Equation(s) for film/paper curves

[alanrockwood]

A quick little history.
To find the speed of film researchers Jones, et al 'discovered' that if the shadows fall on a part of the toe curve that is tangent to a line one-third of the slope, that was the minimum exposure for an excellent print. They made multiple prints and showed them to a panel of observers.
When they did this, there were no personal computers and this 1/3 slope or 1/3 G or 'fractional gradiant' was not so easy to find outside of using pencil rulers and paper graph.

So...three approximations of this 1/3 G (fractional gradient point) were developed that could be found on the toe curve without as much trouble or without a computer:
0.1
Delta-X
W-speed

The three methods were compared and found to be roughly similar in the approximation of 1/3 G speed point.

It turns out 0.1 was eventually chosen as the method to use by modifying it slightly to specify a certain slope the film needs to have (so called 'ASA Triangle') before taking the measurement. This became ASA and was nearly copied directly for the ISO standard and also embraced by many Zone system users.

This whole history of film curve analysis would have probably played out differently had computers been available.
So, as it stands ISO is firmly established, but it is fun to look back at how it came to be.

Realize that in all this research the GOLD STANDARD is "print judgement." Yes, objective viewing of the prints by a panel of observers TRUMPS ALL THE MATH!

View attachment 297328
View attachment 297329

It looks like three of the methods (0.3xGbar, deltaX, and W) are about equally good. The 0.1 method is worse, both in terms of a bias and greater variability.

 

[snusmumriken]

It looks like three of the methods (0.3xGbar, deltaX, and W) are about equally good. The 0.1 method is worse, both in terms of a bias and greater variability.

Glad you said that, I had the same thought.

 

[ic-racer]

It looks like three of the methods (0.3xGbar, deltaX, and W) are about equally good. The 0.1 method is worse, both in terms of a bias and greater variability.

When that paper was published they were ONLY looking at the 0.1 point of the film without specifying the contrast or development time for that film. If one were to check 0.1 point with a specific contrast development for the film, then 0.1 is very good. That is the ASA definition that got turned into the ISO definition.
However to make this determination you need a family of film curves to select the correct one to measure the 0.1 point. An advantage of Delta-x and W-speed is that you don't need to make the family of film curves (ie different development times).

Also see ( https://www.photrio.com/forum/threads/speed-point-0-1-over-base-plus-fog-or-1-3-of-gamma.183061/ )

 

[alanrockwood]

When that paper was published they were ONLY looking at the 0.1 point of the film without specifying the contrast or development time for that film. If one were to check 0.1 point with a specific contrast development for the film, then 0.1 is very good. That is the ASA definition that got turned into the ISO definition.
However to make this determination you need a family of film curves to select the correct one to measure the 0.1 point. An advantage of Delta-x and W-speed is that you don't need to make the family of film curves (ie different development times).

Also see ( https://www.photrio.com/forum/threads/speed-point-0-1-over-base-plus-fog-or-1-3-of-gamma.183061/ )
View attachment 297727

One thing I have never quite been able to figure out is what is the relationship between H in the figure and an ordinary light meter reading of a blank image.

 

[Nicholas Lindan]

One thing I have never quite been able to figure out is what is the relationship between H in the figure and an ordinary light meter reading of a blank image.

There isn't one. The H axis is a floating logarithmic scale indicating relative exposure. What's it relative to? Well, whatever amount is needed to expose the material adequately for making the graph. The axis may indicate light intensity, exposure time or a combination of the two.

 

[Stephen Benskin]

A quick little history.
To find the speed of film researchers Jones, et al 'discovered' that if the shadows fall on a part of the toe curve that is tangent to a line one-third of the slope, that was the minimum exposure for an excellent print. They made multiple prints and showed them to a panel of observers.
When they did this, there were no personal computers and this 1/3 slope or 1/3 G or 'fractional gradiant' was not so easy to find outside of using pencil rulers and paper graph.

So...three approximations of this 1/3 G (fractional gradient point) were developed that could be found on the toe curve without as much trouble or without a computer:
0.1
Delta-X
W-speed

The three methods were compared and found to be roughly similar in the approximation of 1/3 G speed point.

It turns out 0.1 was eventually chosen as the method to use by modifying it slightly to specify a certain slope the film needs to have (so called 'ASA Triangle') before taking the measurement. This became ASA and was nearly copied directly for the ISO standard and also embraced by many Zone system users.

Delta-X won out. The ISO triangle just insures the films have the same value for ΔD and consequently ΔX. The problem comes when people use a fixed density without using the equation when determining film speed from film developed to different contrast levels outside of the ISO standard.

Delta-X is slightly more accurate than the fractional gradient method when compared to the print judgement speeds.

