f_i (**x**) = f( k_i q (
**x**
)
+n_q_i ) +n_f_i
where includes sensor noise,
and includes image noise due to
quantization, compression, transmission.
(For precise definitions of these two noise sources, see [7].)

In the presence of noise,
each picture provides an estimate of the actual quantity of light
falling on the image sensor:
q_i (**x**) = 1k_i
f^-1(f_i(
**x**))
where is an estimate of the actual exposure constant ,
and is an estimate of the true camera response function ,
assuming
[7].

Multiple estimates of the actual quantity of light falling on the
image sensor may be combined as follows:
q(**x**)
=
_i c_i q_i(**x**)
_i c_i

Photographic film is traditionally characterized by the so-called
``Density versus log Exposure'' *characteristic
curve*wyckoff[9].
Similarly, in the case of electronic imaging,
we may also use logarithmic exposure units, ,
so that one image will be units darker than the other:
(f^-1(f_1(**x**))) =Q =(
f^-1(
f_2(
**x**
)
)
) - K
The existence of an inverse for follows from
a semimonotonicity assumption.
Semimonotonicity follows from the fact that we expect pixel
values to either increase or stay the same with increasing quantity of
illumination, ^{1}.
Since the logarithm function is also monotonic,
the problem comes down to estimating the semimonotonic function
and the scalar constant .

The unknowns ( and )
may be solved in a least squares
sense^{2}.