Film curve plotting and fitting

[ic-racer]

I really don't undersstand why people don't just use excel charts. All you need to do is enter your zone densities once you have set up a chart. Very easy to copy for each test you do and reading off numbers to calculate slope at any two points on a curve is pretty simple.

I could not get excel to solve a polynomial that it gave me for my dataset. Do you know how to get it to solve for the 0.1 point from a polynomialt? How about getting it to calculate and show a fractional gradient tangent?

 

[Mahler_one]

Agreed....basically, the BTZS method tiltody.

Sandy....you are ,to my understanding, also pretty much summarizing the method that Phil Davis and others have suggested, correct? The ES of the paper and the DR of the negative are determined by using step wedge densities- as Mr. Davis and others have written.

 

[Photo Engineer]

I could not get excel to solve a polynomial that it gave me for my dataset. Do you know how to get it to solve for the 0.1 point from a polynomialt? How about getting it to calculate and show a fractional gradient tangent?

IIRC, if you have the equation of the curve at any given point, then the fractional gradient tangent is the linear equation resulting from the derivative of that curve. The curve of a normal photographic material fits into a family of curves called cubic splines.

http://en.wikipedia.org/wiki/Spline_(mathematics)

http://en.wikipedia.org/wiki/Spline_interpolation


PE

 

[tlitody]

I could not get excel to solve a polynomial that it gave me for my dataset. Do you know how to get it to solve for the 0.1 point from a polynomialt? How about getting it to calculate and show a fractional gradient tangent?

Can't remember. I went through similar process ages ago and can't remeber what I did.

I think you want this software:

http://s91928265.onlinehome.us/curveexpert/

version I have was free. I think you get 30 days with this one but its not expensive and you can play with functions and formulas till the cows come home.

 

[ic-racer]

IIRC, if you have the equation of the curve at any given point, then the fractional gradient tangent is the linear equation resulting from the derivative of that curve. The curve of a normal photographic material fits into a family of curves called cubic splines.

http://en.wikipedia.org/wiki/Spline_(mathematics)

http://en.wikipedia.org/wiki/Spline_interpolation


PE

Certainly; and it would be nice if Excel would come even close to solving it

 

[Photo Engineer]

Certainly; and it would be nice if Excel would come even close to solving it

I believe that you can use the VB plug in for Excel and write a macro that way which will solve the equation. I have not done it lately with Excel, but I have seen it done and have done some simple things such as write an Excel calculator that runs like the Windows version.

PE

 

[ic-racer]

Just to add some fuel to the fire.
I prefer the 'coffee can' method of speed determination.

This works well with landscape/ LF stuff where you can do the test at the actual shutter speed/aperture combo you normally use with a LF shutter/lens. Compensates for flare.

Paint the inside of a coffee can black. Put the can out in a typical scene. Bracket. Select the negative with the coffee can interior that is around 0.05 log or so as your minimum exposure and base your future exposures on that. Use a reflected average 'view meter' reading of the scene( if you spot the low zones they low zones won't be accurately represented due to flare bla bla bla ).

 

[Photo Engineer]

Flat black or glossy black?

PE

 

[ic-racer]

I believe that you can use the VB plug in for Excel and write a macro that way which will solve the equation. I have not done it lately with Excel, but I have seen it done and have done some simple things such as write an Excel calculator that runs like the Windows version.

PE

Ahh, an Excel macro. Though the last time I really got into Excel programming it seemed easier to just write a better software package in C++ (not a Bill Gates/Microsoft fan and never was. I thought his BASIC for the Altar was useless because you needed a CRT screen to use it....).

 

[sanking]

Agreed....basically, the BTZS method tiltody.

Sandy....you are ,to my understanding, also pretty much summarizing the method that Phil Davis and others have suggested, correct? The ES of the paper and the DR of the negative are determined by using step wedge densities- as Mr. Davis and others have written.

That is correct. I learned Zone system first, later BTZS. A step wedge is enough to determine paper ES but for film testing you also need a transmission densitometer, and some method of exposing the step wedges that is consistent and repeteable.

Film curves in BTZS are plotted with a program called Winplotter. It was originally written for a MAC but ported over to the PC much later.

Sandy

 

[tlitody]

Can't remember. I went through similar process ages ago and can't remeber what I did.

I think you want this software:

http://s91928265.onlinehome.us/curveexpert/

version I have was free. I think you get 30 days with this one but its not expensive and you can play with functions and formulas till the cows come home.

