About incident light expsure metering

This is a second stream of thought about incident light exposure metering.

In incident light exposure metering, we measure a property of the light that illuminates the subject. (I am being intentionally imprecise at this point as to what property that is). It is distinguished from the more-familiar reflected light exposure metering, in which we measure the luminance of the scene (averaged over the field of view, or some part of that, or some region larger than that).

Although we rarely hear it said in this way, the objective of incident light metering is:


That is, it seeks the have the image of a black cat on a coal pile always look like a black cat on a coal pile, not a gray cat on an ash heap (as would typically happen with reflected light exposure metering).

If the entire subject in which we are interested were in a plane, then the property we would want to measure of the incident illumination would be its illuminance upon the plane of the subject.

A photographic exposure (f-number and shutter speed) based on that, taking into account the sensitivity of the film or digital system, will fulfill the objective stated above in blue, regardless of the orientation of the subject plane with respect to the lens axis (assuming that the illuminance was measured as upon that plane), and regardless of the direction from which the illumination came, or how it was composed of illumination from different sources at different angles.

How can we measure the illuminance upon that plane? Imagine a meter that has a flat receptor and responds to the total luminous flux that strikes that its surface. If we go though some neat stuff with a lot of cosines, we will find that the meter's response is proportional to the (average) illuminance upon its receptor. If we position the receptor so it is at the subject (or where the subject will be for the actual shot), and parallel to the plane of the subject, what the meter perceives is the property needed to determine the needed photographic exposure to attain the objective in blue.

Neat! Photographers of billboards or building fronts now know how to proceed.

But not portrait photographers.

If the entire subject does not lie in a plane (a human face decidedly does not, for example), and the illumination is not uniform over all directions of arrival that illuminate parts of the subject that will be included in the shot, there is no photographic exposure that will actually fulfill the objective in blue. Thus in such a case, it would be futile to inquire as to what property of the illumination should we measure to guide the photographic exposure in order to fulfill the objective in blue.

Does this mean that we cannot use exposure metering in such a situation? Not in a definitive way.

If the light comes from two of three discrete sources (three photographic floodlights, for example), could we separately measure the illuminance from each, and adjust that so it was equal for the three sources, and then base our photographic exposure on that value?

Well, luminance must be determined as upon a plane of some orientation. What plane would that be for each of our three floodlights?

Well, from the one aimed at the model's right ear, perhaps the plane of the right side of her face. But the right side of her face is not likely a plane - perhaps more like part of a cylinder.

And in fact let's consider an exposure taken with only this light source in effect (with it set to some arbitrary potency), and let's for the moment imagine that the model's whole face has a uniform reflectance (kabuki makeup, for example).

But we find that the luminance of the different parts of the right side of her face will not be uniform. It will be highest for the part (by the ear perhaps) that is (mostly) locally perpendicular to a line from the source - where the luminous flux density of the arriving light causes the greatest illuminance.

But further to the front (perhaps below her right eye), the light from that source strikes the face "obliquely", thus the illuminance it causes is less, and so so the luminance of the light reflected from it will be correspondingly less. This is in fact the classical "shading" result.


So we can see that there would no plane "against which" we could determine the illuminance from the one light source so that from that measurement we could determine a photographic exposure that would completely fulfill the now infamous blue objective for the right side of the model's head (even if we were willing to make three measurements, one for each light source, to do the total exposure planning job).

[continued]

Part 2

[Part 2]

Now let's look a little at the meter. Of interest here is the acceptance pattern of the meter's receptor. By that we mean a description of the relative importance the meter gives to luminous flux arrival from different angles as an ingredient to its overall reading (I have not said "of illuminance"). Said another way, it would be the relative reading it would give for a beam of light arriving from a single direction (and "wide" enough to illuminate its entire receptor).

Forgetting for a moment about meters and just thinking about an illuminated plane, the illuminance caused on that plane by a beam of light with a certain luminous flux density (a measure of the "potency" of a beam of light at some point in its journey) is the product of the luminous flux density and the cosine of the angle the direction of arrival of the beam makes with a line perpendicular to the surface at the point where we are interested in the illuminance - that angle being called the angle if incidence of the beam.

If the meter is to perceive luminance, it turns out that its response to an arriving beam of light of some luminous flux density must be proportional to the cosine of the angle of incidence of the beam - an arrangement that is called a cosine acceptance pattern.

Now, if the receptor responds to the total of the luminous flux incident upon it, it will respond to illuminance (it is just a proxy for the "target" surface), it must exhibit a cosine acceptance pattern. And yes, an (ideal) flat receptor will exhibit a cosine acceptance pattern.

If we plot the relative "sensitivity" of the receptor vs. the angle of incidence of the light beam, we will get this plot for a cosine acceptance pattern:


It doesn't go beyond 90° in either direction because beyond those angles the beam of light would be coming from behind the meter and would have no effect.

We often plot this in polar form, which would look like this:


The significance of this is that for any angle, A (reckoned from the origin, at the center of the graph, with 0° being straight to the right), the distance r to the point on the curve where a line at that angle intercepts it is the value of the pattern at that angle.

It turns out that if the value r is the cosine of the angle A (as for the case we are considering), the curve will be a circle!

