[fitsbits] CRPIX clarifcation

[Mark Calabretta]

On Tue 2008/05/27 14:42:55 -0400, Jonathan McDowell wrote
in a message to: fitsbits at nrao.edu

Dear Jonathan,

>- since we have talked above about floating point pixel coordinates
>and the implication that they define an interpolated coordinate system,
>we should make sure that this interpolation is unambiguous.

There are two aspects to the interpolation:

  1) The value of the world coordinate at fractional pixel coordinate.
     FITS WCS deals with this.  The interpolated value is unambiguous,
     though since an image can have multiple world coordinate
     descriptions simultaneously it cannot be said to be unique (think
     equatorial and galactic coordinates).

  2) The interpolated data value.
     FITS has nothing to say about this and probably never will.

>Tradition has been that the WCS corresponds to the center of the pixel
>on the sky, I believe.

This is an important but quite separate issue.  I have appended a recent
email to the iaufwg that may clarify why.

Regards,
Mark

>>>

From: Mark Calabretta <mcalabre at atnf.csiro.au>
To: Steve Allen <sla at ucolick.org>
CC: IAU-FWG <iaufwg at nrao.edu>
Subject: Re: [iaufwg] Voting result on the FITS Standard 
In-reply-to: Your message of Tue 2008/05/20 11:06:58 MST
             <20080520180658.GA9914 at ucolick.org> 
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Date: Wed, 21 May 2008 11:16:21 +1000


On Tue 2008/05/20 11:06:58 MST, Steve Allen wrote
in a message to: IAU-FWG <iaufwg at nrao.edu>

Dear Steve et al.,

>> Nearest neighbor is an interpolation technique which is appropriate
>>for binned data.
>
>This might be interpreted in a way that assumes a metric for the pixel
>coordinates.  I assert that it is an egregious error to apply a metric
>to the pixel coordinates.  A metric only exists in for suitable types

Two distinct ideas need to be kept well separate:

  1) Fourier sampling theory (get out your copy of Bracewell),

  2) Data visualization.

To be concrete, assume that we have a 2D sky brightness distribution
of some sort.  According to (1), finite instrumental resolution and
other effects such as seeing are modelled by convolving the true
function by the instrumental function (which could vary from point to
point), this is the observed distribution.  We then sample this by
multiplying by a shah function - i.e. a regular array of delta
functions.  The numbers that we get are the data that we store in the
FITS file.

Fourier sampling theory tells us under what conditions we can completely
recover the observed function by interpolation - basically it has to be
band-limited and we need to sample it at above the Nyquist rate.
However, that is largely irrelevant.  All we need to know at this stage
is that it *does* make sense to interpolate.  That is what takes us from
the "integer array indices", which locate the individual delta functions
in the shah function, to what we misleadingly call "pixel coordinates".

Misleading because so far, there are no "pixels" anywhere.  All we have
are measurements of the observable function taken at precise points on a
regular grid.  Note that these points are precise points with zero
extent.

Suppose now that we want to visualise this sampled function.
Unfortunately, being infinitely narrow in extent and infinite-valued,
delta functions are difficult to visualise.  In 1D they are normally
represented by a spike on a 2D graph at the proper location with a
height that corresponds to their finite area.  This accords nicely with
our intuition but unfortunately this technique doesn't work so well in
2D.  Instead we use something called a "pixel" which is short for
"picture element".  Each delta function in the shah function is
represented by colouring the finite area between it and the delta
functions surrounding it.  Normally this area is square and has the
delta function located at the centre of it.  It is a mistake to think
that these little squares have any significance outside their intended
purpose, i.e. visualization of the value of a delta function.

Now consider the above in the context of binned data.  We may consider
the true function to be composed of a set of delta functions that
correspond to events of some sort that have a location and unit value.
For concreteness, think of photons striking a CCD array or measurements
of the weights of a sample of astronomers.  The instrumental broadening
function is a top-hat function.  The collection of observed delta
functions is convolved by this top-hat function and then sampled by a
shah function in the usual way.  Interpolation in some fashion on the
resulting histogram might (or might not) be interpreted as a measurement
of the observed function at a different point.  That is, what you would
have measured if you had repositioned the histogram bins a fraction to
one side.

If anyone has any FITS data that can't be thought of in these terms
I'd be interested to hear about it.

Regards, Mark