Film curve plotting and fitting

Photo Engineer

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Just a personal aside

Guys;

After sitting back and reflecting on things here in this thread I realized something. I spent 15+ years doing this type of work for paper and film and another 15+ for emulsions for a total of 32 years. I used to bring work home and work to 2AM with curves and derivatives of curves, speeds, emulsion blends, dye stability and emulsion keeping tests - you name it.

Then I retired and began to enjoy photography once again. I used to walk in the parks here and just take pictures. This was something I had lost in the flurry of many many numbers and curves.

Then, in 2005 or thereabouts, I began making emulsions and it was curves and math all over again. I don't enjoy pure photography as much as I used to.

So, you must realize that this is reflected in my answers. I know that the engineers need to do this, but I am amazed that others want to do this when I think I would prefer taking pictures.

Now, some have asked me how to do this or that, and it takes me days and weeks to come up with answers. Why? Because all of my programs and tools were left behind at Kodak. I have dozens of textbooks but the detailed reports are at EK as well. And finally, well, after doing it all those years I would express it this way... "How do you tell someone how to ride a bicycle?" It is more difficult than you imagine just using words. And, when you get right down to it there are not all that many textbooks or detailed articles.

So, I have really no significant software nor literature database here to work with than the textbooks I have mentioned, and now reading them critically I find omissions or what appear to be contradictions everywhere I look.

I can only tell you what I use, why I use it, and to the best of my ability I can extract curves from the texts that point in the direction of what I have experienced personally.

PE
 

ic-racer

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Taking pictures YES!

Actually that is why I'm intersted in having the computer to a quick analysis for me to quickly test for the minimum info I need from the film (speed/contrast) so I can start shooting and processing a film that is new to me.

I tested film in graduate school for classes and curiosity, then did no testing for probably 20 years, unitl I noticed my favorite films and developers were getting harder to get. Even this last spring I ran out of t-max developer and the local store had it on back order. I had to process a bunch of film in Rodinal 1:25. Even though I am good at guessing, I ruined a batch of 4x5 with severe underdevelopment. Now that I have assimilated a quick testing method, I hope this won't happen again.
 

RalphLambrecht

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Photo Engineer said:
... And, when you get right down to it there are not all that many textbooks or detailed articles. ...
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Ron

We all appreciate you sharing your experience. You bring a wealth of knowledge to this forum. Thanks for that.

I highlighted a part of your statement, which I always wondered about. It seems, we are all reading from the same textbooks (probably the reason why Nicholas' curves matched yours). In essence, we are all referring to the same 50-year-old research, written by a handful people, most of which were employed by Kodak. This doesn't make it wrong, but where is the the parallel research from Ilford, Agfa, Fuji and others? They must have done it too, because sharing wasn't always as easy as today. Why is it that the Kodak research is so overwhelmingly sited, and other research is hardly ever mentioned? Language barriers? Lack of publications? More US APUG users than others?
 

RalphLambrecht

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Nicholas

I had no problem fitting the s-curve portion with a 4th order polynominal. Even a 3rd order was very good. But I'm sure you knew that anyway. This would not work if you included the entire data set, of course, but there is no sense in including the horizontal lines, because they contain no relevant exposure data. As you can see, not a problem with polynomials as long as you stick to the relevant data from toe to shoulder. And much better than a normal distribution curve, by the way.
 

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ic-racer

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I'd still like to see anyone post a solution ( handwritten on paper or an actual spreadsheet, or the steps involved in solving it, etc.) to any of these nice gaussian, polynomial or spline curves, solving for y = 0.1, asdpgoldenberg demonstrated for the curve fit from the original post for this thread

x = (ln(exp(D/a1) - 1)-a3)/a2
Click to expand...

He is using "D" as "y",

so it would be x = (ln(exp(0.1/a1) - 1)-a3)/a2



However (Ralph, correct me if I am wrong) I could not get DeltaGraph to do a "user specified" curve fit like the one mentioned in the original post (D(x) = a1*ln(1+exp(a2*x + a3)) ) to get the a1, a2 and a3 values. Did you try it also?
 
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Anon Ymous

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If you use cubic splines, then finding the point where D = 0,1 is easy. You only need to find which of the 3rd degree polynomials has the solution, based on the knots and their values, and then solve a 3rd degree equation.
 

Photo Engineer

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Kodak research was in Rochester, Harrow England and at Pathe in Paris, France, later moved to Chalon su Saon. (can't do the accent mark here)

There was also a small research unit in Australia.

Anyhow, we always exchanged scientists for weeks, months or years when I was at Kodak. I have spent at least a week each on several occasions at Harrow and Chalon. And, these fine researches did make huge contributions to internal and external publications. Just look at Sir Bob Hunt who got many awards from his work on color.

