# curve fitting reciprocity data

Discussion in 'B&W: Film, Paper, Chemistry' started by Murray@uptowngallery, Oct 11, 2005.

1. ### Murray@uptowngalleryMember

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This is probably for PhotoEngineer or other technophiles.

I am playing around with a JavaScript exposure calculator of pinhole or night exposures including reciprocity failure.

Rather than using a reciprocity 'look-up table', I am trying to do some curve fitting to generate an equation so a table can be produced by the web application in response to the user's choice of film, filter, aperture, etc.

I have been using Curve Expert 1.3...maybe 1.37.

I thought I'd ask here in case this wheel has already been invented.

The problem I am having is good overall apparent curve fitting with the errors or residuals being the worst at the shorter exposures, where it matters more!

Part of the problem may be the bumps & irregularities that are apparent in commonly used correction tables...defying a clean equation.

Don't try and dissuade me from the application, just the method.

Thanks

Murray

2. ### Lee LMember

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I think the thread most in line with what you're seeking is here: http://www.apug.org/forums/showthread.php?t=11566&page=12&pp=10&highlight=reciprocity

Gainer offers a formula there, and it's discussed along with the Schwarzschild formula.

From Covington Astrophotography for the Amateur (ISBN 0-521-62740-0):

Actual Speed = rated speed*t^(p-1)
where t=exposure time in seconds
and p is a constant (the Schwarzschild exponent) which differs from film to film, from about 0.65 to 1.0, depending on film type and characteristics. This formula is best for longer exposures.

Covington's more accurate formula:
Actual speed = rated speed*(t+1)^(p-1)
where the terms are the same as the above equation and results are more representative of true film behavior at shorter exposures.

This second formula arranged to isolate the term for exposure time:
Corrected exposure time = ((t+1)^(1/p))-1

See Robert Reeves Wide Field Astrophotography p. 221 e.p. for a testing method involving use of an ND 3.0 filter. (Willman Bell, ISBN 0-943396-64-6)
This is a slight variation on Covington's method (p. 181 e.p. from his book mentioned above).

There are some general guidelines in Covington for an assumed Schwarzschild exponent (the "p" term in the equations above) given film type and speed. I haven't seen a list of tested Schwarzschild exponents for the more common recent films. Reeves has a good table for older films: http://www.robertreeves.com/filmtest.htm

Hope this helps,
Lee

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3. ### Photo EngineerSubscriber

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Murray, Gainers formula is not wrong per-se. It supplies a starting point. You can see my posts in that thread and elsewhere.

The problem arises because film can change in both speed and contrast as a function of reciprocity and therefore the change is not uniform across the entire curve. Therefore, you can be mislead if you just use one equation for all films or one point on a film to predict all changes.

I have found that the only way to test it is to look at three or more points on the curve and then determine what part of the H&D curve best fits my condition and expose for that condition / portion of curve.

Sorry I can't be of more help.

PE

4. ### gainerSubscriber

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Gainer's equation is one that engineers try when they don't know the mathematics of the underlying process. We plot data in various ways. I plotted all the data I could find, including an extansive set of tests on current films by Howard Bond. The starightest line was on log-log paper, which indicates that an equation of the form:

log(Tc) = log(a) + b*log(Tm)

or:

Tc = a*(Tm)^b.

Tc is an adjustment to the measured Tm. If you try to fit the entire exposure time, Tc + Tm, you'll have a hard time. What I found was that the same coefficient b can be used, within experimental accuracy, for all the films I plotted. The data I got from Howard's article was the time required to produce normal exposure for a wide range of measured exposures, and he commented that for the films and conditions that he tested, there was no need to worry about change of contrast. Change of film speed is accounted for when you find by experiment the corrected time.

I suggest you try separating the measured exposure time from the total and plot the difference against the measure time on log-log paper, or plot the log of the difference against the log of the measured time on linear paper if you can't find log-log paper. Don't try to account for all the jabs and quiivers. Working backwards to find the best exposure time is very tedious and subject to all the errors of any exposure determination.

If you linearize the equation by the log transformation, you can do a simple linear least-squares curve fit of all available data. Use the b coefficient you get from that analysis to find a suitable a coeficient for each film in your data base. I found no more than 1/3 f-stop deviation from a line with b = 1.62 when I adjusted the a coefficient for each film. Two of us using a light meter to measure the same scene, might well be more than 1/3 f-stop different in the end.

5. ### Murray@uptowngalleryMember

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As Mongomery Burns would say, "Ex-cellent"!

I also just found equations a moment ago at Chris Patton's pinhole site.

Murray

6. ### Murray@uptowngalleryMember

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Tc = a*(Tm)^b

is exactly what I could see in comparing 'clean' behaving films that had obvious 'scaled' fractions of an f-stop per decade of time. This was two clues in one and worked for all the modern color 35mm & 120 films I started with.

When I got to Tri-X/Plus-X I had to ask what the heck was going on and look for tools/info!

I found something interesting, related but tangentially.

I have some Aerial Surveillance Plus-X (Kodak 2402?). The data sheet claims no reciprocity correction is needed up to something like 1000 seconds.

This in unusual considering it's a cousin of 'terrestrial' Plus-X.

One problem with trying to make any sense of the data for this film is the wide range of methods, processes & chemistry options given, all of which are alien to the us civilians.

It suggests that much 'gospel' film data is just a glimpse into the same multi-dimensional possibilities, but it's generally kept simple so people can USE it.

Thanks again.

Murray