Neil said:
Could it be that Don was referring to interference patterns resulting from
the overlaying of multiple frequencies? The analogy to scanning would be
the overlaying of the scanner's frequency on the target's frequency, and
in such a case, interference patterns certainly result.
I am sure that this *is* what Don was referring to, however there is a
significant difference between the continuous waves, as use in
heterodyne systems, and sampling systems which use a series of delta
functions. Aliasing is analogous to heterodyning, but not the same as
it.
I have some problems with this analogy, because it requires too many
qualifiers to be accurate. If the two layers are from the same piece of
muslin, *and* the muslin is of high quality such that the aperture grid
formed by the weave is a consistent size, then this is more akin to the
phase shifted heterodyning of two signals of the same frequency.
Since they are at different distances from the viewer - and in the case
where the layers are actually in contact that difference will be very
small - they cannot produce the same spatial frequency on the retina,
and hence aliasing does result which is not simply phase shifting.
I don't
see it as a good example of scanning issues, because the likelihood of
both the scanner and subject frequency being the same is fairly low,
especially for silver-based negatives.
The aliased frequency is simply the difference between the sampling
frequency and the input frequency - they do not have to be the same or
even nearly the same. However differentiation of the aliased output
from the input is obviously easier to perceive when the input becomes
close to the sampling frequency. All that is necessary is that the
input exceeds the Nyquist limit - and that is a much greater probability
irrespective of the medium of the original.
Further, if the two muslin pieces are from a different weave or of low
quality such that the apertures vary in size, then it's not really a good
example to use to represent either sampling or Moiré, though it can be an
analogy to the heterodyning of two frequency modulated (FM) signals.
I disagree. The critical difference between heterodyne and sampling is
that a heterodyne multiplies two continuous level signals - the input
and the reference - at all levels, and all levels contribute to the
output. In sampling, the input is either sampled (multiplied by unity)
or ignored (multiplied by zero) - there is no continuous level between
the two and nothing between the two contributes to the output. In the
analogy with muslin, the pattern on the lower layer is either passed by
the apertures in the upper layer (multiplied by unity) or blocked by the
weave (multiplied by zero), which is much more akin to sampling than the
continuous level heterodyne process.
However, looking through the (high quality) muslin at another subject may
be a good example of both the visible Moiré problems *and* aliasing caused
by sampling. All one has to do is imagine that each square of the muslin
grid can only contain a single color. If the subject has a regular
repeating pattern, Moiré will result if the frequencies of that pattern
are not perfectly aligned with the frequency and orientation of the muslin
grid, and aliasing will result from re-coloring portions of the aperture
to conform to the single color limitation of sampling.
You seem to be limiting your definition of aliasing to situations where
the input frequency extends over many samples, which is certainly useful
for visualising the effect but not necessary for its occurrence - as
grain aliasing clearly demonstrates. Single grains usually extend over
less than one complete sample pitch.
While I understand your complaint, I think it is too literal to be useful
in this context. Once a subject has been sampled, the "reconstruction" has
already taken place, and a distortion will be the inevitable result of any
further representation of those samples. This is true for either digital
or analog sampling, btw.
That is simply untrue although it is a very popular misconception - *NO*
reconstruction has taken place at the point that sampling occurs.
Reconstruction takes place much later in the process and can, indeed
usually does, use completely different filters and processes from those
associated with the sampling process, resulting in completely different
system performance.
An excellent example of this occurs in the development of the audio CD.
The original specification defined two channels sampled at 44.1kHz with
16-bit precision and this is indeed how standard CDs are recorded. Early
players, neglecting the first generation which used 14-bit DACs or a
single 16-bit DAC multiplexed between both channels, reproduced this
data stream directly into the analogue domain using a 16bit DAC per
channel followed by a "brick wall" analogue filter. However, the SNR
and distortion present in the final audio did not meet the theoretical
predictions of the process. Initial attempts to resolve the inadequacy
involved the use of higher resolution DACs to ensure that the
reproduction system did not limit the result. Still, the noise and
distortion present in the output fell well short of what should have
been possible. Then the concept of a "digital noise shaping
reproduction filter" was introduced, such that the data on the CD was
digitally filtered and interpolated to much higher frequencies which
were then converted to analogue and filtered much more crudely, the new
sampling frequency being several orders of magnitude beyond the audio
range. Suddenly, improvements in measurable audio quality were
achieved, with the results much closer to theoretical predictions. This
was subsequently followed by Matsushit (Panasonic/Technics) introducing
MASH (Multi-stAge noise SHaping), a high bit depth (21-26bits depending
on the generation) digital filter with only a 3.5-bit Pulse Width
Modulation DAC per channel and ultimately by the Philips "Bitstream"
system where only a 1-bit Pulse Density Modulation DAC was required. In
these latter systems, which are now virtually generic in all CD players,
the full theoretical limits of the original specification were actually
met.