In any event, these companies did publish in the SPSE journal but not so much in text books. In many cases, they relied on EK research and then took advantage of it, although to be fair both Ilford and Kodak benefited from the WWII intelligence reports on Agfa by learning about gold sensitization.

Mees himself admits that he will not disclose some things in his book and Haist and Mees both left out much information on emulsion making and testing. There is so much more involved than is in any textbook, and when you get to color it becomes a maze of matrix math added to calculus!

So, this field is rather close mouthed about formulas, procedures and just about everything from start to finish including testing.

IC has shown a particular problem with his post. The use of Rodinal with a film has to be done on a trial and error basis to find a sweet spot. I have the same problem as I work on desiging a new film developer. I can eat up over $100 just finding the correct development time of a given film in a new developer. That is why I designed a paper developer first. It was far less expensive and a lot easier to test.

In any event, I was trying to show where I was coming from and that is that the curves are not necessary if you use something "known" like TriX in HC110 or the like.

Thanks for your kind comment. It was appreciated. Oh, and when I spoke with Grant the other day, he asked me to say "Hi" to you.

PE
 

ic-racer

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Anon Ymous said:
If you use cubic splines, then finding the point where D = 0,1 is easy. You only need to find which of the 3rd degree polynomials has the solution, based on the knots and their values, and then solve a 3rd degree equation.
Click to expand...


Easy for you

What software are you using to solve a 3rd degree polynomial? Or how are you making it 'easy?'

Or, which one of the methods here is the 'easy one?'
 
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Anon Ymous

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ic-racer said:
Easy for you

What software are you using to solve a 3rd degree polynomial? Or how are you making it 'easy?'

Or, which one of the methods here is the 'easy one?'
Click to expand...

Look here if you wish to see some methods. On the other hand, if you do find the 3rd degree polynomials required for the cubic spline, then there will be one of them where density will be 0,1 in its subinterval. I don't remember the name of the algorithm, but you split the subinterval ( say [xo, x1]) in a way that the f(x0) < 0,1 < f(x1). After some recursion, xo and x1 will be very close, practically a point. There's no need to be more accurate than that.

EDIT: Obviously, the last method is applicable regardless of the nature of the polynomial, or function in general.
 
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Stephen Benskin

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Photo Engineer said:
I know that the engineers need to do this, but I am amazed that others want to do this when I think I would prefer taking pictures.
Click to expand...

I know I'm never happier then when I'm shooting, but I don't think it has to be one or the other.

Ralph
Why is it that the Kodak research is so overwhelmingly sited, and other research is hardly ever mentioned?
Click to expand...

I remember reading in the Mees biography that when they were setting up the research department Mees and Eastman wanted to make Kodak the photographic authority by first doing pure research at a time most companies didn't. And to publish their findings whenever possible and to publish them using common language for the widest possible assessiblity. Plus, Kodak had Mees and Jones.
 

Ian Grant

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Well Kodak certainly published many papers in the British Journal of Photography until the mid to late 1930's. Mees knew George Brown the Editor well before he joined Kodak, he'd contributed material while at Wellington & Ward and they'd have met socially. While the links remained after Brown's death they became less important.

Ian
 

ic-racer

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Thank you for your expertise on the subject. But it still seems like a lot of work with manual calculations.

Are you able to show specifically how to solve the 4th order equation in Ralph's post ((there was a url link here which no longer exists)) and solve it for 0.1 for us??? If you can't read the exact numbers from the post just make some up. Or if you want Ralph could probably re-post it more clearly.
 

Anon Ymous

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IC, if someone is able to do some programming, then it's a "code once, use whenever you wish scenario". Frankly, IDK about 4th degree polynomials, I'm not a mathematician. But think about the second method I mentioned. Perhaps I wasn't clear, but it is a viable solution if you just want to know at which point density has reached 0,1 and you're able to do some basic programming.
 

Nicholas Lindan

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Nobody is questioning the ability to fit one or more polynomials to the central portion of the HD curve.

My question is: What value does a polynomial fit provide?

*******************************************************

On other points:

1) Here is the comparison of the polynomial fit Vs the HD curve, when one looks past the inflection points:



The inability to model the under and overexposure regions of the HD curve is one of the drawbacks of using a single polynomial. The behavior outside the area of interest is a problem with all polynomial fits when a polynomial is not intrinsic to the process. This problem can be mitigated by using a spline fit to the data using multiple 3rd order polynomials, one polynomial between every two data points. Some of the overshoot of the polynomial curve is likely the result of rounding as I copied the poly coefficients from the screen shot and they only gave 4 digit precision.

2) The diagram posted from the 3rd Edition of The Theory of the Photographic Process edited by James - the one where the derivative looks like a mesa, isn’t quite accurate. The derivative does not have a sharp corner at point A’. In any case, this graph is from chapter 20, “The Interpretation of Sensitometric Results” by Tupper. The same diagram is in the Revised Edition, Pg 870, as Fig. 299.

