Scanning MATH

Discussion in 'Digital Photography' started by John Eppley, Mar 5, 2004.

  1. John Eppley

    John Eppley Guest

    Hi Bob: Some additional "info" for you to enjoy....or put you to sleep. I
    found it in my newsgroup this morning.


    The main limiting factor to any digitizing process is its sampling
    frequency.
    The sampling frequency is not the same as the maximum useful frequency that
    can be digitized, however.
    That frequency is called the Nyquist frequency, and in most cases is exactly
    half of the sampling frequency.
    We usually think of frequency as being in the time domain, but in the case
    of regularly spaced CCD sensors the Nyquist frequency is a spatial
    frequency; that is to say, a sinusoidal variation of luminance with
    distance, given in cycles per millimetre.

    For CCD arrays, the Nyquist frequency in cycles/millimetre can be expressed
    as: Nf = 1000/2p ; where p is the pitch or spacing between pixels in
    microns. (1 micron = one thousandth of a millimetre)

    [A scanner CCD resolution of 2700 dpi gives a pixel pitch of 9.4 microns and
    a Nyquist frequency of ~53 cycles per millimetre. Cycles per millimetre is
    close enough to the old photographic resolution standard of line-pairs per
    millimetre to think of the two as interchangeable. Therefore a film
    resolution of 53 line pairs per millimetre represents the maximum useful
    detail we can get from such a scanner]

    Any attempt to capture image detail with a spatial frequency slightly
    greater than the Nyquist frequency will result in a spatial or dimensional
    distortion of that detail. i.e. individual image points will be either
    stretched, shrunk, or displaced to fit the sensor matrix, and if such fine
    detail covers any appreciable area, then visible aliasing will occur.

    However, once the frequency, or fineness of detail, reaches a point where
    it's an exact multiple of the Nyquist frequency, then the spatial distortion
    is minimised and the phase of the detail with respect to the sensor matrix
    will determine the output of any individual sensor. This results in the
    signal (the image detail) being artificially enhanced, averaged, or
    attenuated, dependent only on its spatial relationship to the sensor, and
    whether its frequency is an odd or even harmonic of the Nyquist frequency.
    In other words, a transform takes place, and the CCD sensor acts as a phase
    detector, rather than performing its designed function of detecting
    amplitude. This is a most undesirable state of affairs, since it can result
    in false brightness values being assigned to pixels. Also, the phase
    relationship of the image to the sensor array is an unstable condition,
    requiring only a very small positional change of input to give a drastic
    change in output: This may even result in the scanner becoming sensitive to
    mechanical vibration. (another possible cause of spurious effects, perhaps?)

    Although the series of Nyquist frequency harmonics is theoretically
    infinite; in practise there is a natural attenuation of signal amplitude
    with frequency, and so the signal will reduce to zero after passing through
    only perhaps two or three of the phase sensitive nodes of the sensor array.
    From this we can deduce that the critical aliasing region for image detail
    is from the Nyquist frequency, up to a factor of 2 or maybe 3 times above
    it. Measurement shows that film grain size and clumping lies almost entirely
    in this critical region for the commonly used sensor spacing of 9 microns or
    so.

    From the above, it's reasonable to expect that any phase effects will be
    most noticeable in areas where there isn't much low frequency signal (large
    scale image detail) to cause a disruption of phase in the higher image
    frequencies. This seems to be what happens in practise, where grain effects
    are much more noticeable in areas of low contrast or continuous tone.

    The classic solution to aliasing is to introduce filtering at, or just
    below, the Nyquist frequency, such that the signal is severely attenuated
    above it, and thus can no longer interfere with the sampling frequency. This
    is a fairly simple thing to do with conventional time-domain signals, but
    optical spatial filtering is a different matter. Achieving the necessary
    sharp cut-off with conventional optics is far from easy, and established
    techniques involve using Lasers in conjunction with elaborate optical
    systems and carefully dimensioned aperture 'filters'. Obviously,
    incorporating these components into any affordable scanner is not really an
    option, but there is still much that can be done to alleviate the problem.

    Custom design of the lens and illumination system could go a long way toward
    reducing the effect.
    It's possible to design lenses with a fairly well regulated MTF
    characteristic, which could intrinsically reduce the image contrast above a
    specified frequency.
    An easier, but less elegant way to reduce high frequency contrast is simply
    to de-focus the optical system slightly.
    Paradoxically, it may be found that scanners with poorer focus, or inferior
    lenses, actually perform better in terms of reduced aliasing.
    One area that doesn't seem to have been explored is the use of different
    shaped apertures, other than circular, in scanner lenses to control the
    image spot characteristics. This used to be routinely done in process
    cameras to get better screen definition in halftone separations.
    Another obvious solution would be to increase the resolution (number of
    pixels per inch) of the CCD sensor itself, such that the Nyquist frequency
    was pushed up beyond the natural granularity frequency of most film types.
    Perhaps yet another avenue that could be explored would be the electronic
    filtering of the analogue data from the CCD or CMOS array; before the signal
    was passed to the A/D converter.

    These are design considerations which IMHO should be given a high priority
    by the development team of any future state-of-the-art CCD or CMOS film
    scanner, now that the basic technology is fairly well established.

    [Footnote: Because grain size is fairly constant, regardless of film format,
    it follows that the number of ppi shouldn't be reduced when scanning larger
    formats, if the aim is to recover all the information that large format film
    is capable of. This seems to have been quite overlooked in the past, and has
    serious implications for those involved in digitally archiving historically,
    or otherwise important images, from large format negatives. Unfortunately,
    image archivists seem as blissfully unaware of the problem of aliasing as
    photographers in general.]
    John Eppley, Mar 5, 2004
    #1
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