Film curve plotting and fitting

[dpgoldenberg]

What would I like to see in software??

Just as I'd like software that will give me the 0.3G point from a dataset, I'd also like a computerized determination of the "Contrast Index" like one would get from the good old fashioned clear plastic "Contrast Index Gauge" http://www.tpub.com/content/photography/14208/css/14208_56.htm

Well, that was what I was trying to offer when I first started this thread!

 

[ic-racer]

Well, that was what I was trying to offer when I first started this thread!

EXACTLY!

 

[ic-racer]

How many curves are you planning on measuring?

I'd say I fire off a control strip with every 10th roll of film. I have probably about a hundred of curve datasets on the computer right now.

 

[ic-racer]

Delta-X Criterion gives you the 0.3G point and for the rest of the curve plotting, I'll send you a paper which I think covers it.

Yes, and I think it is a nice tool.

But that brings me to the question I have been wanting to ask you. To figure out your "Delta-X" table, you needed to actually calculate the 0.3G values, right? How did you do that? By hand with graph paper, or using the computer?

 

[ic-racer]

Well, that was what I was trying to offer when I first started this thread!

So, getting back to the original post.

1) How are you defining "Gamma" in the original post?

2) Did the issue Steve mentioned about the '0.3G' and '0.1 crossing point' should give about the same speed with a gamma around 0.6. Did that issue ever get resolved?

Also, Ralph mentioned Delta Graph Pro. I'm already using the 1992 version and just downloaded the updated "free trial" to see if there really is new functionality.

 

[ic-racer]

I use a commercial program called DeltaGraph with great success. It's highly customizable and can solve any equation format I have given it so far. It's available for Windows and MacOSX.

http://www.rockware.com/product/overview.php?id=83

Ok, I just downloaded the newest version and it acts just like my old version. Can you tell me how you can get it to solve any equation?

Under the "Curve Fitting" dialog you can get equations for a number of different curve fits, but I can't get it to solve any of them. I can't even get it to give an X-intercept from a linear regression of the straight portion (clever easy speed point per PE) without re-plotting the data set backwards from the standard H&D curve.

Also, the spline function is only for 'looks.' It provides no statistical weighting of the points (the line always crosses all points) and has no associated equation.

 

[Stephen Benskin]

Yes, and I think it is a nice tool.

But that brings me to the question I have been wanting to ask you. To figure out your "Delta-X" table, you needed to actually calculate the 0.3G values, right? How did you do that? By hand with graph paper, or using the computer?

The ΔX value is determined using only ΔD which is the the Y intersect 1.30 log-H units from Hm. I use Visual Basic for my plotting programs, so I just wrote a sub to calculate ΔX and the ΔX speed point. The numbers in the table for the article were generated using Mathcad. Is that what you were looking for?

 

[dpgoldenberg]

So, getting back to the original post.

1) How are you defining "Gamma" in the original post?

Gamma represents the slope of the linear region. Because my function is only meant to represent the toe and linear region, the derivative of the function approaches a constant value, and gamma is calculated as that limiting value.

2) Did the issue Steve mentioned about the '0.3G' and '0.1 crossing point' should give about the same speed with a gamma around 0.6. Did that issue ever get resolved?

In my original post, there were two sources of confusion regarding the fractional gradient method. The first is that the "fractional gradient" that I was calculating was the ratio of the slope a given point to the limiting slope (gamma), rather than using the average gradient as usually specified. I should probably call that ratio "fractional gamma", and it will always be less than or equal to the traditional fractional gradient. The other, issue has to do with how far the fg=0.3 pt is displaced from the exposure indicated by the meter. I was treating the D=0.1 and fg=0.3 points the same way, but, as Steve points out, the fg=0.3 point will be about 1 stop to the left of the D=0.1 point, and this accounts for most of the difference.

The point, again, is that the parameters derived from the fit function can be used to set a speed by whatever criterion one wants.

