where *n* is the number of discrete intensity levels, for 8-bit
images, *n*=256.

To find the mapping function,
,
we invert the function
to obtain
.
Since the domain and the range of the
functions of this form are identical, the inverse mapping is trivial and is
found by cycling through all values of the function. However, due to the
discrete nature of these functions, inverting can yield a function which is
undefined for certain values. Thus, we use linear interpolation and assume
smoothness to fill undefined points of the inverse function according to the
values of well-defined points in the function. Thus, we generate a fully
defined mapping
which transforms a uniform histogram
distribution to the distribution found in histogram *G*(*i*). The mapping
can then be defined as in Equation :