# Film curve plotting and fitting

Discussion in 'B&W: Film, Paper, Chemistry' started by dpgoldenberg, Jun 20, 2010.

1. ### dpgoldenbergMember

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Hi,
Over the past year or so, there have been at least two long threads here on the topic of plotting characteristic curves and fitting smooth functions to them. I have been playing with something along these lines, and I thought that there might be some interest in it. I am posting this in the hopes that someone might find it useful (or amusing) or might have some suggestions. I'll try to keep it fairly short, but would be happy to answer questions.

My idea is to fit the measured densities to a relatively simple mathematical function with a small number of parameters. After a bit of playing around I found this function:
D(x) = a1*ln(1+exp(a2*x + a3))
where D is density, x is exposure (on a logarithmic scale) and a1,a2, a3 are constants. ln is the natural logarithm and exp is the number e raised to the power in the parentheses.

The basic features of this function are that it begins at zero with zero slope, and the slope increases monotonically to a limiting value. It can thus describe the toe and linear region of a characteristic curve, but it is not suitable for a shoulder. My sense is that most modern films don't have much of a shoulder, and the toe and linear regions are, in any case, the important parts for assessing speed and contrast.

So far as I know, the equation and the parameters don't have any physical significance, but they are related to the shape of the curve in convenient ways:
1. The limiting slope of the linear region is a1*a2. If the exposure values are plotted as log base 10 values, this gives gamma. If (as I prefer) the values are plotted as log base 2, gamma is calculated as a1*a2/log10(2).

2. A fractional gradient (fg) can be defined as the gradient at any point divided by the limiting slope. This can be calculated as:
fg= 1/(1+exp(-a2*x - a3))
and can be solved for x to determine the exposure for a given fractional gradient:
x = (ln(f/(1-f))-a3)/a2
This can be used to calculate a speed point based on a defined fractional gradient (e.g. 0.3 or 0.5)

3. If the position of the toe is defined as the point where the fractional gradient is 0.5, this point is where:
x = -a3/a2

4. The width of the toe can be defined as distance between x values where fg = 0.1 and 0.9 (other values could be used, of course):
w = 2*ln(9)/a2

5. The equation for density can also be solved for x to give the exposure for a given density:
x = (ln(exp(D/a1) - 1)-a3)/a2
This can be used to calculate a speed point based on a minum density (e.g. 0.1)

The potential value of all of this is, I think, that it provides a fairly objective way of analyzing data and deriving parameters.

So far I have implemented this scheme in a Python script, using publicly available modules for the most of the math and plotting. I am attaching a sample plot, for TMY-2 developed in Pyrocat HD. In addition to the plot, the program outputs the following information:

15 min @ 73 F (blue)

gamma = 0.64 +/- 0.02
toe width = 2.55 +/- 0.76 stops
Exposure index set from density threshold (Zone I density = 0.10):
Speed point = -2.67 +/- 0.14 (log2exposure)
Corrected EI = 159.31 +/- 15.23
Avg. gradient (log10) = 0.60 +/- 0.02
Zone I density = 0.10
Zone V density = 0.81
Zone VIII density = 1.38

Speed point = -3.38 +/- 0.12 (log2exposure)
Corrected EI = 260.80 +/- 20.89
Avg. gradient (log10) = 0.55 +/- 0.02
Zone I density = 0.04
Zone V density = 0.67
Zone VIII density = 1.25

The data for these curves were measured from camera exposures. My usual practice is to define log2(exp) = 0 as the exposure indicated by the meter. The program takes the EI used for the meter measurement, and calculates corrected EI values using defined values for fractional gradient or density threshold (Zone I density). It then calculates densities for Zone I, V and VIII based on the corrected EI.

I hope that this will be of interest to those with a weakness for the more geeky end of photography! In its present form, the program is not particularly user-friendly, but I am happy to share it. If there is substantial interest, I may also be willing to put in a bit more time to make it friendlier.

David

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2. ### Photo EngineerSubscriber

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The straight line portion is the only significant part of these curves, (-2 to 4 log E) and the slope is the only part of that that matters beyond that!

PE

3. ### RalphLambrechtSubscriber

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It would be nice to have an equation format that works with film and paper. This, of course, requires adequate shoulder representation. I have posted non-linear equations in the past, which can do that, but they are not fully automatic. They do need a bit of manual tweaking of the parameters for hard paper curves.