The older free version can be had from:

Dead Link Removed

 

[ic-racer]

Rather than PM this question, perhaps others can benefit.

To Steve, I can convert your "Delta-X" table into something to use to get a fudge factor for my speed point as development increases or decreases.

So, first I converted the Delta-D values back to a slope (easier for me because my software does a linear regression on the straight line portion in the area I designate)
Then I 'null out' the table on my known good 0.1 value at my usual development gamma. So, that would be a Delta-X of .22 (for my usual good gamma of 0.65).

Now on my converted table where you would have .975 for the Delta D, I have 0.65 for my slope. And for the Delta-X of 0.22 I make that zero and subtract 0.22 from the rest of the Delta-X values in the chart.

Now I have a table of fudge values to add (or subtract) from my 0.1 determination on any film that gets processed to a gamma other than 0.65.

 

[Lee L]

SciDAVis is a free, open source curve fitting program. You can provide your own formulae, do scripting, etc. Runs on Windows, Mac, linux, etc.

http://scidavis.sourceforge.net/

Lee

 

[RalphLambrecht]

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

 

[Stephen Benskin]

. And for the Delta-X of 0.22 I make that zero and subtract 0.22 from the rest of the Delta-X values in the chart.

Now I have a table of fudge values to add (or subtract) from my 0.1 determination on any film that gets processed to a gamma other than 0.65.

I can follow you up to here and it sounds good so far. Do you mind walking me through these last two steps?

Maybe you could simply do a conversion speed table for each gradient above and below 0.61 that already incorporates the Delta-X equations. It would be similar ISO standard which already uses the Delta-X criterion but only for a single gradient.

 

[dpgoldenberg]

I really don't understand why people don't just use excel charts. All you need to do is enter your zone densities once you have set up a chart. Very easy to copy for each test you do and reading off numbers to calculate slope at any two points on a curve is pretty simple.

I do think that there are some advantages to fitting a smooth function from which the parameters can be derived. Especially at the low-density of the curve, the relative errors in the individual measurements are likely to be significant. It's especially hard to estimate slopes from differences in small numbers with relatively large uncertainties. The approach I am suggesting effectively smooths out these errors and uses all of the data, including the more accurate data from the flat and linear regions of the curve, to help define the shape of the curve. The key, though, is that the smooth function has to be a reasonable representation of the underlying "real" function.

Spline functions, as suggested by PE, can be useful, but they are forced through each data point. If you have closely-spaced data points of high quality, the spline is probably the best way to interpolate. But, I think that it is likely to be problematic with the sort of data that we amateurs are likely to produce. This is a general point that I think is worth considering: the schemes that were developed for standardizing film speeds assumed the use of sophisticated and well-calibrated instruments. I, at least, don't have that sort of resource, and I don't really want to invest in the time its use would require.

Even if one wants to stick to the linear region (something I am becoming increasingly convinced of), it still might be useful to be able to characterize the toe, which raises a question for the straight-liners: How straight does the "line" have to be? In other words, how close to the limiting slope (gamma) does the tangent to the curve have to be at the point where we start using it: 90%, 80%, 50%?

I've modified my program so that it will output the densities and the ratio of the tangent slope and gamma (what I call "fractional gamma" to make the distinction with the traditional fractional gradient) for each zone, given the calculated exposure index. Here are the results for the two curves that I posted earlier (with quite different toe regions):

In each case the EI was set by the D=0.1 criterion.

Delta100_FX-37
gamma = 0.77 +/- 0.03
toe width = 5.04 +/- 0.86 stops

Exposure index set from density threshold (Zone 1.0 density = 0.10):
Speed point = -3.52 +/- 0.26 (log2exposure)
Corrected EI = 114.58 +/- 20.27
Avg. gradient (log10, from Zone I) = 0.57 +/- 0.04

Calculated Zone System Densities and fractional gamma
Zone Dens. Fract. Gamma
I 0.10 0.31
II 0.20 0.52
III 0.34 0.72
IV 0.53 0.86
V 0.74 0.94
VI 0.96 0.97
VII 1.19 0.99
VIII 1.42 1.00
IX 1.65 1.00

