Thanks Ted,

This was very clear and very useful.  I would just like to add a couple
more complications to the "How many grey-levels do we need?" story: Poisson
statistics and the display CRT.  I know that Ted knows all about these two
items but he had already written a lot.

Statistics:

Ted talks about "grey levels" as if they existed in an absolute manner and
this is reasonable as long as the photon signal that they represent
involves a lot of photons (DIC images?).

However, in fluorescence microscopy, this is not always true. Suppose that
the brightest signal in your picture (the one stored at 255) represents
only a signal from 100 photons. Then, because Poisson Statistics limits the
accuracy with which we can measure this number to its square root (or +/-10
photons) then, although we may argue forever whether or not we can see the
difference between a signal representing 98 photons and one representing
100 photons (a 2% difference)  it is pointless to do so because the signal
itself doesn't know what its value is to an even greater uncertainty.

CRT

Just before the end of the process of "seeing," we  often have the  image
displayed on a CRT.  However, in general, we do not know the relationship
between digital signal in the image memory and the brightness of the
corresponding spot on the screen.

The relationship between the voltage on the control grid in the CRT and the
corresponding spot is known and is nonlinear. It is proportional to the
current in the electron beam and this varies with the (grid voltage) to the
three-halves power.  In other words, bright parts of the signal are spread
out more in intensity that they would be in a linear display (a factor that
matches our logarithmic eye sensitivity and affects the Poisson problem
seen above). The problem is that we don't know what the manufacturer of the
CRT driving circuits may have done to "correct" for this nonlinearity. The
may have done nothing or they may have use a "Gamma" correction circuit.
Gamma refers to an intentional nonlinearity introduced to "correct" (or
accentuate) some of those above.

Gamma can make a lot of difference when one is making a display system
where the most important sales criterion may be "readable text".

The nonlinearity/Gamma problems are also present in printers and other hard
copy devices but it is MUCH harder to measure or get specifications on
these.

To summarize:

1.  Don't worry about grey levels if the small number of photons are
producing statistical noise that is much greater than the step-sizes that
you can record

2.  The nonlinearities of the system used to actually view the signal also
need to be included in  your consideration.  These may be somewhat
"correctable" using a gamma control (if available) but make sure that you
don't make things worse. (i.e., make a lot of fuss about something that
turns out to be noise.)

Just thought that I would muddy the waters a little more.  And I didn't
even mention what we are really interested in: "How our mind "recognizes"
"features" in the presence of noise." Maybe someone else will do that one?

Sorry!

Jim P.


>Thanks to Hans for the great note. I'll add a couple bits to the
>conversation.
>
>I believe that the general perception of "64 gray levels is enough" derives
>from basic psychophysical studies which showed that an average person could
>discriminate a roughly 2% change in intensity. At first, it's tempting to
>then say "ok, so 6-bits is fine". But it's also wrong when applied to a
>video digitizer.
>
>Here's the error in the conclusion...
>
>As Hans noted, our visual system works on percentage changes. A video
>system, typically, works linearly, so that gray level 100 is twice the
>number of photons as gray level 50 and gray level 50 is twice the photons of
>25 (this assumes that everything is linear and calibrated). Following this
>for a range of values yields the following table:
>Gray value      Percent change for 1 gray level change
>1       100%
>2       50%
>3       33%
>5       20%
>10      10%
>50      2%
>100     1%
>200     1/2%
>
>If one assumes that the "2% change" rule is appropriate, then one would
>conclude that a one gray level change from 10 to 11 would be very
>noticeable, whereas a change from 100 to 101 should be invisible or at least
>very difficult to discern.
>
>Human studies show that we are in fact capable of seeing a range of 11
>orders of magnitude!(fig 4-3, first edition of Video Microscopy) which
>amounts to a range from 0.0000001 lumens/sq foot (illuminated by a bright
>star) to 10000 lumens/sq. foot (illuminated by the full Sun). Given a 2%
>detectable change, how many distinct brightness levels is this?
>
>0.0000001*1.02 = 0.000000102
>0.000000102*1.02 = 0.00000010404
>etc.
>1.02^x = 10e11
>x=1163
>
>In order to have a video system capable of producing the full range of human
>detectable brightnesses, it would have to have 1163 distinct brightness
>values, each one 2% different in value.
>
>To be more practical, assume that one wanted a 100 fold range of
>brightnesses, how many distinct brightnesses are needed?
>Same calculation:
>1.02^x = 100
>x = 232
>
>In fact, then, with 232 gray levels appropriately distributed, one could
>represent a 100x range of brightnesses in which each successive gray level
>is just noticeably brighter than the last.
>
>How many gray levels is this if one uses a typical, linear, digital video
>system?
>
>Assume that we are comparing systems with a comparable dynamic range, that
>is, the brightest value is 100x brighter than the darkest, non-zero
>brightness.
>
>Our typical 256 gray level, 8-bit system would give a change from gray level
>1 to 2 of: 1/256 * 100 = .256 = 25.6%. Definitely not enough!
>A 12-bit system, with the same gray level change from 1 to 2 gives:
>1/4096*100 = 0.024, pretty close to the required 2%.
>This is probably why dedicated graphic artists all demand 12-bit imaging
>(36-bits for color) for optimal graphic design.
>
>Conclusion:
>The human visual system works on a relative scale
>Digital imaging systems usually work on a linear scale
>
>In order for a linear system to represent brightnesses as subtly as our
>vision system can perceive, one needs considerably more gray levels than one
>might expect. In fact, a 12-bit system has just about enough gray levels to
>display a 100x range of brightnesses.
>
>I hope I've helped to clarify this situation. I'll probably prepare a more
>detailed version of this article for Microscopy Today if people find such a
>discussion useful.
>
>-Ted Inoue

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Prof. James B. Pawley,    (on Sabbatical)       Ph.  61-2-9351-7548/2351
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"A scientist is not one who can answer questions but one who can question
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