So we see here the "polar" plot of the acceptance pattern of a meter with an ideal flat receptor, one whose indication will in fact be the illuminance of the arriving illumination upon the plane of its receptor.

This will be true, incidentally, even if there are multiple beams of light from various directions.

Now we can get back to the portrait photographer's dilemma. Remember, there is no single exposure that will produce the theoretical ideal exposure metering objective in blue in Part 1. Thus no single measurement, regardless of how fancy the meter is, that could tell us the exposure we need - there is no such.

Bur suppose we leaned back a little, and said, OK, we will never get uniform exposure of a real-shaped subject. Is there something that we could measure - one number, please - that would recommended an exposure that would be as close to the blue objective as could possibly happen?

Well, to pursue that, we would have to contemplate in our research all possible combination of illumination of various luminous flux densities arriving from different angles. Don't sign me up for that.

But in the late 1930's, a cinematographer, Don Norwood, wrestling with this problem, evidently had an intuition that if we made a meter with an acceptance pattern that fell off less at angles of around 90° than the cosine pattern, it might (strictly empirically), in many cases we actually encounter in a studio, give a reading that would give a photographic exposure that would give a good compromise exposure result over all of a subject's face.

He evidently came to consider a meter in which the receptor was in the form of a hemisphere (which indeed had a pattern rather different from a cosine pattern - we'll see it in a little while). Evidently tests with it confirmed his intuition.

Note that I have not said that this would give, in the general case of a non-planar subject, under non-uniform illumination, a measurement that would lead to a photographic exposure that would result in (for a subject of uniform reflectance - our kabuki player) a uniform exposure result across the entire face. There is no such photographic exposure uniform from all angles. I just said that such a meter (with an appropriate overall calibration constant) evidently gave a very good compromise result in a lot of practical situations.

I don't mean to at all demean this development. I just need to point out that this new (in 1938) meter design could not solve an insoluble problem.

Later, Norwood concluded that essentially the same response given by his spherical-receptor meter could be closely approximated, less-expensively, with a flat receptor covered by a thin spherical translucent diffuser. And we have all seen pictures of these - on a Norwood Director exposure meter - peering out from some famous actress' cleavage in the hands of some famous assistant cinematographer in a "movie being made" still shot.

[continued]

Part 3

[Part 3]

ISO standard ISO 2720, which covers free-standing photographic exposure meters, provides (in the case of the incident-light type) for two acceptance patterns, which they relate to "flat" and "hemispherical" receptors.

For the flat receptor, they say that the pattern should approximate the "cosine" pattern. We have already looked at that.

For the hemispherical receptor, they say that the patterns should approximate the "cardioid" pattern. What is that?

Well, its curve in the form usually stated in generic analytical geometry work is given by:


In our case, we can consider r to be the relative response of the meter in the direction A (measured from its axis, or "boresight"). But because we want the maximum relative response (on axis) to be 1.0, and the maximum of the function just stated is 2.0, we usually scale this to:


Plotted in rectangular coordinates, this looks like this:


Considering this to be a meter acceptance curve, how can there be any response for angles beyond ±90° ("from behind the receptor")? Remember, this is for a hemispherical receptor. Indeed illumination from "a little behind" can strike the receptor. But of course for angles approaching 180°, the response suggested by this theoretical curve will not happen.

Now, we look at this on a polar plot:


(This is not scaled, so the maximum is 2.0, not 1.0. Sorry. I lifted this figure and was too lazy to reconstruct it from scratch, scaled. All those cosines are so tedious, you know.)

This is in formal geometric terms an epicycloid of one cusp. Somebody back in history thought that it looked like the shape of a heart, and so it was given the name cardioid ("having the form of a heart"). Go figure.

Now, can we expect a hemispherical receptor to have this response. No. The response we would expect it to have, plotted in blue on the rectangular coordinates plot of the cardioid response (itself black), is seen here:


Wow! Will that do the trick? Would an actual cardioid be "better"?

What trick? Remember, there is no meter acceptance pattern that will cause the meter to recommend a photographic exposure that will theoretically be ideal for an entire person's face under a situation of non-uniform illumination.

So why does the standard specify the cardioid response? Beats me. My own cynical guess is that a cardioid response is often used in many technical fields, such as for the acoustical response of a microphone, or the electromagnetic response of an antenna. So it probably seemed "more scientific" than the blue curve that Norwood's meters may well have exhibited. Or maybe they didn't.

And remember, there is no right answer. The problem is insoluble. And different meter patterns produce different compromise results (at best) in different circumstances.

But then, all exposure metering is like that!

What does the standard say are the uses of the two patterns? I paraphrase it thus:

• For the "flat receptor" pattern (cosine): to determine actual illuminance if that is what you want to know; to make measurements of individual sources in a multi-light studio situation for "light balancing" purposes. (Note that this will not solve the insoluble problem.)

• For the "hemispherical receptor" pattern (cardioid): For general purpose photographic exposure determination.

So treat the dome of your favorite incident light exposure meter with respect - but not with reverence.

Well, thank you for listening.

No cosines were injured in making this presentation.

Next week: "The love life of the lesser dodo".

Best regards,

Doug