According to Weber's law, from the beginning of the century, the ratio
of the just noticeable difference 
and the
luminance L is constant, and equals 0.02 for a wide
range of luminances. Nowadays there are better descriptions of just
noticeable difference, and it is clear that it is not constant but
depends on the adaptation level, and can be approximated using Weber's
law just at certain adaptation levels.
The mapping function proposed by Greg Ward in [Ward94] relies on the
work of Blackwell conducted in the early 1970s. Using a briefly
flashing dot on a uniform background Blackwell established the
relationship between adaptation luminance, 
, and just noticeable
difference in luminance 
as:
That means that if there is a patch of luminance 
on the background of luminance it will be discernible, but
the patch of luminance 
, where 
will
not.
A more complex function for the whole range of human vision is used by
Ferwerda et al. [FPSG96], and later by Larson et al. in
[LaRP97]. It accounts for both rod and cone, response, and is
given in equation
2.17.
Ferwerda et al. [FPSG96] and Larson et al. [LaRP97] also
exploit the changes in visual acuity. Visual acuity is the measure of
the visual system's ability to resolve spatial details. It drops off
significantly for low illumination levels. Actually it is about 
at 
and drops off to about 
at 
.
Ferwerda et al. also used the time aspect of adaptation. We are all familiar with the fact that we can not see immediately after entering the cinema if the film has already begun. After some period of time we can see the details again. Using Ferwerda's model it is possible to simulate such time changes of adaptation in computer graphics.