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In article <132@brandx.klinzhai.RUTGERS.EDU>,
webber@KLINZHAI.RUTGERS.EDU (Webber) writes:
> In article <8703252224.AA09472@tcgould.TN.CORNELL.EDU>,
> cheryl@TCGOULD.TN.CORNELL.EDU (cheryl) writes: 
>> The 255 different levels are 255 settings on the VOLTAGES of each
>> of the 3 color guns.  It is a well-known principle of computer
>> graphics that there is a nonlinear relationship between these
>> these voltage settings and the actual intensities that are
>> displayed on the screen.  In fact, it is an exponential
>> relationship.  Standard nomenclature refers to the factor in this
>> exponential relationship as GAMMA, hence the phrase "GAMMA
>> correction."  Since the gamma correction necessary varies from
>> monitor to monitor, most manufacturers of graphics hardware leave
>> it up to the on-site computer graphics wiz to do the gamma
>> correction herself -- or himself as the case may be.
> 
> The nonlinear relationship between the voltage and light energy
> emitted by the phosphors is indeed GAMMA, but this is not an
> exponential relation.  It is a low order polynomial, roughly x to
> the 2.2 (on average maxing around x to the 2.8 for some monitors).
>
> There is an exponential relationship between light energy and the
> neuropsychological perception of brightness, but that is another
> matter (i.e., you would percieve x, 1.02 * x, 1.02 * 1.02 * x, 1.02
> * 1.02 * 1.02 * x, etc. as a linear progression in brightness
> although x is light energy).
>
> Thus GAMMA correction is a monitor hardware internal problem
> whereas the exponential intensity perception is a
> psychoneurological phenomenon.
>
> Although there are doubtless many technical references on this
> material, a good introduction to this sort of thing is:

Bob Webber points out that the relationship between screen intensity
and video signal voltage for a CRT is really a polynomial, not
exponential as Cheryl said.  He's technically correct, but Cheryl's
description is basically right.  Writing it down as a formula,

                                           gamma
        screen_luminance = (signal_voltage)

where '=' really means "is proportional to".  Perhaps this should be
called a "power law" relationship, since "exponential" isn't correct.
The value of "gamma" varies with the monitor.  Also, note that this
is only an approximation - the response of a real monitor is more
complex.

This non-linearity of the monitor is usually corrected for by loading
the frame buffer's colour lookup table with a table that generates
the correct signal voltage for each possible pixel value.  Making use
of the approximation above, for a pixel intensity I in the range
[0..1], and a video DAC whose inputs are in the range [0..DACMAX],
the DAC input (and thus lookup table entry) should be

        table_entry = round_to_int(DACMAX * I ** (1/gamma))

This is referred to as "gamma correction".
This ensures that the intensity of light coming off the phosphor is
proportional to the intensity that you calculated for that pixel
(except for errors in the "gamma" model of CRT behaviour).


However, none of this really addresses the original question, which
was (paraphrasing) "how do you select a representative set of
intensities for each colour?".  The most common method of having
pixel values represent intensities in a frame buffer with 8 bits per
pixel per colour is to have a pixel value of P represent an intensity
of P/255.

This produces the cheapest pixel encoding to calculate, but it is not
the best use of the bits.  The visual system responds to the ratios
of brightnesses between areas, not the absolute difference - it is
effectively a logarithmic measuring device.  If you display a set of
equal-sized objects with luminances of 1, 2, 4, 8, 16, 32, and 64
foot-Lamberts, you eye will perceive equal brightness changes between
each patch.  Thus, to make the best use of a small number of bits per
pixel, the available pixel values should be spaced evenly in
perceptual brightness, which means that the pixel codes should be the
logarithm of intensity, scaled appropriately and rounded to an
integer.  For example, if you have 8 bits per pixel (per colour), and
want to encode an intensity range of [0.01, 1], the pixel value is
given by

        P = round_to_int(255 * (1.0 - log(I) / log(0.01)))

It doesn't matter whether "log" is base-10 or base-e.  Note that you
have to decide in advance how much intensity range you want to
encode, and that intensities below this range produce negative pixel
values that you have to clamp to 0.  As long as the number of DAC
bits is about 2 or more greater than the number of pixel bits, log
encoding is the "optimum" method of encoding intensity in pixel
values.

When you only have 8 bits per pixel for all colours, you have to
divide them carefully.  The simplest method is to use 3 bits for each
of red and green, and 2 bits for blue (since blue does not appear as
bright to the eye as red or green, it contributes less to overall
perceived brightness).  Thus you get 8 distinct values of red and
green, and 4 of blue.  Intensities are converted to pixel bits using
the formula above (with 255 replaced by 7 or 3) and then shifted and
ORed together.

An alternative that provides more equal resolution in all colours is
to encode red and blue as integers in the range [0-5] and green in
the range [0-6].  Since 6*7*6 < 255, this will work.  The pixel value
is encoded by multiplying:

        P = (R*7 + G)*6 + B

Using 7,7,5 instead of 6,7,6 will also work.

Whatever way the pixels are encoded, the decoding is done by the
colour lookup table.  For each entry in the table, calculate the red,
green, and blue intensities represented by that particular pixel code
taking into account the way they were encoded (linear or logarithmic)
and packed (shifts or multiplies).  Then calculate the DAC input
needed for each of the RGB intensities using the gamma correction
formula above, and you have your colour table entry.