"Cosine fourth" falloff - tutorial article updated

In a photographic system, for a given object luminance (brightness), the image illuminance on the film or equivalent declines as we move outward from the center of the image as a result of the geometric optics involved. The result is a relative darkening of the image toward its borders. If we consider a lens having certain ideal properties, it can be shown that the decline in relative illuminance goes very nearly as the fourth power of the cosine of the angle by which the object point is off the camera axis (measured in "object space" at the center of the entrance pupil of the lens).

I have just released to my technical information site, The Pumpkin, an updated version of my tutorial article, "Derivation of the "Cosine Fourth" Law for Falloff of Illuminance Across a Camera Image", available here:

http://doug.kerr.home.att.net/pumpkin/index.htm#CosFourth

This article derives the "cosine fourth" relationship from fundamental optical and photometric principles.

In this new version, the presentation has (hopefully) been improved and some collateral matter, not really needed, has been removed. I have also expanded my discussion of a major "alternate" result presented by some other authors.

I have also expanded the discussion of a related phenomenon that makes the effect of falloff worse in digital cameras.

Doug,

In a camera with a curved image plane, would the light distribution then be even?

Asher

Hi, Asher

Asher Kelman said:

In a camera with a curved image plane, would the light distribution then be even?

Click to expand...

No. Note that all the cosines come into play from the geometry in front of the lens.

If the image plane is curved (and of course it always is in a real lens) and if the film were curved to match (can't really be done), but the magnfication turned out to be constant, then the relationship I derive would still apply.

Best regards,

Doug

Doug,

It is at least counterintuitive to me that it would make no difference. There's no difficulty in having the film in that curved shape. Just a coated glass dish is very easy to make

If there's a pinhole only, then does your formula still apply?

Asher

Hi Doug,

Have you some values to put into the formula? a worked example? How big an effect on the final image does this play, compared to other aspects of lens theory or design, both in theory and in practice?

You have some interesting topics on your site.

Best wishes,

Ray

Hi, Asher,

Asher Kelman said:

It is at least counterintuitive to me that it would make no difference.

Click to expand...

Oh, indeed, and properly so. I didn't mean to imply that a change in the film surface shape would make no difference in falloff. Perhaps what would have been better said is that, if (through differences in the lens designs) the magnification would be constant in both "cameras", then (assuming the other preconditions held as well) the cos^4 relationship would hold for both. And of course the definition of "magnfication" on a curved film requires some special attention (as cartographers have to recognize when devising different map "projections" of a spheoidal earth).

(I assume that by curved film you do not mean film that is curved in only one axis but rather film that is "doubly curved" (that is, has some sort of bowl shape), matching the actual 3-dimensional shape of the image plane (typically a Petzval surface).

As a practical fact, I think it is not possible to attain both precisely uniform magnfication (freedom from geometric distortion) and a flat image field (such that the entire image of a planar object would be in perfect focus over a planar film surface). I have to review my aberration theory, though, about that.

Regarding the pinhole situation, I think that the basic derivation still holds. A pinhole camera does, for one thing, exhibit freedom from geommetric distortion (for a planar film surface and, of course, planar object). If the pinhole is in a negligible-thickness plate, then its projected area, off-axis, will in fact vary as the cosine of the angle of observation (one of the premises of the cos^4 derivation). (I'm ignoring any contribution to all this of diffraction effects, which I'm not able to cope with at the moment!)

Hi, Ray,

Ray West said:

Have you some values to put into the formula? a worked example?

Click to expand...

I should really put some of this into the article. Perhaps next time.

As one example, imagine a 50 mm lens on a full-frame 35-mm format, with focus at a substantial distance. The corner of the object field is about 23.4 deg off axis. The cosine of the angle is about 0.917, and its fourth power is about 0.709. Thus, under the cosine fourth model, the relative darkening at the corner of the image would be almost exactly 1/2 stop.

With a 35 mm lens on the same format, the corner of the object field is about 31.7 deg off axis. The cosine of the angle is about 0.851, and its fourth power is about 0.523. Thus the relative darkening at the corner of the image would be almost 1 stop.

