David J. Littleboy said:
The 4870 is a 2400 dpi scanner. As Epson explains, it uses two 2400 dpi CCDs
offset by 1/2 a pixel width. So the _optical_ resolution (at least as
defined by the pixel size: whether the lens can resolve that well (47 lp/mm
or so) is an open question) is 2400 dpi.
Rubbish! You have not described anything in the above paragraph which
defines or restricts resolution at all! Indeed, you have the
description wrong - the separation is half a pitch, not half a width -
they are not the same! Whilst optical resolution is a function of pixel
size, it certainly is not a function of pixel pitch!
Furthermore David, your entire premise, which you have been peddling in
these parts and others for at least 6 years, is that the optical
resolution limit is determined by twice the pitch of the pixels, giving
one pixel per black line and one per white line in the test - your 47
lp/mm quoted above. Nothing could be further from the truth, and very
simple tests can demonstrate this. The staggered CCD can resolve well
beyond 47 lp/mm and the lens required on the scanner clearly needs to do
so as well. A 100% fill factor CCD element of a 2400dpi CCD can resolve
up to 94.5cy/mm and a practical fill factor of, say 80% area, is capable
of resolving even more than this.
To demonstrate the resolution of the CCD, take two sheets of graph paper
(or draw this on Powerpoint or some other package). On one of them draw
a row of squares with 2cm sides (or two units of whatever size squares
are on the paper) with each square butted up against the next - this is
your model CCD. On the other sheet, draw a square wave of 4cm high and
4cm low, 8cm period - this is your resolution test pattern - high being
equivalent to "white" and low being "black". In fact, the test pattern
should be a sine wave, but square waves are easier to work with as you
will see later and the results, whilst slightly different in amplitude,
are almost identical in terms of boundary conditions.
Now place your model CCD under the test pattern - in any relative
position - and work out what signal each element in the CCD produces. If
the element is entirely underneath a high section of the pattern then it
will output a high signal, if it is entirely underneath a low section of
the pattern it will output a low signal. Some elements will fall
underneath the transition between high and low and will output a signal
which is proportional to the amount of high and low sections of the test
pattern it occupies. What you should see is that, no matter what
position you place the CCD in relative to the test pattern, the output
amplitude is still the same - peaks at 100% high and troughs of 100%
low.
Repeat this with a test pattern which is 3cm (or units) in each high and
low (ie. a period of 6cm), and you will find almost the same results -
no matter where you position the model CCD relative to the model test
pattern the output amplitude will still be the same, however this time
there are more intermediate samples where the signal is between the peak
high and low positions.
Repeat this again, with a test pattern which is 2cm (or units) in each
high and low (ie. a period of 4cm). In this case, in almost every
position the output signal is an intermediate case between 100% black
and white, but an output signal from the CCD which is representative of
the test pattern is still produced. There are two special positions
however. One of these corresponds to where the CCD centres fall exactly
below the centres of the high and low sections of the test pattern,
resulting in 100% signal on each pixel. The other is where the centres
of the CCD lie exactly on the transition of the test pattern between
high and low. In this position, and this position only, the output from
each pixel in the CCD is 50% between black and white and no
representation of the test pattern is produced.
Nevertheless, in almost every position of the model CCD relative to the
model test pattern an output representative of the test pattern is
produced. Clearly the CCD is still capable of resolving the test
pattern, except in the one case where its centres happens to line up
with the transitions. On average, the test pattern is still well
resolved.
Repeat this again, with a test pattern which is 1.5cm (or units) in each
high and low (ie. a period of 3cm). In this case all of the outputs
from the CCD elements, irrespective of the relative position of the CCD
and the test pattern, are at intermediate levels between black and
white. Nevertheless, each pixel, irrespective of the relative position
of the test pattern and the CCD, produces a different output signal. The
overall output signal does not have the same period as the test pattern,
but it is related to it. In short, even though the individual lines on
the test pattern (the high or the low sections) are smaller than the CCD
pitch, they are still capable of stimulating an output signal which is
related to the test pattern.
Repeat this again, with a test pattern which is 1cm (or units) in each
high and low (ie. a period of 2cm). In this case, no matter what the
relative position of the test pattern is to the model CCD, each and
every CCD output is exactly 50% between the black and white levels. The
output contrast in every single position is zero. This is the lowest
resolution limit of the CCD.
What you might care to notice if you actually implement the above
demonstration, is that this lowest resolution limit of the CCD (for
there are many and output signal again appears at reduced level for even
finer test patterns, such as 0.75cm widths of high and low section or a
period of 1.5cm) is actually DOUBLE what you call the optical resolution
above. With a 2400dpi CCD with 100% fill factor (as modelled by the 2cm
squares butted up to each other), this limit occurs at 2400 lp/inch,
which is 94lp/mm, not the 47 lp/mm you refer to above. Your condition
for optical resolution limit corresponds to a test pattern with a 4cm
(or other unit) period which, as demonstrated, is very well resolved
indeed. In short, the optical resolution defined by the pixel size is
actually twice the value that you claim it is!
For test patterns which are between the resolution limit that you claim
for pixel pitch and the true lowest optical resolution limit the output
of the single linear row CCD is an aliased representation of the test
pattern itself. This aliasing is imposed by the sampling density, NOT
the size of the pixel. By introducing a second row of identically sized
CCD elements that are offset by half a pitch (and you are free to repeat
the above demonstration with this arrangement if you need to prove it
for yourself) the aliasing is eliminated and the output signal
equivalent to the true optical resolution of the CCD pixel size is
reproduced exactly.
This oversampling (actually critical sampling rather than true
oversampling) of the CCD size is exactly what the Epson scanner does, it
is *NOT* an interpolation by any stretch of either imagination,
linguistics, nomenclature or semantics.
It "oversteps" this 2400 dpi
optical scanner to generate a mechanically interpolated 4800 dpi image.
NO interpolation is performed at all and, excluding the use of an abacus
or other mechanical computer to implement the necessary calculations,
there is no such thing as "mechanical interpolation"!
There are people (here and elsewhere) who claim that this "overstepping"
improves resolution slightly.
I don't think *anyone* claims that this improves resolution at all, not
even slightly. However many claim that it does improve performance and
image quality, in particular by ensuring that the scanner is limited by
its true optical resolution and not some artefact inducing digital
limitation. It does, however, result in a true optical resolution which
is lower than the Nyquist limit of the sampling system - coincidentally
meeting the requirement of all sampled systems.
In the case of the Epson 2450, 3200, and 4870,
this theory appears leaky.
Only because for the past 6 years you have failed to grasp the basic
premise of what the optical resolution of a CCD element actually is!
These scanners sample at sufficient density such that the true optical
resolution of the combination of CCD element size and the lens can be
achieved. As such their optical resolution (as opposed to sampling
density) is comparable to the same resolution limits as other
photographic systems such as lenses, telescopes, film and paper.
This is entire issue is a consequence of the misuse of the terms
"resolution" and "optical resolution" which is generic throughout the
entire scanner industry and certainly not restricted to the Epson
scanners that you cite, nor indeed even to Epson.