 

[RalphLambrecht]

But wait, for 20 pages on a thread on "film curves" you think we should be plotting them on graph paper

To do it on graph paper, you have to plot all the points, which takes longer than typing in the numbers in the column. Then line up the thing.

Then you have wasted a nice piece of paper. Remember the 'paperless' office...

This is a serious pet peeve for me in that I have essentially supercomputing power to run some 'Flash Player' to display ads, but I can't have the computer make some straight forward 'computations'

Who's talking about graph paper? Print your computer curves and use the plastic contrast gage to determine gamma. The paperless office is a myth. You are not still falling for that one, are you?

 

[RalphLambrecht]

Ok, I just downloaded the newest version and it acts just like my old version. Can you tell me how you can get it to solve any equation?

Under the "Curve Fitting" dialog you can get equations for a number of different curve fits, but I can't get it to solve any of them. I can't even get it to give an X-intercept from a linear regression of the straight portion (clever easy speed point per PE) without re-plotting the data set backwards from the standard H&D curve.

Also, the spline function is only for 'looks.' It provides no statistical weighting of the points (the line always crosses all points) and has no associated equation.

Easy enough, see attached!

 

[ic-racer]

Easy enough, see attached!

It will fit an equation to a curve, but it won't solve the equation. (or at least it won't for me).

Lets say it gives you this result:

y = 5x^3 + (10*x^e-3)/ 2x^2 + 3x

So how are you using the program to solve that to find the x value for y = 0.1??


Or lets say it gives you this from a linear regression of your H&D curve:

y = 0.738*x + 2.3838

How are you getting it to solve for the x-intercept?

 

[ic-racer]

Who's talking about graph paper? Print your computer curves and use the plastic contrast gage to determine gamma. The paperless office is a myth. You are not still falling for that one, are you?

Actually I have tried that but the older version of the software 'scales' the axes so they are not 'square' and it won't work. Maybe the newer version fixes that. BTW where did you get your clear gauge??

 

[RalphLambrecht]

It will fit an equation to a curve, but it won't solve the equation. (or at least it won't for me).

Lets say it gives you this result:

y = 5x^3 + (10*x^e-3)/ 2x^2 + 3x

So how are you using the program to solve that to find the x value for y = 0.1??


Or lets say it gives you this from a linear regression of your H&D curve:

y = 0.738*x + 2.3838

How are you getting it to solve for the x-intercept?

That's not what Deltagraph is for. I guess you could use Excel for that. Deltagraph best-fits a function format to a set of existing data points. This allows you to get an equation, representing the data points, and with this equation you can calculate intermediate point, gradient, intercepts etc.

It works perfectly for film and even s-shaped paper curves with the format that I have supplied.

 

[RalphLambrecht]

Actually I have tried that but the older version of the software 'scales' the axes so they are not 'square' and it won't work. Maybe the newer version fixes that. BTW where did you get your clear gauge??

Deltagraph allows you to scale each axis to any dimension you want. The older version did that too.

I drew the CI meter myself and printed it on clear plastic. The geometry is simple. Would you like a pdf copy?

 

[ic-racer]

Deltagraph allows you to scale each axis to any dimension you want. The older version did that too.

I drew the CI meter myself and printed it on clear plastic. The geometry is simple. Would you like a pdf copy?

That is nice.

I just realized in my old version the y(x) vs. x buttons are always grayed out.

On the newer version by swapping the two I can get my X-intercept speed point easily now (from the linear regression slope of the first 11 points above 0.1)

Steve sent me a pdf of a paper showing a very nice least squares fit to a sigmoid curve. The equation is too complicated to be fit by the DeltaGraph "user defined" function. But I realized in all the test curves I was playing with, most of them are pretty darn straight. In fact, after assimilating most of this thread I have settled on the following film analysis plan:

1) Graph the film test strip datapoints in DeltaGraph Pro
2) Calculate my "development index" (new catch-phrase) as the linear regression of the first 11 points above 0.1 (that area is usually always pretty straight in the B&W films I have tested in the last 10 years)
3) Get the actual number from the y=mx+b result. So "development index" = "m" from the equation
4) Then use the "y(x) vs x" buttons to flip the equation and the read off the speed index as the x-intercept, which is "b" in the equation.

