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 .)
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 .
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 curvewyckoff. 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 sense2.