A dome of its own: incident light metering

I often say in discussions of exposure metering that an incident light exposure meter responds to the illuminance upon the subject (or upon the place where the subject will be for the shot).

In fact, that is imprecise. A more complete description would be that the meter responds to the illuminance upon the plane of the receptor of the meter. We'll shortly hear more about the significance of this. It involves those pesky cosines.

But that's not exactly so either, a further matter of those pesky cosines.

The illuminance upon a plane caused by an arriving "beam" of light of some luminous flux density (the measure of the ongoing potency of a "beam" at some place it reaches) is the product of the luminous flux density and the cosine of the angle that the direction of arrival of the beam makes with a line perpendicular to the plane at the point of interest (the angle of incidence).

For a flat subject surface that is "ideally diffusing" (that is, is a Lambertian reflector), its illuminance (what we can think of, somewhat imprecisely, as its brightness) is:

• Proportional to the product of the illuminance on the surface and its reflectance.

• The same for any angle of view.

• Not affected by the angle of arrival of the beam that produced the illuminance beyond the effect of angle of incidence we earlier mentioned. That is, if we had one beam that arrived at a zero angle of incidence (perpendicular to the surface), and another beam that arrived at a 45° angle and had 1.414 times the luminous flux density, then both would deposit the same illuminance, and we could not tell the difference between the two cases, for example by examination of the luminance of the surface as seen from different angles.

Now if in fact all our subjects had flat and Lambertian surfaces, for exposure planning, we would want to know the luminance upon that surface or upon a hypothetical plane with the same orientation.

And we could measure that with an exposure meter that in fact responded to true illuminance, by placing it at the point of interest with its receptor parallel to the plane of the surface.

In a meter that responds to true illuminance, every "component" of the arriving light is "weighted" by the cosine of its angle of arrival, so as to mimic the definition of illuminance.

Said another way, the angular sensitivity pattern of the meter follows a cosine pattern. If we plotted this, the polar plot would be a circle with a point on the circle at the origin ("at the meter") and its diameter extending out along the meter's "boresight".

But for most of our work, the subject is not a flat surface (in glamor photography, we certainly hope not). And the various surfaces of the subject may or may not be close to Lambertian in their reflective behavior. ("No, she said she was an Episcopalian".)

So this whole lovely theory falls apart in practice.

It's not just the theory that falls apart, but the actual photometrics. Suppose we consider a model, facing the camera, whose face is mostly illuminated by a beam from her right left. Its angle of incidence on the right side of her face is smaller than its angle of incidence on the front of her face, so we might expect a greater illuminance on the right side of her face than on the front. An thus we might thus expect a greater luminance of the right side of her face than the front.

And of course this i hardly a surprise to the studio photographer, who may have put that light source where it is to produce that exact effect. And the sophisticated studio photographer may in fact use an incident light exposure meter, of the type that measures true illuminance (we'll talk in a bit about what kind that is) to make separate illuminance measurements of the illuminant on both sides of her face and on its front, each principally coming from a separate source, to fully judge the effect to be gotten.

But in a less sophisticated situation, the photographer may want to just make one incident light measurement. Clearly, that can't tell us about the differences we will get in exposure on the different aspects of the face (which may not have the relationship we like from an aesthetics standpoint, but clearly we are not up to dealing with that precisely).

Various workers (and Don Norwood, the developer of the famous Norwood Director series of studio exposure meters is often credited as a key player in this) found that in many cases (an important phrase in all discussions of exposure metering) a good overall compromise exposure may be determined based on the observation of a meter whose receptor did not have a "cosine" directivity pattern but rather a one something like:


where A is the angle off the boresight direction. This polar plot is an epicycloid of one cusp, which some have very fancifully thought looks something like the iconic representation of a heart - this shape thus came to be called a "cardioid" ("heartlike") pattern. It is, however, as one author commented, actually much more like the cross section of an apple without its stem.

The "good exposure result in many cases" would come from uniformly orienting the axis of the meter toward the camera (as we would do with the other kind of meter to place its receptor in a plane facing the camera, as for the near side of the face. (Hey, if this is to be a "simple" procedure, lets keep it truly simple.)


How do we make a meter have such a directivity pattern? One technique (proposed and refined by Norwood) is to place a translucent hemispherical shell over the meter's basic flat receptor. (Of course, some further details need to be attended to.)

And in fact the iconic studio exposure meter, The Norwood Director, has as its hallmark a prominent hemispherical diffuser.


Now, in many of the photographic exposure meters offering an incident-light mode, the hemispherical differ is part of the standard configuration for that mode, and so these partake by default of the "Norwood strategy".

Especially in the more sophisticated types (several of the Minolta machines, for example), there is an alternate "flat" diffuser (maybe needing to be separately purchased) that, mounted in place of the hemispherical diffuser, will give the meter very nearly the classical cosine pattern. Often the manuals will advocate the use of this when making separate measurements of the illuminance from multiple sources in a studio to plan "ratio lighting".

And thus the loop is closed to where I came in.

Best regards,

Doug

I have taken a quick look at the math behind the concept of the dome receptor for an incident light exposure meter. I started with the explanation of its benefits in Don Norwood's patent (1940 - US 2,214,283).