A better diagram, from the Revised Edition edited by Meese, pg 173, Chapter 5 “The Relation Between Exposure and Density” by Burton, is provided below:



This diagram represents one emulsion with one population of silver grains. Real films and papers are made from multiple emulsions and have a significant straight line portion that is made up of a succession of overlapping curves. This is the justification for splitting the bell shaped derivative curve at it's peak and splicing in a horizontal straight line segment.

3) The HD curve is an 'S' shaped curve over the interval of interest. The derivative of all 'S' curves is a bell curve. The Gaussian distribution is the simplest bell curve and the Error Function (erf) is the simplest 'S' curve. This alone justifies its selection as the curve to use for modeling the HD curve. It is interesting, though, that the physics of density formation does indeed produce a Gaussian distribution.

4) The HD curve over it's entirety isn't really an 'S' curve. In fact, it better resembles:

(c) Antoine de Saint-Exupéry, “The Little Prince”
 
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RalphLambrecht

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Nicholas Lindan said:
Nobody is questioning the ability to fit one or more polynomials to the central portion of the HD curve. ...
Click to expand...

Nicholas

You did in post #271:

A polynomial (except for 0 and 1st order) can not fit a straight line. As an HD curve has a straight line section and two straight line asymptotes, using one polynomial for the entire thing is impossible.

The two straight line asymptotes are irrelevant for what we are discussing her, and again, we're talking about one polynominal not several, but if you agree that the relevant portion of the HD curve can be fit by one polynominal, maybe we can drop the case?
 

Photo Engineer

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I have a Kodak publication from research here somewhere called Sensitometry and Densitometry. For some reason it is not on the shelf where I keep these books so I have been unable to rely on any of it during this discussion. I hope I find it soon.

In any event, while searching, I came across this in James and Higgins "Fundamentals of Photographic Theory", page 185. I thought you all might be interested in this.

"It should be noted that ASA speed relates to the minimum exposure which will yield a negative from which an excellent print can be made. Except under the most favorable conditions, an exposure somewhat greater than this should be used for normal photographic work......."

PE
 

RalphLambrecht

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A great quote from another book I cherish and I'm proud to own, but again, we're all reading the same books!

By the way, I got the 2nd edition, 2nd printing and could not find this quote on page 185.
 

Nicholas Lindan

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RalphLambrecht said:
The two straight line asymptotes are irrelevant for what we are discussing here
Click to expand...

The Lone Ranger and his heretofore faithful Indian companion Tonto are hiding behind rocks. The Apache are circling to attack:
Lone Ranger: "How are we going to get out of this, Tonto?"
Tonto: "Who is this 'we', White Man?"

The asymptotes ARE what I am discussing. The one to the left extends to infinity (it's infinite as the horizontal axis is Log E). Good thing, too, otherwise unexposed film would be black.

RalphLambrecht said:
if you agree that the relevant portion of the HD curve can be fit by one polynominal, maybe we can drop the case?
Click to expand...

The sticking point here is 'relevant portion'. If that means 'that portion that can be fit with a polynomial', well, yes - it's a tautology and thus unarguable.

But, the point is that nobody has ever said that it can't be fit.

My questions is: So What? It is like saying you can use a french curve to draw it - there is obviously no argument that you can't.
 

RalphLambrecht

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ic-racer said:
Easy for you

What software are you using to solve a 3rd degree polynomial? Or how are you making it 'easy?'

Or, which one of the methods here is the 'easy one?'
Click to expand...

Maybe I can help. Deltagraph can solve for y=f(x) and x=f. So, if you are willing to put up with a mediocre x=f fit, you'll get your answer by solving the attached equation for 0.1, using Nicholas data as an example. But, you could always do it graphically with the y=f(x) function.
 

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Nicholas Lindan

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RalphLambrecht said:
We are trying to fit a curve through relevant data
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Well, we have a disagreement, that appears will never be settled, as to what is relevant. I find the HD curve goes from no exposure through to and past solarization and reversal. Exposure past that which results in DMax is common, as is exposure under the DMin point. The sections of the HD curve that deal with under and over exposure do exist and do have to be dealt with. As an example, we all make flashing exposures below the DMin exposure.

RalphLambrecht said:
and you are including no exposure data without density and unrealistic density beyond Dmax
Click to expand...

?????????????? I think there is a grammatical problem here - the sentence makes no sense.
 

Photo Engineer

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RalphLambrecht said:
Ron

What's the chapter name, and how many pages into the chapter is the quote?

(I'm trying to find it in my edition)
Click to expand...

It is "The Determination of Sensitivity" and is located right under figure 11.2d, the idealized characteristic curve divided into 3 portions based on exposures given to pictures on another page. It is the 7th page in the chapter.

They begin the discussion of average gradient on the page following.

If you have a problem, I can scan the page in.

PE
 
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