So now I'm happy the software spits out both a 'development index' and a 'speed index' for relative film comparisons. Realize I did not mention anything about meter calibration or exposing scenes etc.

 

[Photo Engineer]

IC and others;

Did you read my anecdote about Cape Canaveral? Have you heard of the phrase "you press the button and we do the rest"?

You labor at finding the best speed, but the speed and development conditions for Fuji, Kodak and Ilford films are posted on their web sites and in many cases, on the box.

Doing all of this experimentation is nice, but it does not produce better pictures. What is needed is development of the eye and the mind in the sense of art to get good pictures. Even a bad tone scale can yield a prize winner if it is the right picture.

We are losing that sense here. Actually, math is great after the fact but often not before the fact when you are a photographer. The reverse is true when you are a photo engineer BTW. So, all of this work keeps you from taking pictures.

Go take pictures and experiment that way!

PE

 

[ic-racer]

So, I need to thank Steve for the paper.
In the paper he sent me the Kodak researchers used this "s" equation:

y = f + c* F(x+h)

where if you can get a computer to do a least squares fit of that s-shaped curved to your data you can just read off the contrast and speed as follows:
f = fog
c = contrast
h = speed
F( ) = the relationship y = F(x) where y = 1/ (1+ e^-x)

Their premise was that any lateral shift of the curve was related to speed changes (defined as "h") and that any change in slope of the curve is related to contrast (they defined as "c" above).

What I realized is the my 'curves' were almost always nearly straight. So, I just used the same underlying premise stated above and applied a least squares fit to a line, rather than an 's'

 

[ic-racer]

PE, thanks for pointing that as I wanted to make the point that:

1) I agree that the speed info from the major manufacturer's is good info (in fact that is how I 'calibrate' my sensitometer, and I use name-brand film for the comparison)
2) Name brand film is expensive and can be hard to obtain.
3) Non-name brand film may not have a web site or useful info with respect to speed or development
and of course...
4) the development info, even from reputable sources, like Kodak, is still just a starting point, even they say so. Once you have determined a favorite contrast level, I find applying that to unknown film easiest by test wedge exposure.

So, in this age of 'film of the week,' home testing allows one to check unknown film against name-brand or one's 'favorite' film with respect to relative speed and development.


Real life example: I'm just finishing up 100 sheets of 8x10 TMY I got for $2 a sheet. In my opinion its one of the best B&W films ever made, but I don't make enough $$ in my day job to justify $6 a sheet for more right now. So, I have Shanghi film on my radar screen.

Once I get the film, by checking a test wedge on a piece of the film I will quickly know a) how many stops slower it is than TMY and b) how close the development info on the digital truth site (if any exists) comes to my favorite development contrast and c) if it has a nice stright line curve.

 

[ic-racer]

I noticed a very surprising thing last night. When going back and checking a bunch of TMY (ISO 400 in case anyone does not know) film curves, most of them give me a 'personal speed point' with my personal graphing convention and unique sensitometer light output, of almost exactly 2.7.

This is a chance coincidence, but as you may or may not know 27DIN is easily converted to ISO400

 

[Photo Engineer]

IC;

I understand your POV. Having said that, there is a simple way to avoid much of the rest of the issues by taking that curve in post 218 and using it as your 400 speed aim. Then a quick exposure with the film of the week, using the same conditions for everything will show you whether you need more or less exposure revealed by threshold speed, or more or less development revealed by contrast. In fact, an exposure series on one strip of film will eliminate the speed guesswork and narrow it down to development.

Much quicker and simpler requiring no calculations and one "aim" curve for each speed / contrast condition.

