## 33064Re: Nyquist was Re: ngc7331 7 hrs Lum

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• Oct 2, 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.
Agreed. Good old Fourier analysis...
I 'simplified', and shortened what could become rather a long comment, to
give the general 1/3rd seeing 'rule'. It works better than might be
thought, because the fundamental, is not actually at the frequency
represented by the 'seeing', and with the normal 'seeing' measures, it
results in the inclusion of the next harmonic for the image date, giving
quite a close overall approximation.

> 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).
Yes.

> 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