Subject: Re: WCS problems on Optical Telescopes.

From: wmo at wansor.jpl.nasa.gov (Bill Owen)
Date: 3 Oct 1997 16:07:56 GMT

In article <DWELLS.97Sep26171353 at fits.cv.nrao.edu>, dwells at nrao.edu (Don Wells) writes:
> ... A Chebyshev polynomial is a sufficient representation of the pixel
> regularization map; the number of terms needed is essentially the same as
> the number of spline coefficients that would be needed for the same
> precision of representation. The Chebyshev expansion, which is orthogonal
> and has the equi-ripple property, can be converted to the equivalent
> conventional polynomial for interchange. I expect that the alternative FITS
> keyword notation which I mentioned in the BoF at Sonthofen will support up
> to 9999 polynomial coefficients per axis.

True, but it's not the only possible set of basis polynomials.  I use the
Legendre polynomials, because if the reference stars are uniformly distributed
the resulting model coefficients will be uncorrelated.

-- Bill Owen, wmo at wansor.jpl.nasa.gov

---

From: Don Wells <dwells at nrao.edu>
Date: Thu, 9 Oct 1997 11:19:43 -0400 (EDT)

"BO" == Bill Owen <wmo at wansor.jpl.nasa.gov> writes:

BO> In article <DWELLS.97Sep26171353 at fits.cv.nrao.edu>,
                dwells at nrao.edu (Don Wells) writes: 
>> regularization map.. The Chebyshev expansion, which is orthogonal
>> conventional polynomial for interchange..

BO> True, but it's not the only possible set of basis polynomials.  I use the
BO> Legendre polynomials, because if the reference stars are uniformly
BO> distributed the resulting model coefficients will be uncorrelated.

Yes, the Legendre polynomials are better for astrometric numerical
mapping over rectangular fields than are conventional polynomials,
because the orthogonality property means that the correlation between
the coefficients in the least-squares regression approaches zero as
the number of reference stars approaches infinity. Of course, for any
finite number of reference stars, even if uniformly distributed, the
correlation is nonzero (but smaller than the correlations between
coefficients of the equivalent conventional polynomial). I have never
understood why many/most astrometrists persist in using conventional
polynomials in their numerical mapping regressions.

An interesting question is the (dis?)advantages of using Chebyshev
polynomials instead of Legendre polynomials in astrometric numerical
mapping of rectangular fields. Both sets of polynomials are
orthogonal, so both have the uncorrelated-coefficient property. I
prefer Chebyshev polynomials because I consider the form of their
error-of-fit (the Chebyshev equi-ripple/minimum-maximum-error
property) to be desirable in astrometric numerical mapping
applications.  Comments?

For circular fields there is no question about the ideal functions to
use: the Zernike polynomials are the only reasonable choice among sets
of polynomials which are orthogonal on the unit circle. Indeed, when
doing least-squares fits of radial distortion functions in optical
systems, such as telescopes with corrector lenses, I believe that we
should use the radial polynomials from the Zernike set instead of
conventional polynomials.

The choice of which polynomials to use for FITS _interchange_ is
independent of the choice of which to use for regressions. At
ADASS'97, when I suggested transmitting Chebyshev coefficients in FITS
headers, I immediately got a comment from the floor (from Rodney
Warren-Smith?)  saying that it would be better to use conventional
polynomials because many programmers will be confused/intimidated by
Chebyshev polynomials.  A least-squares fit using a set of orthogonal
polynomials (Legendre, Chebyshev, Zernike,..) can always be recast as
a conventional polynomial for FITS interchange purposes, with no loss
of accuracy. The use of conventional polynomials will be easier to
describe in FITS standards documents, and will leave all of us free to
make our own choices of numerical mapping functions for regressions.
-- 
  Donald C. Wells         Associate Scientist         dwells at nrao.edu
                    http://fits.cv.nrao.edu/~dwells
  National Radio Astronomy Observatory                +1-804-296-0277
  520 Edgemont Road,   Charlottesville, Virginia       22903-2475 USA