Estimation with unknown exposure values

Generally the relative exposure values $K$ are not known a-priori. Thus simultaneously solving for $K_i$ and $F^{-1}$ is an optimization problem:

• Initially, pairwise comparagrams are constructed and unrolled, each assuming an arbitrary value of $K=1$ (e.g. $k=2$). The resulting $F^{-1}_i$ function estimates for each pair of images will have the same overall shape, but each be scaled differently, apart from noise.
• Next these estimates of $F^{-1}_i$ are used to estimate the relative $k_1$ values. Without loss of generality, $K_1=1$ may be assumed, and $K_i \forall 1<i<I$ may be found.
• Next the estimates of $F^{-1}_i$ are consolidated (averaged together) by using the above estimates of $K_i \forall 1<i<I$ to register them. This registered and averaged $\hat{F}^{-1}$ is now used to re-estimate the $K_i$ values, by comparing ratios: K_i = F^-1 - Q where $Q=\log(q)$ is the logarithm of the reference quantity of light. (At this point, this first guess is often good enough to stop, but may undergo successive refinements by continuing as follows...)
• Finally, if desired, images are sorted in increasing order of exposure (least to greatest), and a final estimate of $\hat{F}^{-1}$ is made across all pairs of images, using the estimates of $K_i$.
• If desired, this process may again be repeated, e.g. using that estimate $\hat{F}^{-1}$ to again estimate $K_i$ and so on.