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33035Nyquist was Re: ngc7331 7 hrs Lum

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  • Douglas B. George
    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
      than your screen size.

      Doug

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

      Doug George
      dgeorge@...

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

      25 Conover Street
      Ottawa, Ontario,
      Canada, K2G 4C3

      Phone: (613) 225-2732
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