Equation(s) for film/paper curves

Ian Leake · Jan 31, 2022

I would like to use software to model film/paper curves.

Does anyone know of a reference to math equations that may help with this?

Thanks in advance

DREW WILEY · Jan 31, 2022

Model what? Alleged film speed points? Simulating actual curves mathematically using calculus (making something easily seen in a simple graph hell to interpret instead)? Or some kind of downloadable software which plots those curves visually for you when you enter your step wedge density readings? (doable and realistic). No need for you to cook up your own equations. But if you do want to go that route, recognize that density units and their plotting is done logarithmic. There's not a lot to it if you have an appropriate densitometer to begin with. But for film you need a B&W transmission densitometer; for prints a B&W reflection densitometer, or else a unit combining both functions. Interpreting the results is something else.

Ian Leake · Jan 31, 2022

DREW WILEY said:
Model what? Alleged film speed points? Simulating actual curves mathematically using calculus (making something easily seen in a simple graph hell to interpret instead)? Or some kind of downloadable software which plots those curves visually for you when you enter your step wedge density readings? (doable and realistic). No need for you to cook up your own equations. But if you do want to go that route, recognize that density units and their plotting is done logarithmic. There's not a lot to it if you have an appropriate densitometer to begin with. But for film you need a B&W transmission densitometer; for prints a B&W reflection densitometer, or else a unit combining both functions. Interpreting the results is something else.

It's easy to measure and plot a curve manually, but I'd like to go further than this by using software to generate a reasonable model of the curve in software that I can then use for plotting and mathematical analysis. It's not too difficult to model part of the curve in isolation, e.g. the toe, but I'd prefer to have the whole curve as a single equation.

Nodda Duma · Jan 31, 2022

Hey Ian,

I’ve found no satisfactory polynomial to match, nor does a power series expansion work well to create a mathematical equation. Even then, I don’t think the coefficients would tie to any real world characteristics. Fundamentally, measured densities are a statistical sample and I think from a pure mathematical perspective prevents a simple solution. This does not preclude mathematically describing the photo-electro-chemical reaction (quantum mechanics), but at a macro scale statistical distributions drive the shape of the curve.

The best I’ve done is interpolation on a smooth curve fit of measured data points to establish the important points for sensitometry analysis and characterization of the emulsion. I did this in excel.

radiant · Jan 31, 2022

I think this is an interesting question.

I've actually tried this myself, not too seriously and quit because it started to become too complex to understand when things didn't work.

I think it would be sooo helpful to beginners to play around with this kind of software and see how different things affect to end results. It wouldn't need to be 100% accurate, just enough to get the gut feeling how the whole process works.

Ian Leake · Jan 31, 2022

Nodda Duma said:
Hey Ian,

I’ve found no satisfactory polynomial to match, nor does a power series expansion work well to create a mathematical equation. Even then, I don’t think the coefficients would tie to any real world characteristics. Fundamentally, measured densities are a statistical sample and I think from a pure mathematical perspective prevents a simple solution. This does not preclude mathematically describing the photo-electro-chemical reaction (quantum mechanics), but at a macro scale statistical distributions drive the shape of the curve.

The best I’ve done is interpolation on a smooth curve fit of measured data points to establish the important points for sensitometry analysis and characterization of the emulsion. I did this in excel.

That's pretty much where I got to... It just seems strange to me that after 120+ years of research, there wouldn't be a mathematical model of these curves.

snusmumriken · Jan 31, 2022

Apart from smoothing functions to make a smooth curve between your data points, there are families of curves (logistic, quadratic, etc) that can be made to conform as closely as possible to your data by tweaking their parameters using regression techniques. Forgive me if I’m teaching you to suck eggs. I suspect what you may really be asking is which family of curves (=model) is theoretically justified, so that the parameters might actually represent real features of the development process, rather than being mere tweaking factors?

snusmumriken · Jan 31, 2022

Sorry, I was typing as your later post came up. I think you might get a good fit with a logistic curve.

ic-racer · Jan 31, 2022

I'd prefer to have the whole curve as a single equation.