4. ### Stephen BenskinMember

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I can't speak to the formulas, but with an average gradient of 0.60, the two speed methods should be in close agreement, or if you are using the formula for FG from before 1960, the FG speed should be one stop lower than the 0.10 fixed density method of 0.10. The FG speed point isn't where the exposure is supposed to fall, but the point from which the film speed is calculated. The FG method uses a constant. The current value for the constant is 0.4. In actuality, the fixed density method is similar in that the point where film speed is calculated isn't necessarily where the exposure is designed to fall. The fixed density method has a constant of 0.8 which shifts lowers the film speed by 1/3 stop.

Applying Zone System values to either of these two methods is also problematic because it is a method onto itself and uses different parameters; however, although not as much with the fixed density method as with the FG method.

5. ### dpgoldenbergMember

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Thanks to those who offered comments. I'll just reply briefly to all three:

Photo Engineer: Perhaps I didn't choose the best examples for my curves. I am attaching another plot with two curves for Delta 100, one developed with FX-37 and one with Pyrocat HD. These examples show that my fitting function can do a pretty good job on quite different shapes in the two region. FX-37 gives a much broader toe.

Ralph: I haven't given much thought to shoulders. But something that might work is to fit the curve in two parts: First fit the two and linear regions with my function, and then use a sort of mirror image of this function to fit the shoulder, forcing the linear part from the shoulder to fit the linear part from the first fit.

Stephen: I should explain that I am not really trying to match the standardized methods for establishing speeds, but rather doing simple testing for my own purposes, in a zone-system fashion. As I indicated in my original post my "fractional gradient" is based on the limiting gradient (gamma), not an average gradient as used in the old official fractional gradient methods. This should be slightly more conservative than the official fractional gradient method. To get my EI, I am just taking the speed point from the threshold density (or my fractional gradient) and setting this to give 4 stops underexposure from the meter reading (Zone I).

I have seen references to the constants you mentioned, but haven't seen a very clear explanation. Could you point me to a reference?

Thanks,
David

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6. ### Photo EngineerSubscriber

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David;

If you wish a good uniform representation of the scene, the fact still remains that the straight line portion of the curve should be used otherwise you compress the scale due to the change in slope (decrease in contrast).

The accepted figures are a slope of about 0.6 for a negative film (often corrected to about 0.65 for flare) and about 2.5 for a grade 2 paper. Of course, with the paper, the toe and shoulder are important. These control fractional gradients.

PE

7. ### dpgoldenbergMember

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PE,
Thanks. Now I see and appreciate your point, which I didn't at first. I am just suggesting that my curve-fitting method provides an automatic method to define the curve, which can then be used in any way to decide whether the contrast is appropriate and where the exposures should lie. The mathematical function can be manipulated to calculate the speed by just about any criteria (though some, like the traditional fractional gradient method would require an iterative procedure). Not that I am suggesting that anyone throw out their graph paper, rulers and French curves . . .

David

8. ### Stephen BenskinMember

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The best material is at the source. I have the first one in a pdf file so if you want to send me your email address, I'd be happy to send it.

Nelson, C.N., Safety Factors in Camera Exposures, Photographic Science and Engineering, v.4 no. 1, Jan-Feb 1960.

Nelson, C.N. and Simonds, J.L., Simple Methods for Approximating the Fractional Gradient Speeds of Photographic Materials, Journal of the Optical Society of America, v. 46 no. 5, May 1956.

If you want to understand more about the fundamental concept of film speed, check out Loyd Jones' seminal papers from the 30s and 40s.

It's also important to understand the standard model of exposure.

Connelly, D, Calibration Levels of Film and Exposure Devices, The Journal of Photographic Science, v. 16, 1968.

Stimson, Allen, An Interpretation of Current Exposure Meter Technology, Photographic Science and Engineering, v. 6 no. 1, Jan-Feb 1962.

There are a couple good books you might also want to look into.

Photographic Sensitometry: The Study of Tone Reproduction by Todd and Zakia.

Exposure Manual by Dunn and Wakefield. Appendix B

Photographic Materials and Processes, by Stroebel, Compton, Current, and Zakia.

The Manual of Photography, by Jacobson, Ray, Attridge, and Axford.

