[fitswcs] Status of WCS negotiations

[Don Wells]

Mark Calabretta writes:
 > 
 > On Wed 1998/07/15 14:36:27 -0400, Don Wells wrote
 > >that we need to implement support for the function widely. I say
 > >'pixel correction function' because I myself favor perturbing the
 > >pixel coordinates before applying the linear transformation. ..
 > 
 > ..  I would also call on
 > optical astrometrists to specify more general optical plate solutions ..

I recommend that the TAN projection be augmented with radial terms
analogous to ZPN. It is true that ZPN is sufficient to represent all
projections plus radial distortions, because the difference between
TAN and ARC is a power series, but the fundamental geometry of most
optical imagery is TAN, and therefore TAN should appear explicitly in
our notation. The primary non-TAN distortion in most optical cameras
is radial, due to corrector and field-flattener lenses. Our notation
should give explicit support for this most common case.

 > ...  Note that plate solution
 > projection types can co-exist with Pat Wallace's pixel regularization table
 > (if this is what optical astrometrists want).  ..

The six term 'plate solution' of traditional astrometry is exactly
equivalent to the CRVALi and CDij keywords. The tangent point of
traditional astrometry is exactly equivalent to the CRPIXi keywords.
The result of the linear transformation is called 'standard
coordinates' in traditional astrometry, and is exactly equivalent to
the G&C concepts. It might be a constructive exercise for one of us
(Pat?) to write up a one or two paragraph exposition of these facts,
with appropriate references to the traditional literature, for
inclusion in the G&C paper as background tutorial material.

Traditional astrometry adds polynomial terms to the six term (affine)
plate constant solution to represent various distortions of
complicated cameras. The distortions can even include things like
color (spectral index) terms, which represent lateral chromatic
abberation (I hope that FITS is never required to convey such
information). These terms are exactly equivalent to CDij keywords of
higher order, which we could agree to define. My current opinion is
that this would be a bad idea, but Doug Mink may want to argue in
favor of it, because his current astrometric implementations are done
in this style.

I recommend that we define a pixel correction matrix/function rather
than adding terms to CDij or to PROJk. Originally, four or five years
ago, I argued for a functional form. Doug Tody convinced me that we
might agree on a correction matrix more easily. This dialog between
Doug and me led to the appendix-A "Pixel Regularization Image"
proposal in the existing G&C draft. Later we realized that we need to
apply different corrections to multiple IMAGE extensions in a single
FITS file, and it was not obvious how to associate multiple
regularization IMAGE extensions with the multiple IMAGE extensions.
This led to wanting the correction to be in the header as
keywords. Representing the correction as function coefficients will
need fewer keywords than would representing it as table elements, and
so I switched back to arguing for a functional form. At first I wanted
to use an orthogonal polynomial, but when I said that in the FITS
session at Sonthofen, somebody responded that simple polynomials would
be easier to describe to implementors, and I accepted that argument.
So the concept which I believe makes most sense for us is to agree to
use a simple polynomial pixel correction function with origin at
CRPIXi, and with the convention that the correction is zero at
CRPIXi. Probably we should add the additional convention that the
first order terms should be nearly zero (i.e., their information
should be carried by CDij).

I contend that a functional form which perturbs the pixel coordinates
before the CDij/CRVALi transformation to standard coordinates is
mathematically equivalent to the higher-order terms of the plate
constant solutions of traditional astrometry.

One potential problem with incorporating the pixel correction function
into the projection formulae is that it might not properly support the
long-slit spectroscopy case. In long-slit spectroscopy a camera
records a 2-D image in which one axis is angle along a line on the sky
and the other axis is spectral. This is really a 2-D slice from a 3-D
space--the angle axis is a row of pixels from a celestial coordinates
projection, usually TAN plus possible distortion terms. The 3-D CDij
can represent the position angle of the slit on the sky, and the skew
terms can represent the refraction.  The slit should be orthogonal to
the spectral dispersion, but there is always some tilt, which will
also appear as skew terms of the 3-D CDij. Thus this notation can
represent the primary geometric properties of the long slit. However,
real slits are not straight, real telescopes often have corrector
optics before the spectrograph slit and real spectrograph cameras have
distortions. These facts imply that there is work for a pixel
correction function to do. I hope that we can agree on a correction
function notation which not only supports the direct imaging case, but
also supports the long-slit case.

-Don