 

[alanrockwood]

There isn't one. The H axis is a floating logarithmic scale indicating relative exposure. What's it relative to? Well, whatever amount is needed to expose the material adequately for making the graph. The axis may indicate light intensity, exposure time or a combination of the two.

My understanding is that H is the exposure in lux seconds, which implies that H is not a floating number. (Reference: https://en.wikipedia.org/wiki/Film_speed) However, I don't know how to convert a light meter reading to lux seconds.

 

[Stephen Benskin]

My understanding is that H is the exposure in lux seconds, which implies that H is not a floating number. (Reference: https://en.wikipedia.org/wiki/Film_speed) However, I don't know how to convert a light meter reading to lux seconds.

Metered exposure is 8/ISO. Black and white speed point is 10x below that. 8/125 = 0.064. Speed point for125 = 0.80/0.0064.

 

[Nicholas Lindan]

My understanding is that H is the exposure in lux seconds, which implies that H is not a floating number. (Reference: https://en.wikipedia.org/wiki/Film_speed) However, I don't know how to convert a light meter reading to lux seconds.

The X-axis is really in watt-seconds (WS). Lux-Seconds is a photometric unit related to the response of the human eye. Imagine characterizing an IR film in LxS - in the infrared the LxS reading is always 0.0 as the eye can't see IR. But, as LxS are easier to measure, and the two are related, for a given light source, by a constant multiplier LxS is used as a unit of measure. This multiplier, when applied to the logarithmic H/X axis just shifts things to the right or left; the shift is of no consequence as the H values are all relative.

But how many WS is plotted on the H axis is another matter. You can plot a characteristic curve from the exposure of a strip of photographic paper to a test strip f-stop/log10 sequence starting at the white point (almost but not quite entirely white) and proceeding in f-stop/log10 intervals. When the H-axis is plotted the tick marks are really each test strip patch. Yes, the paper is exposed in 'lux-seconds', and they are plotted in a logarithmic scale, but where the zero point is depends on the material being exposed and how bright the light source is and it's spectra. These are elements that are irrelevant to the curve where only the shape matters.

If you want to calibrate an HD curve to your equipment pick a density in the middle of the curve and determine the exposure that results in that density. Then you can label the X-axis in absolute units for your setup.

Some meters, Weston Master series?, have a chart on the back that relates meter readings to lux.

 

[alanrockwood]

Metered exposure is 8/ISO. Black and white speed point is 10x below that. 8/125 = 0.064. Speed point for125 = 0.80/0.0064.

So, to be sure I understand what you are saying, does this mean that

H=8/iso?

It looks to me like 8/iso is a constant. Where does the meter reading enter into the equation?

 

[alanrockwood]

The X-axis is really in watt-seconds (WS). Lux-Seconds is a photometric unit related to the response of the human eye. Imagine characterizing an IR film in LxS - in the infrared the LxS reading is always 0.0 as the eye can't see IR. But, as LxS are easier to measure, and the two are related, for a given light source, by a constant multiplier LxS is used as a unit of measure. This multiplier, when applied to the logarithmic H/X axis just shifts things to the right or left; the shift is of no consequence as the H values are all relative.

But how many WS is plotted on the H axis is another matter. You can plot a characteristic curve from the exposure of a strip of photographic paper to a test strip f-stop/log10 sequence starting at the white point (almost but not quite entirely white) and proceeding in f-stop/log10 intervals. When the H-axis is plotted the tick marks are really each test strip patch. Yes, the paper is exposed in 'lux-seconds', and they are plotted in a logarithmic scale, but where the zero point is depends on the material being exposed and how bright the light source is and it's spectra. These are elements that are irrelevant to the curve where only the shape matters.

If you want to calibrate an HD curve to your equipment pick a density in the middle of the curve and determine the exposure that results in that density. Then you can label the X-axis in absolute units for your setup.

Some meters, Weston Master series?, have a chart on the back that relates meter readings to lux.

I think there is an error in what you wrote. Watt seconds doesn't make sense on its own. There must be a unit of area somewhere, such as meter^2.

 

[alanrockwood]

I found more information on the relationship between light meter reading and the amount of light per unit area on the subject. The reference is a paper titled Exposure Metering Relating Subject Lighting to Film Exposure, written by Jeff Conrad. Here's an equation from the paper.

Here's a definition from the paper

The author also gives the following definitions for the quantities in the paper. "where EV is the exposure value, S is the arithmetic ISO speed, and K is the meter calibration constant." And for Nikon or Canon equipment K=12.5

One thing that is missing from the equation is the reflectance of the subject. What if we assume the average reflectance equals 12.5? That might (possibly) relate to the constant, K, in the equation.

 

[Nicholas Lindan]

I think there is an error in what you wrote. Watt seconds doesn't make sense on its own. There must be a unit of area somewhere, such as meter^2.

