous column)
EXTNAME = 'FUNCTION'
HDUCLASS = 'ASC '
HDUCLAS1 = 'FUNCTION'
HDUCLAS2 = 'COMPOUND'
COMMENT
COMMENT Mission and instrument are relevant when units include counts
COMMENT and pixels
6

COMMENT
MISSION = 'AXAF ' /
TELESCOP = 'AXAF ' /
INSTRUME = 'ACIS ' /
DTYPE1 = 'CORE.X0' / Function parameter
DVAL1 = 0.0 / [pixel]
DTYPE2 = 'CORE.Y0' / Function parameter
DVAL2 = 0.0 / [pixel]
FUNCNAME = 'ACIS—PSF' /
FNAXIS = 2 /
FTYPE1 = 'ENERGY' /
FTYPE2 = 'THETA' /
FNAXIS1 = 2 /
FNAXIS2 = 3 /
MODEL = 'Gauss2D[Core]+Gauss2D[Halo]'
NPARAM = 8 /
PARAM1 = 'CORE.SIGMA' /
PARAM2 = 'CORE.NORM' /
PARAM3 = 'CORE.X0 ' /
PARAM4 = 'CORE.Y0 ' /
PARAM5 = 'HALO.SIGMA' /
PARAM6 = 'HALO.NORM' /
PARAM7 = 'HALO.X0 ' /
PARAM8 = 'HALO.Y0 ' /
ENERGY THETA CORE.SIGMA CORE.NORM HALO.X0 HALO.Y0 HALO.SIGMA HALO.NORM TRW—ID
1.5 0.0
6.4 0.0
1.5 1.0
6.4 1.0
1.5 2.0
6.4 2.0
7