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The general case: multiple images with unknown K
It can be seen, from (12)
that changing the value of changes only the amplitude
(range scaling) of . Therefore,
may first be solved using a ``generic'' value of ,
so that the shape of can be found up to another single
unknown scalar constant.
Subsequently, when working with more than two images,
a function may be found for each image pair, and since all
the estimates of 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 for the entire image
sequence. Then the individual values may be found by comparing
each to the averaged .
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
values (in which case comparagrams with like values may be averaged,
to reduce the number of all possible image pairs).
Since can only be determined up to an unknown offset and gain constant,
without loss of generality, the 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 values returned after one iteration appear in Fig 6.
Figure 6:
Estimated exposure values, 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.

Next: Weighting the solution by
Up: Estimation with unknown exposure
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Steve Mann
20020525