The Theory of the Photographic Process, 3rd edition.

David, I'm not so concerned with the shadow gradient you are using. I wanted to point out that while the original Fractional Gradient method had the speed point one stop to the left of the fixed density point of 0.10, the film speeds weren't 1 stop faster, but one stop slower. Your film speed for your gradient method is 2/3 of a stop faster than the fixed density method for the same film test. Drawing curves is great, but interpreting them correctly is key.

Speed points aren't necessarily about where the shadow exposure is supposed to fall but a point that can be defined and creates the best and most consistent quality with the greatest variety of films. Prior to 1960, the difference between the FG speed point and where the shadow exposure was supposed to fall was around two stops.

I would also like to warn you that there are discrepancies between tone reproduction / sensitometry and the Zone System. Trying to correlate them without factoring in the discrepancies can introduce errors interpreting the curve data. The second group of papers can help with the standard model of exposure. Spoiler alert. One of the biggest differences is that the Zone System uses a 4 stop range between the metered exposure and shadow density and tone reproduction theory (read ISO and the standard model) uses a 3 1/3 stop range. Another is flare. The ISO speed method, the fractional gradient method, and standard model all factor it in, the ZS doesn't. Flare is also an important factor when determining negative density range and average gradient.

Steve

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9. ### dpgoldenbergMember

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Steve,
Thanks very much for all of the reading suggestions and comments. I'll send you a personal message with an e-mail address. I think that I now at least understand the offset between fg=0.3 and D=0.1. I was treating them equivalently as the point for the shadow density, so it makes sense that my fg speed was faster than the D=0.1 speed. My naive take on it was that the two offered a reasonable bound for a practical EI.

David

10. ### RalphLambrechtSubscriber

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That's not so naive. A fixed minimum density value does offer a reasonable approach for a practical EI, but I don't think it should be as low as 0.1. You can achieve similar results to the fractional gradient by moving the speed point higher up on the characteristic curve. 0.17 for example is a very reasonable speed point for modern films. Of course, that is not Zone I but Zone I.5.

11. ### Photo EngineerSubscriber

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Ralph;

Very good comments. I would like you to consider that color negative film has a dmin (fixed minimum density) of about 1.0 to blue light and yet maintains speed. It can be done if you just make the right design choices. I've seen excellent films with a dmin that would shock you, but as long as the dmin is not entirely due to fog, and if dmax is adjusted, the film is just fine.

Remember that most people forget that dmin and fog are different.

PE

12. ### Stephen BenskinMember

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It seems to me that whatever density is chosen for the shadow density with the Zone System, that density is Zone I. Otherwise, it implies that there are specific densities for specific Zones.

David, the reason .3G was chosen was that it gave the most consistant results over a large range of film types. Other gradients were tested, including the minimum useful gradient. By using .3G, the shadow gradient isn't fixed by is connected to the overall gradient. The use of any fixed density outside of the ISO parameters is problematic with consistant results for the very reason that the shadow gradient is not related to the overall which results in a broader range of results.

13. ### RalphLambrechtSubscriber

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I don't think so. Shadow density does not automatically mean Zone I. Zones I to III are shadow zones and have different densities.
I have always questioned the choice of picking Zone I as a speed point, because Zone I is of no pictorial value. It's way to dark to see detail and offers only a hint of tonality. To me, the pictorial range starts with Zone I.5, and I see no practical need to set the speed point any lower than that.
By picking a speed point at Zone I but exposing for Zone III, we are fooling ourselves a bit. The higher up on the curve we set our individual speed points, the less we are influenced by the tonal compression in the shadows of Zone II, and they are more important than Zone I.

I don't question the value of setting the ISO standard to 0.1>b+f as the universal measuring stick to compare film speeds, but I feel more comfortable with an EI determined by a higher shadow density value, because it helps me to push the film toe into an area of tonality where it doesn't interfere with pictorial values.

Also, as we've communicated before, selecting a higher speed-point density results in similar EIs as the fractional gradient method.