You are quite right: Watt-Seconds/Meter^2, though often ergs/cm^2 is used.

 

[Stephen Benskin]

I found more information on the relationship between light meter reading and the amount of light per unit area on the subject. The reference is a paper titled Exposure Metering Relating Subject Lighting to Film Exposure, written by Jeff Conrad. Here's an equation from the paper.

View attachment 297813

Here's a definition from the paper

View attachment 297814


The author also gives the following definitions for the quantities in the paper. "where EV is the exposure value, S is the arithmetic ISO speed, and K is the meter calibration constant." And for Nikon or Canon equipment K=12.5

One thing that is missing from the equation is the reflectance of the subject. What if we assume the2$ average reflectance equals 12.5? That might (possibly) relate to the constant, K, in the equation.

You said, I don't know how to convert a light meter reading to lux seconds." What I gave you was the conclusion. This is the exposure all meters aim for or Hg.

Meter calibration and camera exposure are actually a very large and complex topic filled with changing nomenclature and conversion between different units of measurement. It can quickly become overwhelming.

I'm glad you found Conrad's paper. It is really good. I do have a few qualms concerning some of the variables he uses in the examples. Not that they are wrong or that he doesn't point them out, it's just I prefer to use the standard's variable. They are basically a possible null type position for the variables. For instance, the variable in the standard for distance to object is the lens is focused at infinity. As a side note. this is something to be aware of with hand held meters. They have to assume conditions of the camera and optical system. If the conditions differ, the exposure exposure will be affected. Through the lens metering eliminates this problem as it sees the actual value of the transmitted light.

With Conrad, in eq 12 and 13, the changing of the off axis variable (He could have changed the value of a few others. I haven't read the paper in some time.) places Hg at 11.38x Hm instead of 10x. This is absolute an example of the type of conditions that can exist, but when learning a process, I believe it's less confusing to go with the standard values. Speaking of confusing, I'm used to the nomenclature from the original scientific papers. "A" equaled aperture then and now it's "N".

I'm happy to help out where I can. Along those lines, I've uploaded Connelly, D., Calibration Levels of Films and Exposure Devices, The Journal of Photographic Sciences, Vol. 16, 1968. Also a couple pages of something I through together a number of years ago call Defining K. You can let me know if you wish to see more or you can search for the rest of it somewhere in this forum. BTW, that comment you included with the screen shot of Conrad's paper that mentions something about 12.5 being somehow connect with average Reflectance. That 12.5 is the value of K in cd/m2.

I've have something that might work for you . I'll start fresh in a new post.

Stephen

 

[Stephen Benskin]

B = Luminance (currently L)
A = Aperture (currently N)
S = Film Speed
K = Exposure Constant = 1.16 cd/ft2 of 12.5 cd/m2
T = Exposure Time

Let's break it down and solve for B for Sunny 16.

We have a value for B (L) and for K. Let's plug it into the following equation and solve for Ev.

I'm using S=100 because as far as I've seen, exposure meters use 100 to determine Ev.

To check using the first equation in determining the camera exposure for a Luminance value. The exposure meter is pointed to a subject and reads Ev = 14.644

Now we plug it into the camera exposure equation.

q has many of the same variables as K, expect for those relating to the meter. Again, this contains assumptions about the camera and its optical system. I've left off the value for flare because at this level of exposure, it has very little influence.

Checking the value.

Hg = 0.08 lxs
Hg = Average mean exposure or metered camera exposure

It can also be done in cd/m2

The following example was created differently and the film was at 125 and the shutter was 1/125. The key is that it has Luminance values.

I hope this answers your question.

Stephen

 

[alanrockwood]

Thanks Stephen,

That's a bit to chew on for a while.

 

[Bill Burk]

The short answer to “where is the meter aiming?” is 10x from the 0.1 speed determination point.

So if the film is 400 speed the curve will cross 0.1 above base plus fog when developed to ASA parameters… at log -2.7 meter candle seconds.

The meter point would be 10x that, log -1.7 mcs. Or you can look at the 40 speed point

This scale that I put on all my graphs shows the film speeds and the corresponding mcs that go with them.

 

[ic-racer]

To link ISO meter settings to a film curve created in the lab I do an in camera test with the same film (same roll, actually)

The ISO setting that gives a Zone I exposure of a uniform target measuring 0.1 log D after processing is pinned to the speed point on the H&D curve from that roll of film.

With that information, H&D curves of “unknown “ films can be superimposed over the known films to get an Exposure Index for the “unknown” film in the lab. These initial EIs are pretty good and sometimes don’t need change after the film is put in use.

https://www.photrio.com/forum/threads/testing-some-new-film-shanghai-8x10-how-i-do-it.171836/

 

[Stephen Benskin]

Here is a guide for L in cd/ft2 and cd/m2 for different Ev using 100 film speed.