You won't be able to get the "whole curve" with a conventional sensitometer. You can't increase it's output beyond it's maximum, which, with a 21 step scale, will be somewhere short of the shoulder.

It just seems strange to me that after 120+ years of research, there wouldn't be a mathematical model of these curves.

There are plenty of mathematical interpretations of the H&D curves in the literature. Where did you look to not find any?

Ian Leake · Jan 31, 2022

snusmumriken said:
Apart from smoothing functions to make a smooth curve between your data points, there are families of curves (logistic, quadratic, etc) that can be made to conform as closely as possible to your data by tweaking their parameters using regression techniques. Forgive me if I’m teaching you to suck eggs. I suspect what you may really be asking is which family of curves (=model) is theoretically justified, so that the parameters might actually represent real features of the development process, rather than being mere tweaking factors?

snusmumriken said:
Sorry, I was typing as your later post came up. I think you might get a good fit with a logistic curve.

This is very helpful, thanks.

ic-racer · Jan 31, 2022

You can fit your H&D curve to a polynomial in Microsoft Excel using "Trendline" and "Show equation on chart"

Nodda Duma · Jan 31, 2022

ic-racer said:
You won't be able to get the "whole curve" with a conventional sensitometer. You can't increase it's output beyond it's maximum, which, with a 21 step scale, will be somewhere short of the shoulder.

There are plenty of mathematical interpretations of the H&D curves in the literature. Where did you look to not find any?

OSA has published at least one white paper, but you either have to pay …. or be an OSA member.

Nicholas Lindan · Jan 31, 2022

For a 1st order fit you can use a sigmoid curve - the integral of a Gaussian distribution.

To go a bit further use one sigmoid to model the toe to midsection and another to model the midsection to shoulder - shoulders and toes are not mirror images of each other.. Sometimes adding a straight bit in the middle can help.

Sigmoids work best on very simple single emulsion papers. VC papers can get a bit messier and you can end up with four to six sigmoids - one for each emulsion.

A problem with sigmoids is that they are mathematically intractable. Leading most folks to be happier with polynomial splines.

Sigmoids for film aren't that good a representation: as films are never developed to completion (with the exception of having grabbed the Dektol instead of the D-76 by mistake) there is almost never a shoulder. The best fit I have found is the outline of a humpback whale.

As sigmoids, and Gaussian distributions, pop up everywhere in nature there is a pop culture applying them to all sorts of idiotic holistic crystal navel gazing.

Google "sigmoid curve" for more.

snusmumriken · Jan 31, 2022

ic-racer said:
You can fit your H&D curve to a polynomial in Microsoft Excel using "Trendline" and "Show equation on chart"

Yes, but what would be the interpretation of the equation parameters? Unfortunately Excel won't fit a logistic, but surely that would make more sense given the way the process works? I am of course lacking any expertise in sensitometry.

ic-racer · Jan 31, 2022

snusmumriken said:
Yes, but what would be the interpretation of the equation parameters? Unfortunately Excel won't fit a logistic, but surely that would make more sense given the way the process works? I am of course lacking any expertise in sensitometry.

Everyone should do the Kodak Sensitometry Workbook PDF, it is free on the internet.
The two most common math functions for the H&D curve are contrast and speed.
For speed I do W-Speed and ASA (ISO) speed with a spreadsheet which calculates the data. Also popular is Delta-x speed but I found it harder to program that one.

This is a screen shot of my spreadsheet and a picture of the old-fashioned protractor used in the initial papers.

ic-racer · Jan 31, 2022

The output of my spreadsheet gives a contrast index, 0.1 point, ASA triangle, W-speed and safety factor.

Ian Leake · Jan 31, 2022

Nicholas Lindan said:
For a 1st order fit you can use a sigmoid curve - the integral of a Gaussian distribution.

