Digital BW, The Print

What follows is as it has been explained to me by an Imaging system 
scientist/design engineer. He also happens to be a skilled printer and 
photographer, who does his prints on EPSONs. He also is a regular contributor 
to my (shamelss plug)  1700+ member user list of EPSON photo printer 
afficionados.  ;-)

To cut to the bottomline for those who inveterately turn to the last page 
first.. For current EPSON printers, the max usable and demonstrably better 
image resolution (when sent to the driver) is 720 dpi (assuming the image is 
not being resized by the driver). I'll leave it to my friend to explain why...

Keith

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Well I don't know any image theory that suggests that you should only 
send 1/2 or 1/3 of the printer resolution - unless you are working with 
a basic halftoning algorithm, in which case the maximum resolution 
should be around the square root of the number of dots in each half tone 
cell - so a 3x3 cell would require 1/3rd of the resolution of the 
printer.

However most printers these days, and certainly all of the Epson range, 
use a stochastic dither process similar in principle to the operation of 
a delta-sigma ADC.  The printer places the dot of the closest tone and 
hue to the required pixel values and then the software calculates the 
error between what was placed and what should have been placed.  This 
error is added to the value of hue and tone for the pixel to be placed 
at the next dot position and the process is repeated - the printer 
places the closest dot type it can to the required pixel hue and tone 
and the error is again calculated.  (In practice, the error is 
distributed around the adjacent eight dots in the matrix, rather than 
carried along the line, as it is with digitising an analogue signal with 
a single bit delta-sigma ADC.)

Obviously the printer has a limited set of ink tones, determined by the 
number of droplet sizes and colours available, so it is unlikely that 
any individual drop will result in exactly the tone and hue of the 
intended pixel.  However the error between what is placed and what 
should be placed is distributed to neighbouring dots at the full dot 
resolution of the printer.  Significantly, the error is not reproduced 
spatially, but as a noise in the amplitude of the image.  Consequently 
there is no need for each pixel in the image to be represented by many 
dots in the printer - since the amplitude error in hue and tone can just 
as easily be accommodated by appropriately adjusting adjacent pixel 
values - whether the adjacent pixels have the same tone (such as if they 
were indeed the same pixel) or not.

What this means is that with such a stochastic dither process, the 
maximum spatial resolution which the printer can reproduce is EXACTLY 
the same as the dot pitch - although at that limiting resolution the 
signal to noise of the image will be very low.  In fact it can be as low 
as about 75% of the intensity of the dots, or an SNR of only 1.5dB.  So, 
for example, if only one dot size is available and only 4 colours are 
used then clearly the printer can only place a maximum intensity dot or 
not for each pixel.  The worst error occurs when the pixel is at exactly 
mid level and the printer must decide either to print a dot or not - in 
each case the error being approximately half of the available intensity 
range in the image.  Since the pixel intensity was at mid range and the 
error was equal then the SNR is unity, however the root mean square 
error over all pixel intensities is reduced by the square root of two, 
to 0.7, hence the 1.5dB SNR figure mentioned earlier.

Obviously a signal to noise of only 1.5dB is pretty poor but the error 
distribution means that the spatial integration due to limited resolving 
ability of your eye is perfect - just as oversampling DACs in CD players 
shift the noise up and out of the hearing band, so the dot placement 
algorithm used in modern inkjets also pushes the noise beyond the visual 
resolution limits at normal (and even close) viewing distances.  With a 
10" viewing distance, the human eye can only resolve around 250ppi on 
the page, so the output of a 2880x720dpi printer is averaged over around 
33 dots, increasing the SNR to a more respectable 16.5dB.  Given that 
the human eye can only resolve around 50-100 tones at a fixed 
illumination level, this is not much less than the capability of the eye 
itself.  Clearly at higher magnifications, or closer viewing, the 
shortfalls become obvious, hence the need for multiple drop sizes and 
colours to get near photographic quality output.

So a 2880x720dpi printer can certainly resolve an image of 
2880x720pixels per inch presented to it.  A more practical limit though, 
also exploiting the resolution limit of the eye, is imposed by most of 
the Epson range, which resamples the image to 720dpi in both axes - 
irrespective of the dot size and pitch.  This ensures that the image SNR 
never falls below a certain level (which they have deemed to be 
acceptable) under any conditions.  Consequently there is no point with 
any of the current Epson range in sending anything above 720ppi to the 
printer - it will be resampled, and this resampling may be the source of 
the urban legend that insists on 240ppi as being the limit.  There are 
some esoteric reasons why printing at an integer division of the 
resampling pitch might give more pleasing results - the harmonics of any 
spatial frequencies alias back on themselves - but in practice this 
doesn't make too much difference with real images where the spatial 
frequencies are only present over a limited extent, so aliased harmonics 
don't actually reproduce in the image at all.

Typically I try to print at anything up to 480ppi as a limit but have 
gone up to 600 or even 720ppi on occasions where the image can stand it 
- large areas of near uniform colour (or test images such as resolution 
charts where errors at high resolution are distributed spatially as 
noise).  I try to avoid printing below 200ppi since a keen viewer can 
start to discriminate individual pixels around that level.  The problem 
of printing at higher resolutions has more to do with the source of the 
image than the capabilities of the printer.  In scanned images, for 
example, film grain can be highly accentuated due to the errors at each 
dot placement, making the print look substantially grainier than an 
equivalent chemical print.

Hope that helps.  There are no "magic numbers" - just throw whatever 
resolution you end up with at the printer and let the magic of 
stochastic dithering and spatial noise shaping sort it all out for you. 
:-)

Kennedy