Delta100_Pyrocat-HD(blue)
gamma = 0.59 +/- 0.01
toe width = 2.15 +/- 0.91 stops

Exposure index set from density threshold (Zone 1.0 density = 0.10):
Speed point = -3.57 +/- 0.15 (log2exposure)
Corrected EI = 74.24 +/- 7.85
Avg. gradient (log10, from Zone I) = 0.57 +/- 0.02

Calculated Zone System Densities and fractional gamma
Zone Dens. Fract. Gamma
I 0.10 0.68
II 0.25 0.94
III 0.42 0.99
IV 0.60 1.00
V 0.78 1.00
VI 0.95 1.00
VII 1.13 1.00
VIII 1.31 1.00
IX 1.49 1.00

What is the minimal fractional gamma for the shadows? I'm really not trying to be provocative here, but how straight is straight? Does it depend on how broad the toe is? (I would think so.)

David

 

[alanrockwood]

IIRC, if you have the equation of the curve at any given point, then the fractional gradient tangent is the linear equation resulting from the derivative of that curve. The curve of a normal photographic material fits into a family of curves called cubic splines.

http://en.wikipedia.org/wiki/Spline_(mathematics)

http://en.wikipedia.org/wiki/Spline_interpolation


PE

PE, as you recall we have had the discussion of cubic splines before. Essentially, a cubic spline can be made to give a pretty good fit to any reasonably smooth curve, though it is almost never a truely optimal choice of fitting function.

If there is a functional form which is known to be a reasonable model for the process being fitted to a curve (for example, an exponential decay if one is dealing with radioactive decay, or a Gaussian function if one is dealing with a certain subset of statistical processes) then it is better to use that functional form for the fit rather than a cubic spline because one will generally get a better fit with fewer adjustable parameters. For example, if exposure curves were known to be very similar to an integrated Gaussian function then one could get a pretty accurate fit over the whole curve using a three parameter fit (four if one were to also include the zero-exposure density).

Has there been any research on determining functional forms for the toe and shoulder of the exposure curve? It seems likely to me that the best functional form would likely be something related to some kind of exponential function. It seems quite unlikely that the best form would be polynomial-like (or piece-wise polynomial) such as a cubic spline function. However, it's possible that a cubic spline might be good enough for practical purposes.

In principle the very best approach would be a non-parametric approach, i.e. working directly off the data without fitting a curve to the data. For this approach the points need to be spaced very close together and enough replicates of the experiment performed so the statistical variability in the result in negligible. However, a non-parametric approach would require a rather huge amount of work and is probably not practical.

 

[dpgoldenberg]

Alan,
Have you taken a look at my function? It will only work with the toe and straight-line regions, and there isn't any obvious (to me) physical basis, but it does seem to capture the essential features of the toe and line regions: It begins at zero with a derivative of zero; the derivative increases monotonically over a range defined by the adjustable parameters; and then it approaches a straight line. I don't have an extensive collection of data sets to try it with, but it looks pretty good to me with the data I have tried.

DAvid

 

[alanrockwood]

Alan,
Have you taken a look at my function? It will only work with the toe and straight-line regions, and there isn't any obvious (to me) physical basis, but it does seem to capture the essential features of the toe and line regions: It begins at zero with a derivative of zero; the derivative increases monotonically over a range defined by the adjustable parameters; and then it approaches a straight line. I don't have an extensive collection of data sets to try it with, but it looks pretty good to me with the data I have tried.

DAvid

I had a quick look earlier and it looked like it was fitting the toe very well.

 

[RalphLambrecht]

Alan,
Have you taken a look at my function?...

David

Have you looked at the nonlinear function I highlighted in post #39? It does all you desire for toe, midsection and shoulder. I don't know why anybody would need more.

 

[Photo Engineer]

PE, as you recall we have had the discussion of cubic splines before. Essentially, a cubic spline can be made to give a pretty good fit to any reasonably smooth curve, though it is almost never a truely optimal choice of fitting function.

If there is a functional form which is known to be a reasonable model for the process being fitted to a curve (for example, an exponential decay if one is dealing with radioactive decay, or a Gaussian function if one is dealing with a certain subset of statistical processes) then it is better to use that functional form for the fit rather than a cubic spline because one will generally get a better fit with fewer adjustable parameters. For example, if exposure curves were known to be very similar to an integrated Gaussian function then one could get a pretty accurate fit over the whole curve using a three parameter fit (four if one were to also include the zero-exposure density).

