Let us now consider the brightness perception. Brightness is the magnitude of the subjective sensation produced by visible light. The light intensity can easily be measured, but brightness as a subjective phenomena cannot be exactly measured. Nevertheless, brightness is often approximated as log luminance, or luminance powered to 1/2 to 1/3 depending on the authors. More precise studies showed that there is no one single formula, but rather the brightness-luminance relation depends on the adaptation level and the surrounding light. We will describe the work of Stevens et al. [StSt63] extensively used by Tumblin and Rushmeier in developing their tone mapping operator in this section.
Stevens et al. [StSt63] devised the ``brils'' units to
measure the subjective value of brightness. According to Stevens 1
bril equals the sensation of brightness induced in a fully
dark-adapted eye by a brief exposure to a 5 degree white target of
(
) luminance.
Note that two images with different luminance values can have the same brightness values, and appear to be the same. The reason lies in the adaptation mechanism, and the inability of neural units to transfer high dynamic range signals from the retina to the brain. Actually we are very poor judges of absolute luminances, all that we can judge is the change in luminance, i.e. the brightness.
What did Stevens do? He measured brightness as a function of
luminance and adaptation by using ``haploskopic matching''. That
means he tried to match the brightness when one eye is dark adapted
(standard condition for brightness measuring) and the other eye is
adapted to a test value. Brightness comparison between two eyes was
made quickly, before either could change adaptation level
significantly. Measured brightness is then:
where B is brightness in brils, 
is radiance of target in
millilamberts, 
is threshold of detectable radiance in
millilamberts (this depends on the adaptation radiance), and n and
K are constants, dependent on the strength of the adapting
field. For full dark-adaptation 
, n=0.33, and K=10.
Stevens proposed the next equation from his measurements:
where, assuming 
is adapting, white background luminance in
lamberts and 
is target luminance in lamberts, 
in
dB, where 
and R is the target luminance difference in dB
After substituting S and R expressions in equation 2.19 we
can write the final equation:
where B is brightness in brils, L is viewed (target) radiance in
lamberts, is luminance of white surround and
These complex formulas provided by Stevens are, unfortunately, neither
valid nor accurate when applied to more complex images. They are valid
for laboratory settings only. Bartelson and Breneman
[BaBr67] have measured many test photographs in order to find
appropriate brightness versus luminance function for more complex
images. They have proposed an extended formula for complex scenes:
where 
and 
are parameters dependent
on viewing conditions and are given graphically.