Film curve plotting and fitting

???

A gradient w/o a curve?

The ISO standard doesn't define the film as long toed or short toed, but it has a set of contrast parameters that are required before determining the film speed. These parameters are a function of the Delta-X criterion. As such, they define the 0.3G point for the shadows and a Δ log-H 1.30 and ΔD 0.80 for an average gradient of 0.61.

Before anyone suggests that 0.61 is too high for a normal gradient and therefore produces optimistically high film speeds, it doesn't under the Delta-X criterion. The misunderstanding comes from those who use the fixed density method to determine films developed to higher or lower gradient levels. In other words, they are mixing two different speed determination methods. The fixed density method tends to over rate higher gradients and under rate lower gradients.

Correct me if I'm wrong, but by the time you say cubic splines, to produce a curve for x data points, then you need x-1 3rd degree polynomials. So, you can't have a single 3rd degree polynomial if you're using splines. Anyway, I did some quick and dirty (very sloppy actually) coding to try to make a spline for 7 data points as input. The curves actually looked fine and they pass from all data points (by definition). I tried it with something as simple as f(x) = x*, or x^2, some typical film densities found from some sources and f(x)=sin(x), for x = 0, 45, 90, 135, 180, 225, 270°. The curve looked fine, although I didn't superimpose the true curve for that function (remember, it was just a sloppy attempt). I'm afraid that I'm away from home and I won't be able to give any examples for weeks. Anyway, at some point I'll try to refine my code to produce something usefull.

* Yes, that's not a curve and the result was a perfect straight line. The polynomials weren't 3rd degree in this case either. If ax^3 + bx^2 + cx + d is the general form of the 3rd degree polynomial, a and b were zero in this case.

Try using the 21 data points from the aim curve I posted above.

PE

Photo Engineer said:

Try using the 21 data points from the aim curve I posted above.

PE

Click to expand...

You mean (there was a url link here which no longer exists) PE?

You may use that one, but I posted the paper curve alone here in this thread a few pages back. That is what I was thinking of as it has the entire curve with toe and shoulder. The one you noted is ok but a bit confusing due to the multiple curves. It also does not show the full film curve as the film is usually plotted on a horizontal graph paper that will fit the full latitude.

PE

Oh, (there was a url link here which no longer exists) then.

Yes, less confusing and easier to use.

PE

Photo Engineer said:

Just some thoughts that struck me today....


1. Yes, EK uses a cubic spline to describe all curves for all products. We used either DEC or HP computers in a room full of automated Densitometers. They were plotting curves from several precision 1B Sensitometers. . . . . In all of my work I have never seen the spline fail!
PE

Click to expand...

This is speculation, and I'm sure that PE will correct me if I am wrong, but I suspect that the EK protocol is/was something like this:

1. For a given film/developer (or paper/developer) combination, a standard curve is generated by using very accurate, closely spaced data and fitting a cubic spline to the entire curve. As discussed earlier, the spline is made up of many small segments, each of which fits four points perfectly.

2. For quality control (and fine tuning purposes), data are measured for an emulsion batch and the data fit using the full spline as a "function" that is adjusted with a small number of parameters that shift the position and gradient of the curve. This is the procedure that is described in a paper (by Kodak engineers) that was discussed earlier in this thread:

Description of d-log E curves by specifically chosen parameters
1961, Bayer et al. in
Photographic Science and Engineering Vol 5, No. 1, Jan.Feb

They don't explicitly mention cubic splines, but it would make a lot of sense to do it this way for this purpose. The parameters derived from the fitting procedure would be directly interpretable in terms of speed and contrast, relative to the "standard" curve that is used to construct the spline.

I'm not sure that this a practical approach for amateur sensitometrists (if such people can ever be considered practical) but I would be quite interested in knowing if this is the actual procedure.

David

This begs for the question: Is there an equation format for cubic splines, reflecting Kodak's approach?

Ralph, there is a set of equations, but I don't have them. When I did see it, it was in the format of either FORTRAN code or PL1 or some such.

David, we used either a 21 step tablet or a continuous wedge that was precision cast with carbon. These were read and the data was plotted as discrete points in the case of the step tablet and then a curve was drawn over the points. In the case of the continuous wedge, there was only a single line shown. In virtually all cases, the curve fit the data, but there were outlying points at times due to defects or mismatched blends, etc.

PE

The 1961 Bayer describes how two terms can be added to the logistic growth equation to represent generic speed and contrast for comparison of two films. Then it shows how to do a least squares fit to the equation by two methods. First with an approximation method which can be done by hand and then with an exact method requiring a computer. It has nothing to do with splines.

It is a crude but clever tool for QUICK and EASY comparisons because the relative SPEED and CONTRAST values can be read right from the variables in the resulting least squares curve fit without solving any polynomials!

I've been thinking this over and here are some thoughts which color my responses.

We had a whole lab of people, about 20, who established aim curves. It was not productive to us to debate the aims. We got them with exposure conditions and we were expected to meet these curves. Our goal was defined in terms of a curve and an exposure condition and we did not have a lot of data outside of that. It was compartmentalized.

This is unfortunate now, and here, but is the way it was at EK.

So, I had curves and exposure data and I was required to meet that. I had little beyond that but the reading of computer code and reports from the other lab.

This is what prevents me from giving more information.

PE

Photo Engineer said:

Try using the 21 data points from the aim curve I posted above.

PE

Click to expand...

I made some progress and I've made two versions (there was a url link here which no longer exists) PE. One using 21 points:

1 0,1
2 0,1
3 0,1
4 0,1
5 0,1
6 0,1
7 0,15
8 0,25
9 0,42
10 0,66
11 1
12 1,37
13 1,75
14 1,98
15 2,13
16 2,19
17 2,2
18 2,2
19 2,2
20 2,2
21 2,2



And another one with points 6 to 17.



I used (natural) cubic spline interpolation and the only problem I can find is when combining linear parts with curves and more specifically, at the transition from the linear to the curved part. It's not very obvious at the 21 step attachment, but the curve goes below dmin between points 5 and 6. The problem becomes more obvious when the curve rises quicker.

So, any comments?