It is difficult to test his assertion since it is not accurately stated. In particular, he does not at all utilize any specific photometric quantities (illuminance, luminance, luminous intensity, luminous flux), and so forth, but merely speaks of "amounts of light".

But I believe I recognize the point he is trying to make, so I will first take the liberty of forming his concept into a more precisely defined one.

As I discussed in the previous note in this series, a "classical" incident light exposure metering process would be conducted with a meter that responds to the illuminance deposited by the ambient illumination upon a planar subject surface. (We will assume throughout a Lambertian reflecting surface - otherwise, all hope is lost.) The meter's receptor would be oriented parallel to that surface.

For any given uniform subject reflectance, a photographic exposure based on such a measurement would give a consistent exposure result regardless of the orientation of the subject surface and regardless of the distribution of the incoming illumination from sources at different locations.

But, if the subject has surfaces at different orientations (even of the same reflectance), we may well have quite different exposure results for the different surfaces (right side of face, left side of face, etc).

Now for the tricky part, where Norwood doesn't really rigorously espouse an objective. But let me posit one I think he had in mind. We wish to be able to make a single illuminance measurement that will predict the average luminance of the subject (for a subject of uniform reflectance). This then could guide an exposure that would, on the average across the subject, fulfill our exposure strategy.

Now, apparently in pursuit of this, Norwood speaks of the total amount of light on the subject from the various light sources, and suggests that if we knew that, this would be the proper basis of exposure planning. (He thus completely jumps over those pesky cosines and photometric quantities.) Reading that I get the same queasy feeling I get when somebody speaks of the "loss of light" when using an extension tube.

He then goes on (quite reasonably) to suggest that a hemisphere would be a usable proxy for a subject's face. Since, he says, what we are really interested is the total amount of light landing on the subject's face, if we could capture the total about of light landing on a hemispherical receptor, that would tell us what we needed.

Of course, there would be a scaling by virtue of the different sizes of the face and its proxy, the meter receptor, which would be taken care of if this were being discussed in terms of specific photometric units (not "total amount of light". It isn't, so there seems to be a paradox. But that is easily overcome by a little care with the photometric quantities.

He then says, so let's make a meter that responds to the "total amount of light" landing on a hemispherical receptor. To test the validity of that approach, we need to have a true definition of our objective. I have posited that it is to have the meter indication tell us the average luminance observed by the camera across the subject (for any assumed uniform subject reflectance).

Note that this is the average as across the across the image, not the average we would get crawling over the surface the hemisphere (an important distinction). A region of a certain area "around to the side" would contribute less to the average under my definition than if we were taking a survey by crawling around the hemisphere.

So does the hemispherical measurement scheme fulfill that objective? The general mathematical proof would be rather hairy, but I set up some numerical cases that, if they do not give a consistent result, would show that the proposed relationship must not be correct in the general case. If they did give a consistent result, I would be condemned to actually develop the generalized equations o see if that was just a fluke of the examples I chose to test.

I actually used a cylinder, rather than sphere, as the face model and the meter receptor, so I would only have to integrate in one direction. (Hey, this is tough stuff for an old geezer!)

I took two cases. In one, the face was illuminated head on with illumination of a certain luminous flux density. In the second, it was illuminated from both sides, with beams of that same luminous flux density (the same total luminous flux as in the first case) In each case, the entire "face" was illuminated.

Clearly, the same amount of luminous flux would be deposited on the entirety of the face, and thus on the entirety of the meter receptor, the proxy for the face, in the two cases. So we should expect that the "Norwood" meter would give the same reading for the two cases.

Now the punch line: would the average scene luminance, as seen from the camera, be the same in the two cases? No. For the "lit from both sides" case it would theoretically be 64% of the value for the "illuminated from the front" case.

Now, does this tell us that the Norwood metering approach is not valuable? Not at all. In fact, from these same two cases we can see the limitations of trying to so this with a true illuminance meter. Assuming that it was used to make a single measurement (that's the objective), in this model, for the second case (illumination from the side) it would give a zero reading!

In the first case (face-on illumination), the "true illuminance" meter would give a reading about "1/3 stop" higher than the "Norwood" meter.

However, when it comes to the actual exposure recommendation issued by the meter (a mater of its incident light exposure meter calibration constant), the ISO standard provides for about an 0.4 stop offset to bring the two more closely into agreement.

In summary

There is no doubt but that in many real cases, the Norwood metering approach will give a better "compromise exposure" (and that's all we can ever get in the case of other than a planar subject surface) than a true illuminance measurement.

But we must not think that the "acceptance pattern" of the Norwood hemispherical receptor is a magical one that will allow such a meter to deliver, over a range of situations, an exposure recommendation that would produce an absolutely consistent average exposure for a uniform-reflectance subject.

Epilog

This has been quite a journey for me. I had never looked into the the properties of the dome receptor before, This note was all written on the fly, with an occasional pause of 10-20 minutes to do some cogitating and then some algebra.

I have a new respect for the advantages of the hemispherical receptor, while at the same time realizing that, as I had suspected. it is not any sort of "magic bullet". And I now understand better the significance of different calibration constants prescribed by the ISO standard.

Perhaps it will be of some value to others.

Best regards,

Doug