To go a bit further use one sigmoid to model the toe to midsection and another to model the midsection to shoulder - shoulders and toes are not mirror images of each other.. Sometimes adding a straight bit in the middle can help.

Sigmoids work best on very simple single emulsion papers. VC papers can get a bit messier and you can end up with four to six sigmoids - one for each emulsion.

A problem with sigmoids is that they are mathematically intractable. Leading most folks to be happier with polynomial splines.

Sigmoids for film aren't that good a representation: as films are never developed to completion (with the exception of having grabbed the Dektol instead of the D-76 by mistake) there is almost never a shoulder. The best fit I have found is the outline of a humpback whale.

As sigmoids, and Gaussian distributions, pop up everywhere in nature there is a pop culture applying them to all sorts of idiotic holistic crystal navel gazing.

Google "sigmoid curve" for more.

Also very helpful, thanks. (I hadn't thought about the weird composite curves of VC paper.)

Ian Leake · Jan 31, 2022

ic-racer said:
Everyone should do the Kodak Sensitometry Workbook PDF, it is free on the internet.
The two most common math functions for the H&D curve are contrast and speed.
For speed I do W-Speed and ASA (ISO) speed with a spreadsheet which calculates the data. Also popular is Delta-x speed but I found it harder to program that one.

Thanks

snusmumriken · Jan 31, 2022

[QUOTE="ic-racer, post: 2513957, member: 18971]
The two most common math functions for the H&D curve are contrast and speed.
[/ QUOTE]
Presumably where a shoulder is apparent there must be another parameter that represents something that gets used up - the finite ‘population’ of silver grains available to be exposed and developed?

ic-racer · Jan 31, 2022

snusmumriken said:
[QUOTE="ic-racer, post: 2513957, member: 18971]
The two most common math functions for the H&D curve are contrast and speed.
[/ QUOTE]
Presumably where a shoulder is apparent there must be another parameter that represents something that gets used up - the finite ‘population’ of silver grains available to be exposed and developed?

I don't shoot slide film any more, but there is a much information on exposing slide film out there too (where the shoulder is used for speed determiniation. In that case the sensitometer exposes a complete shoulder, and does not fully outline the darker parts of the curve.
If you are interested in the shoulder of negative films in terms of image quality, that has been explored in depth also. Lots of interesting stuff out there! Most of the articles need to be paraphrased to be shared, unless they are really old. Those in school, may have access to papers on -line for free through the school library.

BTW this does not show the shoulder, just its effects. That is as soon as the highlights get onto the shoulder, print quality diminishes.

DREW WILEY · Jan 31, 2022

The best way to analyze a family of curves is simply to plot a family of curves and then visually compare them. That is why traditional curve plotting paper was translucent; you could overlay them atop a light box for differential comparison. Or color pencil could be used. Too much interpolation or extrapolation ruins the accuracy of the whole concept. That's why I don't fool around with hypothetical speed points or ironed out respective curve sections.

Want to see the full shouldering off, and get beyond 21 step of density, already way past what's usable anyway? Just attach a supplemental carefully measured neutral density sheet and go from there. No big deal.

I've done hundreds, probably thousands, of plots.. But I have zero interest in either dumbing anything down to rote formulas, or bringing alongside redundant mathematical convolutions just for sake of making my old high school teacher roll around in his grave any more than I already made him do. And that guy truly deserved every single spit wad ever thrown at his blackboard. I've also known a number of incredible mathematicians and physicists quite well, and none of them were worth a hill of beans when they picked up a camera.

But is someone happens to enjoy math dog and pony shows for their own sake, by all means, help yourself.

Nodda Duma · Jan 31, 2022

Different people see the world differently, Drew. Nothing wrong with a different way of wrapping one’s head around concepts. To many, mathematical formulas is as second nature as overlaying vellum.