Oversampling, noise shaping, PWM and PDM output were never part of the
original Red Book specification and the improvements apply (and can be
measured) on CDs that were available in 1981 just as readily as they
apply to the latest CDs issued today. The difference is in the
reconstruction, not in the sampling process and the reconstruction is
completely independent of the sampling process.
I was hoping to point you to a longstanding article on the ChipCenter
website but I just checked and it has been removed - perhaps as a
consequence of my complaint about the serious errors it contained. I
have an archived copy if you are interested in reading it though!
Basically, this article, like your statement above, assumed that the
"reconstruction" had already taken place when the signal was sampled
and, after several pages of examples of perfectly valid oscilloscope
traces demonstrating the distortion introduced by sampling whilst
completely ignoring the reconstruction filter, concluded with a table of
the ratio of sampling to maximum signal frequency required to achieve a
certain signal to noise ratio in the data. Note "the data" - the
requirement for an appropriate reconstruction filter applies however the
sampled data is analysed just as much as to the final analogue signal,
and this is not only the crux of noise shaping CD players, but of the
error in the ChipCentre article. This significant but subtle error
resulted in the conclusion, fed to untold millions of electronic
engineers as fact, that 16-bit accuracy required sampling at least 569x
greater than the highest frequency in the signal - some 235x greater
than Nyquist and Shannon require - and 36x for 8-bit accuracy, with the
conclusion that audio CDs cannot reproduce much more than 1kHz at a
marginal SNR! I know many audiophiles who consider CDs to be poor, but
none who consider them to be that bad, but it demonstrates the
consequence of ignoring the reconstruction filter which is fundamental
to the sampling process.
Whether one has "jaggies" or "lumpies" on output will depend on how pixels
are represented, e.g. as squares or some other shape. However, that really
misses the relevance, doesn't it?
Not at all - it is critical. "Jaggies" in an image produced from
sampled data indicate inadequate reconstruction filters. You seem to
have a missplaced assumption that the sampled data itself is distorted -
that can only occur if the input filter is inadequate, which is the crux
of the issue of selecting an appropriate spot size and shape in the drum
scanner.
That there is a distortion as a result
of sampling, and said distortion *will* have aliasing
No, that is completely wrong. A sampled system requires two filters -
an input filter, which is present in the signal stream prior to the
sampling process, and an output filter which is present in the signal
stream after the sampling has been undertaken. Aliasing is a
consequence of an inadequate input filter. This will result in
distortion of the sampled data, however if the filter is adequate then
there is no reason for such distortion to be present.
Ideal sampling itself does *not* introduce distortion - I suggest
reading Claude Shannon's very readable original paper on the topic if
you have difficulties grasping this. Clearly practical sampling will
introduce some distortion since all samples are not in the exact place
that they should be, however in almost all cases this is negligible.
This is achieved through the use of crystal oscillators for the
generation of the sampling frequency, or high accuracy lithography of
the semiconductor industry for the generation of the spatial sampling
frequency, as is the case with scanners.
Jaggies, or output distortion, are a consequence of an inadequate output
filter - just as the inadequate output filters caused poor SNR and
excess THD (the direct audio equivalent of jaggies) in those early CD
players.
which exemplifies
the difficulty of drum scanning negatives, and that appears to be the
point of Don's original assertion.
A properly filtered sampled system exhibits *NO* distortion. Nyquist's
mathematics is not an approximation, it is *EXACT*, which is why
Shannon, who was particularly pedantic in most respects, based his
entire thesis on it, which has well stood the test of time.
Our elaborations haven't disputed this
basic fact.
On the contrary, I hope you now see the difference!
