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Scanning MATH

 
 
John Eppley
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      03-05-2004
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.]



 
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