Ian Leake · Feb 1, 2022

ic-racer said:
Everyone should do the Kodak Sensitometry Workbook PDF, it is free on the internet.

I've read this. It's a great primer but doesn't provide the depth of information that I'm looking for.

You've shown a couple of interesting charts in your posts. Could you give references for these please? Or other references in the literature?

I don't have access to academic databases or company libraries, so am restricted to what I can find on the internet, historical sources and published books. I'm happy to buy books, but given the prices of technical literature I want to know that they have the right information in them beforehand.

alanrockwood · Feb 1, 2022

Ian Leake said:
I would like to use software to model film/paper curves.

Does anyone know of a reference to math equations that may help with this?

Thanks in advance

I have worked on a computational method that seems to often give a reasonable curve shape for the toe region of a negative film and possibly part of the mid-range region as well. It's based on what I think is a semi-realistic physical model. It is amenable to short toe and long toe films. It's a little too involved to explain in a short post.

I have not extended the model to the upper mid range or shoulder regions.

I should mention that my method is best described as an algorithm rather than an equation. (It is probably possible to write it as a single equation that is applied to data fitting, but the equation would be quite complicated, and the method is easier to implement as an algorithm than as a single equation.)

Philosophically speaking (and probably practical considerations as well) methods that are based on realistic physical models, or at least semi-realistic physical models tend to work better than methods that use arbitrarily chosen functions because they tend to be more robust with respect to noise and small defects in the data. It's pretty well known among those who study numerical analysis as applied to real data that for many problems polynomial fits to data tend to be a very bad idea. This is not always the case, but is often the case. Problems with polynomials are that they tend to be very bad at extrapolating outside of the range of data that was collected, and in some cases they can produce wild oscillations in the fit, even within the data range that was collected, particularly if high-order polynomials are used. If a curve is based on a reasonable physical model those issues tend to be much less of a problem.

Here's an example of using a realistic physical model to apply to experimental data.

"Correction for isotopic interferences between analyte and internal standard in quantitative mass spectrometry by a nonlinear calibration function" by Rule, Clark, Yue, and Rockwood, published in Analytical Chemistry, 2013.

The result was very well behaved. A polynomial has the wrong functional form to do a good data fit for this problem, especially if one tries to fit the data over a wide range.

ic-racer · Feb 2, 2022

Ian Leake said:
I've read this. It's a great primer but doesn't provide the depth of information that I'm looking for.

You've shown a couple of interesting charts in your posts. Could you give references for these please? Or other references in the literature?

I don't have access to academic databases or company libraries, so am restricted to what I can find on the internet, historical sources and published books. I'm happy to buy books, but given the prices of technical literature I want to know that they have the right information in them beforehand.

A quick little history.
To find the speed of film researchers Jones, et al 'discovered' that if the shadows fall on a part of the toe curve that is tangent to a line one-third of the slope, that was the minimum exposure for an excellent print. They made multiple prints and showed them to a panel of observers.
When they did this, there were no personal computers and this 1/3 slope or 1/3 G or 'fractional gradiant' was not so easy to find outside of using pencil rulers and paper graph.

So...three approximations of this 1/3 G (fractional gradient point) were developed that could be found on the toe curve without as much trouble or without a computer:
0.1
Delta-X
W-speed

The three methods were compared and found to be roughly similar in the approximation of 1/3 G speed point.

It turns out 0.1 was eventually chosen as the method to use by modifying it slightly to specify a certain slope the film needs to have (so called 'ASA Triangle') before taking the measurement. This became ASA and was nearly copied directly for the ISO standard and also embraced by many Zone system users.

This whole history of film curve analysis would have probably played out differently had computers been available.
So, as it stands ISO is firmly established, but it is fun to look back at how it came to be.

Realize that in all this research the GOLD STANDARD is "print judgement." Yes, objective viewing of the prints by a panel of observers TRUMPS ALL THE MATH!

Equation(s) for film/paper curves

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