## 33035Nyquist was Re: ngc7331 7 hrs Lum

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• Oct 1, 2004
Roger Hamlett wrote:

> Whoa....
> The suggestion to sample at half the 'seeing', is a common _and wrong_
> application of Nyquists theorem...
> Nyquist, shows that for sinusoidal waveforms, the _minimum_ sampling
> 'interval', is half the wavelength, to reproduce a particular frequency.
> When sampling with a camera, the pixel size is the sampling interval, and
> the seeing, may (possibly), be taken to represent the 'wavelength', so
> people then go on to say that you have to use 1/2 the seeing as the maximum
> sample size to not lose significant resolution.
> The first (biggest) problem, is that the image is a 2D structure, not a one
> dimensional structure, and the worst sampling, is diagonally across the
> pixel, not the width of the pixel. Hence ignoring anything else, the
> Nyquist criteria, actually requires you to sample (assuming square pixels),
> at 1/2.8th the seeing, if detail is not to be lost.
> There is though also a second problem. The light curve produced by a star,
> is not a nice sinusoid. The image of a star, will be close to the shape of
> the Airy disk, then 'spread' by a Gaussian noise function. If you do this,
> and then look at the measures of 'seeing', you will see that the edges on
> the star, have much sharper rise/falls than a sinusoid, and to reproduce
> these properly, requires a slightly higher sample rate. If you sit down,
> and calculate the effects, you find that you have to sample at just over
> 1/3rd the 'seeing', to get all the available detail without oversampling.

Actually, to properly represent that shape without significant aliasing
would require much higher sampling than that. Non-sinusoidal shapes are
represented by harmonics that are 2x, 3x, 4x... the fundamental
frequency of the sinusoid.

In the real world, compromises have to be made in terms of image/signal
fidelity and noise. When the sampling is sufficient that the "noise"
due to aliasing is smaller than the noise from other sources (photon
shot noise, read noise, etc.) then you might consider it to be "ideally
sampled".

But of course bright stars have a different SNR than faint ones, so
there probably isn't an "ideal sampling rate" for the whole image.
Instead you have to compromise based upon what you are trying to
achieve. If you are doing profile fitting photometry, you'll want very
high sampling. If you're doing supernova hunting, you might make a
different choice.

In reality, the "1/3 the seeing" guideline is a simply pragmatic choice
of sampling for "pretty picture" imaging. It isn't necessarily the
optimum in any theoretical perspective.

And I'll also point out that the Nyquist Sampling Criterion has more to
it than just the minimum sampling rate. The criterion requires that, in
order to reproduce the original signal/image, that you:

1. Filter out any components that are above 1/2 the sampling rate (this
is impractical in optical systems; however, the Gaussian shape is fairly
close to sinusoidal so the residual error is not huge. It does suggest
however, that if you're undersampling you might want to defocus the
telescope a bit).

2. Apply a "reconstruction filter" when you output the signal/image.
The reconstruction filter provides the necessary interpolation between
the individual samples, to reconstruct the original image.

This last step seems to be completely ignored by astroimagers. Partly
this is due to technological limitations (see below), but I think it's
also due to ignorance of how Nyquist Sampling Criterion actually works.

In the digital domain, the only practical way to perform a resampling
filter is to upsample the data, then low-pass filter it to 1/2 the
original sampling rate. That's what the Double Size function in MaxIm
DL does. (Of course a better reconstruction filter can be had using
even higher sampling and appropriate sin(x)/x correction, like CD
players do, but that isn't practical for imaging.)

The technological limitation I mentioned comes from the fact that you
might not have enough pixels on your computer screen to display the
resampled image. However, it is a useful exercise if you are planning
on printing on a high resolution media, or if your images are smaller

Doug

-----------------------------------

Doug George
dgeorge@...

Diffraction Limited
Makers of Cyanogen Imaging Products
http://www.cyanogen.com

25 Conover Street
Ottawa, Ontario,