Digital BW, The Print

Hi Austin,

I've been thinking a bunch about all this and think
maybe I've got something new to add.

Although I felt that your audio signal processing references
were not obviously applicable, nonetheless they sure contain
a lot of mathematical and descriptive support for your
contention of dynamic range being equivalent to number of
bits of number of steps.  I'm not an audio signal processing
expert but I figured I need to get some idea what its about.

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To start I'd like to revisit the "staircase" model.  Just
to keep it out of the contention realm, lets imagine its
just a staircase in a building.

The most basic relationship is:
Height = NumSteps * StairRise

Pretty simple so let's put in some numbers:
Height = 10 feet = 120 inches
NumSteps = 20 steps
StairRise = 6 inches

Thus 120 = 20 * 6

If we wanted Height = 150 inches we could do:
150 = 25 * 6
or
150 = 20 * 7.5

The idea is: we have 3 variables (Height, Num, Rise) and we
can pick values that satisfy the equation.  However, there
just two independent variables and one dependent variable.
You certainly can pick any of the three variables to be
the dependent -- pick any values for the other two and
calculate the third.  (Obvious, but I just want to make sure
we're thinking the same.)

Now here comes the building department and they say the
building codes require all steps in the building to be the
same and fix it at 6 inches.  Now we have one constant,
one independent variable and one dependent variable.
For any staircase, you get to either pick the Height 
determining how many steps or you pick the number of steps
determining how high the stairs go.  You have only one
independent variable and you can completely characterize
any staircase with EITHER the Height or the NumSteps i.e.
they are "equivalent" descriptions.  Not the same numerical
number but the same information.

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Now let's talk about audio systems. Just simple analog to start.

There are dB's (decibels) just about everywhere. In general
terms dB's are defined as: 10 log10 (power ratio).
  --reference from Higgin's same page as Dynamic Range.

Austin won't need this but others may wish to look at:
   http://www.phys.unsw.edu.au/~jw/dB.html

The important thing is that it compares two audio power levels
using a logarithmic scale.  E.G. every 3dB is a doubling or
halving of audio power. The other very important property
is how it relates to human perception of sound.  We perceive
loudness on basically the same scale.  Each increment of 
dB gives up an equivalent increment of perceived loudness.
The minimum discernable loudness different for humans is
somewhere around .5dB (I couldn't find an actual number so it
might be .4 or .3  or .6, but for the sake of argument I'll
keep using .5dB).  So every .5dB notch up we go gives the
different loudnesses we can hear.  Think about the high-end
amplifiers that instead of having a continuous volume 
control have a control that goes up in notches probably
labelled .5dB or 1dB at a time.  The great thing here is
that all the steps are perceived as the same size and that
size is directly related to human sound perception.

Another aspect of this, is that since dB is just a power
ratio, you can talk about a 3dB signal change anywhere in
the system.  It makes no difference if the 3dB is at the
input to the amp, inside, or at the output.  A dB is a dB
is a dB.  If you use dB values they are universal from
end to end in the audio system.

Now we define "dynamic range".  Again its just a dB measure
of the power ratio:  power of largest signal / power of the
smallest discernable signal.  "discernable" is important in
that we have to have some sound signal detectable above
the noise level -- otherwise you might be tempted to 
consider a totally zero signal that wouldn't do well in
the ratio :) .  If we come up with a 60dB number for DyR,
it means that the loudness sound is 60dB louder than the
quietest sound.  The kicker is that since all the dB's
are perceived equivalently we can say StepSize = .5dB,
Range = 60dB and voila: number of steps = 60/.5  = 120
different loudness levels.  With a fixed step size, the
dynamic range specifies both the Height and NumSteps.
Only one independent variable.

Austin, I hope you agree with all this.  It seems I've done
all this purely from the audio viewpoint.

Moving more quickly now to the digital signal processing
world, I don't know who designed DSP concepts and all their
motivations.  But if you are designing a digital system to 
mimic all the audio systems I describe above, it sure makes
sense to spec it such that you handle down to human
discernable levels and no further. You'd use number of
bits that fit the computer (16 usually) thus determining
number of steps, the step size would be the human reception
level (.5dB) and as the Higgin's shows on the very same
page we calculate the dynamic range. Really its just the
same as the analog with the addition 2^NumBits = NumSteps

I think all this agrees with your statements about
dynamic range.

-----------------------

Now let's look at it in the image world.  

To start: let's take a print with two grays that are very
close but just discernable.  For the sake of argument let's
say they differ in density by 0.03 log density units.  This
is the minimum discernable density step.  This print came
from a negative that has densities on it with in the printing
process produce the density step on the print.  What is the
density difference measured on the megative??  You can't
tell.  The print may have been made on a grade 1, a grade 3
or a grade 5 paper.  It the different cases you could come
up what the negative difference must have been.  Maybe
0.021 or 0.024 or 0.027.  The negative by itself cannot
tell you if the two tones on the print will be discernable.
This is very different than audio: a 3dB difference is
always a 3dB difference no matter what stage you look at.
On prints there is no notion of dB tonal difference.  Its
not a matter of it hasn't been defined yet. As shown in the
example its a matter of being impossible for the concept
to exist. The discernability changes from stage to stage
because of the very nature of how we do imaging.  (In fact
I think its even more intractable than that:  As Martin
observed a minimum discernable density on a single print
varies from the dark portions to the light portions.
Giving rise to the concept of Gamma.)

So for Imaging its just impossible to state one property
"dynamic range" and have it specify both the max /min
and the number of step (or number of bits).  My argument
basically goes back to the staircase model, with imaging
its a two independent variable system.  Whereas with
audio it was possible simplify the analysis to a one 
independent variable system.

---------------------

In our multitude of discussions, I never considered assuming
all the steps would be the same.  And I think your
extensive audio experience lead you to assume there
was "some" fixed step size.  Actually, more accurately, all
the formulas that you brought over had the fixed step
size builtin long, long ago -- no need to ever worry about
variable size steps in audio.

I think this is the root of all our collective confusion.
Maybe if we can agree on why we disagree, we can regroup
and agree to agree on a new view.

Regards,

Roy

Roy Harrington
roy@...
Black & White Photography Gallery
http://www.harrington.com