15. ### Stephen BenskinMember

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Ralph,

You misunderstand what I am saying. The Zone System is a way to visualize a scene. Each scene has a range of luminances. You must place the range on the film. Whether you use stops, logs, or Zones there is a range. Since you begin you luminance range at 0.17, why not just refer to it as Zone I? If I have a seven stop range and I underrate the film by 1/2 stop that doesn't mean that I now have a 1/2 to 7 1/2 stop range. It's still seven stops. The exposure just shifts 1/2 stop to the right. The perception of the values of finished print tones / Zones does not.

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16. ### Photo EngineerSubscriber

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One thing that you keep missing is the fact that if you place the exposure on the straight line portion of the characteristic curve, this gives the best exposure regardless of any other consideration. As long as this is true, over and under exposure is relatively meaningless as long as you print through it. AAMOF, an exposure series that is maintained on this straight line can be printed at the same contrast and with the same highlight and shadow detail by merely increasing or decreasing the printing exposure by the appropriate amount.

Effectively, by this method, the zone system is rendered meaningless as are all other considerations.

Of course, few of us develop film to that exactitude and few use films that have a long straight line characteristic curve. Also, the edge effects change as does granularity, but the highlight and shadow relationships will stay constant otherwise. This works best with B&W films.

PE

17. ### RalphLambrechtSubscriber

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No, no, Steve. First of all, is Zone I really the beginning of the luminance range? Zone I is a range itself, isn't it? So, where in Zone I would you like to put the speed point? I chose the end of it, you chose the center. Secondly, who says I have to put my speed point at the beginning of the luminance range anyway?

18. ### Stephen BenskinMember

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No, I don't keep missing it. You keep thinking I do no matter how detailed and referenced my posts are. Just because I make a point on something specific and I don't address everything having to do with the subject, doesn't mean I don't understand it. I just didn't address it. Please read my posts more carefully.

You also shouldn't be so absolute in your statements. There are too many important considerations that come into play as a limiting factor as to the range of exposure possible - resolution, grain, printing time, and format size. The change in the film speed standards in 1960 are a direct result of those "unimportant" considerations.

Everytime you repeat that the entire exposure should fall on the straight line portion of the curve, I am reminded of Hurter and Driffield's Inertia speed method which as you know is not around any more for good reasons.

19. ### Photo EngineerSubscriber

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Steven;

H&D are still around every time we plot a curve of Density vs LogE. But back in their time, they had no real good emulsions with long straight curves. At that time, they were bowed and therefore contrast varied with LogE and placement resulted in a variable result in prints. Back then, the inertia speed had some reason for utility due to these older style curves. Today, this is not true.

With the bow, placing speed point on a totally variable curve was problematic and quite open to opinion. I've worked with both types of emulsions and seen the speed points determined for them.

PE

20. ### tlitodyMember

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tests I did ages ago indicated to me that you need at least 0.07 difference between zone I and Zone II for zone II to show what it should according to the zone system.
Many film developer combinations don't provide that because the toe is too shallow/long. There are two ways to deal with that, either you set film speed lower to give extra exposure and adjust development accordingly or you aim for a higher zone I value. The latter leaves some room for error so is a valid route IMO.
There is another way and based on the fact that you calibrate to a 10 zone system, then it really doesn't matter if you expose for a highlight or a shadow if you are not modifying with N+ or N- development which most roll film users don't bother to do. Exposing for a zone 7 or 8 will always give you higher density in the lower zones if the subject scale is less than 10 zones. You can correct by increasing print contrast. And since you will then get consistent zone 7 or 8 neg densities, you will get consistent print times with less time adjustment as you change print grades since zone 7 or 8 is the speed point of VC printing filters. Only when subject scale is greater than your calibrated 10 stop scale do you go back to exposing for the shadows to make sure you don't lose any shadow detail.

21. ### ic-racerMember

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Other than equating Zone with Stop, I think Zone is a word without a good definition.*

Is the zone on the print, is it on the negative, is it on the film plane at the time of exposure, is it out in the scene, is it in the photographer's mind...?????

(*Except "Zone 1" which does have a stable definition that most persons are familiar with; density 0.1 log above fb+f)

22. ### dpgoldenbergMember

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Oh my! Now I am naive *and* confused. I'm going to do some more reading and thinking before I get back into the discussion I started!