With a 20 mm lens, the corner of the object field is about 47.2 deg off axis. The cosine of the angle is about 0.678, and its fourth power is about 0.212. The relative darkening at the corner of the image would be about 2-1/4 stops.

Paul van Walree, in this piece:

http://www.vanwalree.com/optics/vignetting.html#optical

shows falloff curves for two famous lenses. He also plots for each the impact of the "cos*cos^3" result for theoretical falloff. Relying on that result, he points out that the relative reduction in image-space off-axis angle in one of the lenses for a given position of an off-axis object (as a result of the more forward location of its exit pupil) leads to it exhibiting reduced falloff. Of course, as you perhaps know from my article, I don't believe that.

Best regards,

Doug

Hi Doug,

Thanks for your reply, and the link. So, in rough terms, for a given theoretical lens, the darkening across the image will vary between 0 and, say a stop or so, but not linearly, but proportional to the 4th power of cos of the angle. For a crop frame camera, and a practical full frame lens, the effect is much less noticeable then if it were a linear, square or cubic relationship. (I am aware of some of the issues between theory and practice, but if I'm right, it explains a lot to me, and if I'm wrong it gives you the chance to explain some more ;-)

Best wishes,

Ray

Hi, Ray,

Ray West said:

Hi Doug,

Thanks for your reply, and the link. So, in rough terms, for a given theoretical lens, the darkening across the image will vary between 0 and, say a stop or so, but not linearly, but proportional to the 4th power of cos of the angle. For a crop frame camera, and a practical full frame lens, the effect is much less noticeable then if it were a linear, square or cubic relationship. (I am aware of some of the issues between theory and practice, but if I'm right, it explains a lot to me, and if I'm wrong it gives you the chance to explain some more

Click to expand...

You've "got the picture".

For example, with the same 35 mm lens I talked about before on a camera with a format size of 22.5 mm x 15.0 mm (e.g., a Canon EOS 20D), the theoretical darkening at the corner would be about 0.4 stop (vs. about 1 stop for the full-frame 35-mm format).

On the full-frame 35-mm camera, we would have this same darkening at a point on the image 13.5 mm from the center (the same offset distance at which the corner of the 22.5x15 mm image lies).

It is fact that we have the fourth power of the cosine at work here that makes such a dramatic change as we change off-axis angle.

Best regards,

Doug

Hi Doug,

You can guess what's coming? a 50mm full frame lens on a crop frame camera, cf an 85mm lens on a full frame camera, cf a 50mm efs lens on a crop frame camera (I guess the efs should make no difference, in theory, but I would expect the construction may cause vignetting more readily.) (efs = canon, no idea equiv for Nikon)

Best wishes,

Ray

Ray West said:

You can guess what's coming? a 50mm full frame lens on a crop frame camera, cf an 85mm lens on a full frame camera, cf a 50mm efs lens on a crop frame camera (I guess the efs should make no difference, in theory, but I would expect the construction may cause vignetting more readily.) (efs = canon, no idea equiv for Nikon)

Click to expand...

I assume that by "full frame" you mean a camera with a format size of 36 mm x 24 mm (although there is no reason it should mean that) and by "crop" you mean a camera that has had its original sensor cut off on two sides?

(I'm just pulling your leg - many here know of my concern over certain terminolgy. To me, "full frame" can mean 8" x 10", the format size of my Century Studio Camera if I use the largest practical film holder and don't close the "half-frame" wings.)

Well, the real answer is, "it depends entirely on the individual lens design".

Best regards,

Doug

Hi Doug,

I speak pretty common - what I guess I was trying to do was to tease from you some practical aspects of the theory. I couldn't quite get my head around the confusion caused by using different lenses as I described. If it's a theoretical lens, then they will all be the same - since it is depending on the angle - its no good, I'm getting a headache....

From a very crude practical test, an led say, with constant voltage supply, a couple of shots, measure with the eyedrop tool in cs2? It will show all the lens correction, conversion corrections, etc., but at the end its the result that counts (usually?).

Best wishes,

Ray