Digital BW, The Print

One other perspective which ought to make you stop and think a little about
the current methodology.  Tyler has nicely espoused the current way of doing
things:  a desire for a nice linear (straight line) progression of density
from dMin to dMax across the full range of pixel values, 0-255 in 8 bit, 0-1
normalised.  So, for EEM for example, this would mean 0 maps to a density of
1.68 and 1 maps to a density of 0.04 and all other points in between sit on
the straight line between the two.  (If I am wrong on this please correct
me.)  In so doing you are defining (or calibrating) the colour space of the
printer.

NO COLOUR SPACE IN GENERAL USE TODAY HAS THIS SORT OF PROFILE. Not Apple
RGB, not Gray Gamma 1.8, not Adobe RGB, not Gray Gamma 2.2, not sRGB, not
LAB.  NONE of these are linear when we plot density vs pixel value.  ALL are
curved in this dimension, such curvature representing their gamma (a fancy
word for rate of change or contrast).  Plot them out.  You have all the
formulas.  Most have been covered in this thread and many are in general use
(eg we typically read LAB with our photospectrometers to then calculate the
density figures which are linearised above).  Those that aren't are covered
on Bruce's site and he even has calculators to make the job easier.

From what I can see (excuse the pun), making pixel value vs density plot as
a straight line has absolutely no basis in "colour" theory.  The only thing
that I have come across in photography that is linear in this dimension is
the measuring of light intensity by a photosensitive cell (a camera "pixel"
for example) and what is the first thing we do with this type of data?  We
apply a gamma, we curve the data.  Why?  Because the eye doesn't work in a
linear fashion in this dimension.  So I find it rather peculiar that people
on the one hand get obsessed with "how the eye sees" and yet, on the other,
ignore this when they make pixel value vs density linear.

The plot that more often than not is linear is log10(pixel value) vs
density!  ALL (with perhaps the exception of LAB, see my note below) the
colour spaces mentioned above ARE linear when we plot log10(pixel value) vs
density.  (As an aside, note that when Kodak plots the response curve of
photo paper it plots log10(exposure) vs density!) If you have a straight
line with pixel value vs density it is NOT linear when you then plot
log10(pixel value) vs density.

A note on LAB:  the LAB space may not provide a perfectly linear plot of
log10(pixel value) to density but it is very close.  If it did Bruce
Lindbloom would have known the gamma embedded in the LAB model.  Instead he
had to undertake an exercise to find which gamma best fit the LAB model.  He
came up with 2.2.