Subject: Re: Interpolation using sinc functions

From: [email-masked] (Frank Valdes)
Date: 10 Apr 1995 08:53:40 -0500

>  In several places (I think even IRAF is offering it these
>  days) people are interpolating data with sinc functions.

IRAF provides sinc interpolation for resampling spectra.  There is, as
yet, nothing for 2D sinc interpolation.

>  The argument for this is that the power spectrum of the
>  new, interpolated (re-binned) data is the same as the original.
>
>  I haven't seen any reference that defends this and I
>  suspect that all the interpolation does is add extra
>  high frequency noise to compensate for the smoothing 
>  inherent in re-sampling.
>
>  Is that the case?  If so, why bother? - it's not often one
>  adds noise to data on purpose.  If not, what's the 
>  physical significance of the noise introduced?

My experience with sinc interpolation concerns its use for one dimensional
spectra and that is what my comments address.  I don't have the specific
reference but the discussion of the advantages of sinc interpolation for
spectra that I read was, I believe, in a PASP article by authors from
Lick Obs.  For my purposes, and what I recall the paper stressed, is
that the advantage of sinc interpolation is if one shifts (interpolates
to a new origin) a spectrum by some partial pixel amount, possibly
one or more times, and then reverses the shifts then one will recover
the identical spectrum.  I did find this to be true, the residuals between
the original spectrum and shifting and shifting back using sinc interpolation
were at least 10x smaller than using some other interpolation such
as a low order polynomial.

This seemed like a very good property for spectral resampling and I
intended to make it the default in the IRAF spectral software.  Then
I and users tried real data.  The problem with sinc intepolation is that
if the data has any sharp features, meaning cosmic rays, the sinc
interpolation rings badly and clobbers a large part of the spectrum
(since the sinc function must be carried out quite a ways to approximate
the infinite extent of the function).  Polynomial interpolators, say
a cubic or quintic, will only ring within a few pixels of the cosmic
ray.  This data destroying feature is the main reason I don't recommend
this type of interpolation though one has the choice of using it if
you know the data is well-behaved.

Why bother?  Well, for spectra you often HAVE to resample to compare
spectra, do arithmetic, and do radial velocities.  So interpolation, sinc
or other, is a necessary evil.  The resampling does not introduce noise,
the noise actually looks smaller (except for ringing at sharp lines and
cosmic rays), but it broadens features (loss of information) and the noise
statistics become correlated across pixels.

By the way, by definition the power spectrum is unchanged (apart from
numerical limitations) since sinc interpolation is a phase shift in Fourier
space.

Cheers,
Frank Valdes
NOAO IRAF Group