PE

 

[ic-racer]

IC;

I understand your POV. Having said that, there is a simple way to avoid much of the rest of the issues by taking that curve in post 218 and using it as your 400 speed aim. Then a quick exposure with the film of the week, using the same conditions for everything will show you whether you need more or less exposure revealed by threshold speed, or more or less development revealed by contrast. In fact, an exposure series on one strip of film will eliminate the speed guesswork and narrow it down to development.

Much quicker and simpler requiring no calculations and one "aim" curve for each speed / contrast condition.

PE

Excellent point. And, in fact, thats the way I have usually been doing it..."Shoot first and ask questions later".

However, with the weight of the 8x10 equipment and cost of a whole sheet of film and need for some daylight (or what ever condition for the exposure) I'm thinking I can do it easier and quicker by doing a test wedge first, then go out shooting.

 

[Photo Engineer]

IC;

I know that feeling as well. Fortunately, you can get quite a few 1" x 10" strips from an 8x10" sheet and you can use them for your experiments. BTDT as well. When I could only make 10 feet of film from an experiment at EK, I made up 35mm strips 12" long and used them for quite a few experiments of this type.

Actually, you can get 4 1" strips and 2 4x5s out of one 8x10 sheet and this can do lab tests and in-camera verification with a 4x5. I've done that as well.

We got a transparent overlay with the aim curve on it, along with an exposure sheet for the 1B Sensitometer. We dialed in those settings and made step wedge exposures and compared with the aim. It worked quite well and saved a lot of time.

PE

 

[RalphLambrecht]

So, I need to thank Steve for the paper.
In the paper he sent me the Kodak researchers used this "s" equation:

y = f + c* F(x+h)

where if you can get a computer to do a least squares fit of that s-shaped curved to your data you can just read off the contrast and speed as follows:
f = fog
c = contrast
h = speed
F( ) = the relationship y = F(x) where y = 1/ (1+ e^-x)

Their premise was that any lateral shift of the curve was related to speed changes (defined as "h") and that any change in slope of the curve is related to contrast (they defined as "c" above).

What I realized is the my 'curves' were almost always nearly straight. So, I just used the same underlying premise stated above and applied a least squares fit to a line, rather than an 's'

I have a question. Does 'h' not change 'y' in the equation above? If yes, it is not responsible for a lateral but a vertical shift. So how can it be speed, which is, indeed, a lateral shift?

 

[ic-racer]

I have a question. Does 'h' not change 'y' in the equation above? If yes, it is not responsible for a lateral but a vertical shift. So how can it be speed, which is, indeed, a lateral shift?

Looks like that probably because the way I typed it and they way they simplified the equation with the F(X) nomenclature. The 'h' is supposed to be part of the 'e' exponent in the full equation (which they never show in full in the paper)

Again, F(X) is the relationship y= 1/(1+e^x)

The paper is from 1961 and they used an IBM 705 to solve for 'h.'

 

[ic-racer]

I did use another graphing program (not DeltaGraph) to draw out some various permutations of y=1(1-e^x) curves with actual values and it is way to much of an "S" shape to fit my film curve datasets. (I can make some screen shots if anyone is interested). I did not do the sum of squares fit from the paper, but just by looking at the values of the "S" curves I made, I could tell that no way would the 'fit' be less than the correlation coefficient of 0.99 I am getting with a straight-line-fit on my film data.

This is a typical example, they really don't have much "S" to them in the range I'm plotting.

(BTW the green in the above graph represents the first eleven points above 0.1 and that is the part of the curve to which the straight line is fit with the least square method in DeltaGraph) (I came up with that personal convention a few years ago because the first eleven points above 0.1 plotted out pretty straight in just about all the films I have ever tested.)
(BTW: Just for clarification, that graph I posted with the colored lines is just to show how straight the curves are. It is not supposed to be analogous to the top one showing a speed change)