David

23. ### Stephen BenskinMember

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Once again, I have no problem with you wanting to place your exposure higher on the curve. I think you know I understand the concept. For those who do ZS speed testing, they are already getting that result. But that was not what I was addressing. I was attempting make a conceptual point that it is only Zone I 1/2 if you begin measuring the Zones at .10 but since you are beginning at 0.17 shouldn't it be considered the new Zone I just for conceptual purposes? I also thought the way you wrote the post it could be interpreted as suggesting specific fixed density values for certain Zones. I wanted to point this out because it is a frequent topic of confusion with many as they learn the Zone System.

Personally, I think the miscommunication here illustrates how needlessly imprecise the use of Zone terminology is as opposed to the use of sensitometric terminology. If the subject is about sensitometry, why not use sensitometic terminology. I don't like to talk in terms like where in Zone I would I like to put the speed point. First, it's a relative concept. Luminance range is luminance range. You place the luminance range on the curve where it can be reproduced. Film speed is a number that works with the exposure meter to calculate a camera setting to place the exposure on the usable portion of the curve.

Second, I know flare pretty much eliminates precise placement of the shadow. Third, it's only pertinent when attempting to equate the Zone System concepts to Tone Reproduction Theory. For me, I use of Zones to in help visualize what I want in the print and not as a replacement for sensitometry. Anyway, that kind of question is kind of an angles dancing on the head of a pin for anyone who thinks in terms of tone reproduction theory.

24. ### Stephen BenskinMember

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Could you be more condescending?

Actually, I believe the inertia method was history by the teens. According to W.F Berg in Exposure: Theory and Practice, "Thus all those speed criteria must be rejected which are based merely on a property of the negative material alone; unless and until it can be shown that this property is intimately liked with the quality of the print. Nevertheless, speed criteria are in practice invariably determined from the behaviors of the negative material alone. Where progress has been made in recent years in fixing a sensible criterion, it has been in the direction of linking print quality with the characteristics of the negative material."

I believe Hurter and Driffield concluded that since the straight-line portion of the characteristic curve had the greatest and most linear reproduction of tones, it was the most desirable area to place the exposure. However, they never did psychophysical testing to prove that theory. Jones did and concluded H&D were in error.

It is the lack of linking print quality with the characteristics of the negative material which is my greatest contention with the many popular exposure systems. Many of the ideas are good, but the conclusion should be considered more a rule of thumb than a true speed or exposure method.

David, here are three more papers you might find interesting.

Allbright, G.S., Emulsion Speed Rating Systems, The Journal of Photographic Science - Proceeding Issue, 1990.

Todd, H.N, Zakia, R.D, A Review of Speed Methods, Photographic Science and Engineering, v. 8 no. 5, Sept-Oct. 1964.

Holm, Jack, Exposure-Speed Relations and Tone Reproduction, IS&T's Annual Conference/ICPS 1994.

The last paper is indispensable but impossible to find. It was given to me by the author. Maybe I'll scan it. It should be shared.

25. ### RalphLambrechtSubscriber

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The only zones we need to be concerned about are the ones in the scene, negative and print.
Strictly, speaking only the zones in the scene are really zones, and they are one stop of luminance apart. Due to material characteristics, they get compressed, stretched shifted (you name it) on film and paper, losing their even increments, are not a stop apart anymore, and ultimately should not be called zones but 'values' as AA did. Nevertheless, most literature (including mine) carries the term 'zones' through the entire tonal reproduction cycle, which is not how AA used it, but it avoids adding another term.

Let me point out again, that a zone is a zone and, therefore, cannot have a single fixed value. Density values for a zone range from x to y, but as you said, most practitioners assume the center of Zone I to be at 0.1>b+f. Common regular values for film and paper (not absolute values, because it depends on material characteristics, as I said) can be found here:

http://www.darkroomagic.com/DarkroomMagic/Darkroom_files/RegularDensities.pdf

26. ### RalphLambrechtSubscriber

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No, you can set the speed point at any density or zone you like. There is no need to set the speed point to 0.1D, in fact, I think there are several arguments against it. Toe shape and shadow compression to be one, the flare issue you mentioned being another.

Yes, for any given material, you can relate densities to zones, and in my experience, many modern films and papers behave similarly enough to publish rule-of-thumb tables. The values are not chiseled in stone, are only valid for N development, but they do help in understanding the relationship between sensitometry, tonal reproduction and Zone System. Minor White and Richard Zakia used them for teaching purposes as far back as 1970.