The general case: multiple images with unknown K

It can be seen, from (12) that changing the value of $K$ changes only the amplitude (range scaling) of ${\bf F}$. Therefore, ${\bf F}$ may first be solved using a ``generic'' value of $K=1$, so that the shape of $F$ can be found up to another single unknown scalar constant. Subsequently, when working with more than two images, a function $F$ may be found for each image pair, and since all the estimates of $F$ should vary only in overall amplitude, the curves may be scaled to the same height and averaged to obtain an average estimate of the general shape of $F$ for the entire image sequence. Then the individual $K$ values may be found by comparing each $F$ to the averaged $F$.

After a crude solution found by averaging, as a first of a succession of guesses, the result may be refined by a least squares fit across all possible pairs of images. Alternatively, a combination of averaging and least squares across all pairs may be made, especially if some pairs turned out to be related by the same $K$ values (in which case comparagrams with like $K$ values may be averaged, to reduce the number of all possible image pairs).

Since $K$ can only be determined up to an unknown offset and gain constant, without loss of generality, the $K$ values are assumed to be on the interval from 0 to 1.

As an example, nine of the eleven test images were used (two were left out), and the $K$ values returned after one iteration appear in Fig 6.

Figure 6: Estimated exposure values, $K_i$ from nine of the eleven test images. We can clearly see, in the estimated values, where two of the images were left out of the sequence.