Has there been any research on determining functional forms for the toe and shoulder of the exposure curve? It seems likely to me that the best functional form would likely be something related to some kind of exponential function. It seems quite unlikely that the best form would be polynomial-like (or piece-wise polynomial) such as a cubic spline function. However, it's possible that a cubic spline might be good enough for practical purposes.

In principle the very best approach would be a non-parametric approach, i.e. working directly off the data without fitting a curve to the data. For this approach the points need to be spaced very close together and enough replicates of the experiment performed so the statistical variability in the result in negligible. However, a non-parametric approach would require a rather huge amount of work and is probably not practical.

Alan;

The cubic spline is in use at EK as it fits the entire curve quite well for film and paper. In addition, plotted as V Log E (Voltage), it works just as well for digital systems.

There are chemical and physical reasons for toe and to some extent, for shoulder to exist, particularly in reflective prints. I have seen the arguments in reports that describe all of these, but I could not reproduce them here so long after the fact. (more on the Dmax question below though)

The response of the human eye does not work the same way due to its adaptation to intense or dim light, and therefore the human eye has a response that is by definition, linear at all light levels.

AAMOF, due to the "linear" response of the eye, a print made to match the eye's response (but perforce with a toe and shoulder) looks flat and therefore as I noted above, the actual "good" print must be higher in contrast than the response of the eye would dictate in order to achieve a pleasing response in a person viewing the print.

But, to take this further, there is a reason for the toe, but there appears to be no definite reason for having a shoulder. If you coat enough silver with enough blends of emulsions, the curve could be continued out as far as you wish. I have seen curves attain a 5.0 density or 6.0 density in films. OTOH, paper prints are limited physically by the internal reflections to a density of about 2.2 or thereabouts. Therefore, rollover in the shoulder is due to either lack of Silver Halide, or internal reflections depending on whether it is film or paper support.

PE

 

[Mahler_one]

I sincerely hope that I am NOT inviting the wrath of all, but I simply must ask the following: Will any of the information so well presented and described by the obviously bright individuals on this thread enable us to produce negatives that are materially different ( or "better" ) than those that we are already making? After all of the mathematics and splines, is it likely that the speed points of any of the films we use be materially different than what we have determined using the BTZS methods ( which no one has stated is wanting or "inferior" to any of the other methods noted ) using the Plotter software? Moreover, if the speed points are different, how much "different" will the negatives made at, for example, a film speed of 100 vs. that of the same film at 90 really be? Once again, simply curious and asking in order to gain information and insight...not being critical of what is being discussed, or of any who are adding information.

If one simply takes PE's interesting point ( and I am inclined to do so ), why not simply "make" one's negative to a slightly greater contrast ( by developing longer ), and make the requisite print? Expose adequately for the shadows according to one's experience with the film being used, and develop to a higher contrast. AFAIK, to determine such information would probably only take a handful of 4x5 sheets and a day in the backyard and darkroom-or one "complete" film test.

 

[Photo Engineer]

Mahler;

What I actually recommended was to overexpose by about 1/3 stop. This has the effect of higher contrast if your original work was on the toe region, because now you would be on the straight line portion which is higher in contrast than the toe. I think that Sandy and Ralph said much the same.

And, this method does not require a lot of tests and curve drawing! However, it generally does not work with reversal films.

PE

 

[Stephen Benskin]

Daivd,

Did you incorporate flare into the exposure points for the Zones? The log-H axis represents the camera exposure or exposure at the film plane and not simply the scene luminance range.

Calculated Zone System Densities and fractional gamma
Zone Dens. Fract. Gamma
I 0.10 0.68
II 0.25 0.94
III 0.42 0.99
IV 0.60 1.00
V 0.78 1.00
VI 0.95 1.00
VII 1.13 1.00
VIII 1.31 1.00
IX 1.49 1.00

A good example of there being a difference between the speed point for the determination of film speed and the placement of exposure is with color reversal film. The speed point is found in the midpoint of the curve. While that point hasn't changed, the speed equation went from 8/Hm to 10/Hm, which is a 1/3 stop adjustment.

 

[Mahler_one]

Mahler;

What I actually recommended was to overexpose by about 1/3 stop. This has the effect of higher contrast if your original work was on the toe region, because now you would be on the straight line portion which is higher in contrast than the toe. I think that Sandy and Ralph said much the same.

And, this method does not require a lot of tests and curve drawing! However, it generally does not work with reversal films.

PE

Ah....understood, and thanks Ron.

Ed