back to fitting functions
Hi,
As I indicated in my original post in this thread, the function that I proposed for fitting film curves was something I had made up, and I didn't think that it had any obvious relationship to the underlying physics. After reading one of the papers that Steve has circulated, I now realize that this function has a long and honorable history in sensitometry!
In the paper that lays out the "delta x" method that Steve favors (Nelson and Simonds, 1956, J. Opt. Soc. Am., 44, 324-333), "my equation" is described as "Luther's equation" and is used as the basis for the method (see page 326, eq. 1) The authors cite the following paper:
Luther, R., 1923, The under-exposure period of the characteristic curve, Trans. Faraday Soc. 19, 340.
I was able to find this paper. It doesn't actually present the equation, but it is clearly related and shows how far back the idea of fractional gradients goes. Luther found that the ratio of the slope of the curve at the "inertia point" to the slope of the linear region (gamma) was very close to 0.5 for many different films, with many different developing conditions (something like 700 combinations.) He then goes on to demonstrate that this relationship is predicted "on a partly theoretical basis" by the theories of the day, which were basically statistical.
A later paper (1945) by Silberstein shows that this relationship, referred to as "Luther's rule" can also be predicted with quantum mechanical models for exposure. Importantly, both the original treatment by Luther and Silberstein's show that the relationship holds even for emulsions made up of mixtures of grains with different sensitivities.
The connection between "Luther's rule" and the equation is that the functional form of the equation requires that the rule be satisfied. I suspect that there was a later, and harder to find, paper by Luther in which he presented the equation. It can also be shown that the Luther ratio is approximate 0.46 for the integrated Gaussian function discussed earlier in this thread. (This forms part of the basis for connecting the theory of exposure to the "rule")
In their paper on the delta-x method, Nelson and Simonds report that the equation worked well with all of the films they tested, developed to different contrasts, and then use it to derive the parameters for the delta-x method.
I doubt that this will convince anyone to adopt the properly renamed "Luther equation", but I was excited to find the historical connection. Robert Luther was a Professor at the University of Dresden, where he set up an institute of photographic science. His students there included Imogen Cunningham and Emanuel Goldberg, another fascinating person in the history of photography.
David
P.S. My apologies for going back to the original thread topic, which now seems to be in an off-topic phase!