Digital BW, The Print

OK got it (on the subway my maths lessons finally came back).

Normalised pixel value = ((10^(-Density))^(1/Gamma)

So if we know the dMin and dMax of our printer/ink/paper combination (by
measuring the density values of paper white and max ink load, respectively)
we can identify, in either terms of normalised pixel values or in an 8 bit
0-255 range (or any other bit rate), how much of the tonal range can be
rendered by the printer.

For EEM which I will say has a density range of 0.043 to 1.65, this range is
approximately, using the 0-255 scale, pixels from 18 to 243, assuming a
gamma of 2.2.  Anything outside this range can't be reproduced on EEM with
today's inks because we can't get the ink black enough and the paper isn't
white enough.  

So what?

Well the problem is we are taught to stretch our scan or initial digital
capture to fill the full tonal range from 0-255.  What we don't do, however,
is compress it back again to the tonal range of our printer before sending
it to print.  

But it still prints fine!

Well yes but this is because a programme like QTR (I am using QTR as the
example here only because it is transparent and simply the one I most
familiar with) has its own set of tables which tell the printer how much of
what ink to put down on paper for a given pixel value and such tables range
from 0-255.  But let's look at the end points of this table.  For example,
in a simple one ink curve file, the 100% black or 0 pixel value point is
defined by Limit_K = x where x is a percentage (assuming no use of BOOST).
If x is set to 95% then the quad file will have 0.95x2^16 = 62,259.  Note
this is the value at pixel value = 0.  Yet from above we know that a pixel
of value 18 (not 0) should be getting this amount of ink! In the table for
this ink curve the value and hence the ink ascribed to a pixel of value 18
is LESS THAN the ink it should get to be effectively portrayed as shown on
screen.

In other words, linearised or not, the printer space has a GAMMA that bears
little resemblance to the one we are using for a workspace.  It is no wonder
that the printer prints images lighter than they appear on screen - remember
all our pixels with value 18 are getting less ink than the should, and so on
and so forth.  Because we are not working in a colour synchronised world
where differences in gamma are managed and there are complicated mechanisms
for dealing with out of gamut colours we will never get a WYSIWYG workflow
if the printer space/gamma differs markedly from the workspace/gamma.

Consequently we are left fiddling with s-curves to try to get the print to
match to some extent what we see on screen.  Worse yet this fiddling is very
imprecise because it is not done with an understanding of the tonal range
the the printer is capable of and not done with an understanding of the
printer's gamma.  

Until we have a workflow in which the printer's gamma is aligned with that
of our workspace and we send an image that is within the tonal range of the
printer you will get a WYSIWYG workflow.  But it is possible to better align
the two - after all we know the characteristics of our workspace and we know
the normalised pixel values that can be reproduced by the printer/paper/ink
combination.  

If, in my example, QTR put the value 62,259 in the 18 slot and 0 in the 243
slot and linearised all points in between to our workspace gamma of 2.2, and
we compressed with a PS curve the tonal range of the image to a range of 18
to 243 then we should get a WYSIWYG result.  The PS curve is easier and more
intuitive to do - we have the end points defined for us and can think about
whether the mid point should be altered or not (followers of Mr Adam's Zone
System would make sure it was put back to 128/128 because they placed it
there at the rime of shooting).

So - any takers?



> From: Steve Kale <stevekale@...>
> Reply-To: <DigitalBlackandWhiteThePrint@yahoogroups.com>
> Date: Thu, 02 Dec 2004 11:10:04 +0000
> To: <DigitalBlackandWhiteThePrint@yahoogroups.com>
> Subject: Re: [Digital BW] Tonal range and linearization
> 
> 
> Hi Peter
> 
> This is much simpler than anything on Bruce's site.
> 
> I do know that in its simplified form:
> 
> Luminance = (normalised pixel value)^gamma
> 
> And that:
> 
> Density = -log10(luminance)
> 
> So if gamma =2.2
> 
> Density = -log10(normalised pixel value^2.2)
> 
> Which means:
> 
> (Normalised pixel value)^2.2 = 10^(-Density)
> 
> And here is where my knowledge of algebra leaves me wanting.  [if my gamma
> were simply 2 then I know I could take the square root of both sides and be
> done]
> 
> If I know a density value, eg 1.65, then
> 
> (normalised pixel value)^2.2 = 10^-1.65 = 0.02239
> 
> I am having trouble with the very last step in this simple algebra.
> 
> By the very unsatisfactory method of interpolation I know the answer is
> normalised pixel value is approx 0.1775.  In other words, a density of 1.65
> - about the max we can get from EEM - is roughly equivalent to a 82.25%
> black in a 2.2 gamma space.  Any black darker than this can not be
> reproduced on the printer.  So if you were to look at your nicely stretched
> image's histogram in PS, you would know that the left-most 17.75% (or you
> prefer to think in an 8 bit 0-255 range... all values from 0 to 45) could
> never be printed as you are looking at it!  All those pixel values are
> simply outside the tonal range of the printer.  A similar calculation can be
> done for the highlights.
> 
> Cheers
> 
> Steve
> 
> 
>> From: Peter <spamme2001@...>
>> Reply-To: <DigitalBlackandWhiteThePrint@yahoogroups.com>
>> Date: Thu, 02 Dec 2004 02:10:58 -0000
>> To: <DigitalBlackandWhiteThePrint@yahoogroups.com>
>> Subject: Re: [Digital BW] Tonal range and linearization
>> 
>> 
>> 
>> --- In DigitalBlackandWhiteThePrint@yahoogroups.com, Steve Kale
>> <stevekale@b...> wrote:
>> 
>>> So I would be grateful if someone could help me with my maths
>> problem and
>>> also for input on the printer gamma point.
>>> 
>>> Cheers
>>> 
>> 
>> 
>> Hi Steve,
>> 
>> If you haven't done so already, check out Bruce Lindbloom's site
>> (http://www.brucelindbloom.com). He has many helpful conversion
>> calculators, equations, and information about color and perception
>> theory.
>> 
>> Regards,
>> Peter.
>> 
> 
> 
> 
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