Exposure Functions
© 18 Apr 2013 Luther Tychonievich
Licensed under Creative Commons: CC BY-NC-ND 3.0
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How eyes and cameras work, and why they sometimes fight each other.


I recently had occasion to observe both the screen of the camera that was filming a lecture and the lecture itself at the same time. On the screen, there was what looked like a spotlight shining on the wall. In the real world, that "spotlight" was all-but invisible, just the faintest light gradation. Why?

The answer lies in exposure functions in both the real world and in our eyes.

You can think of any camera or eye as being made up of a lot of tiny light receptors. Each of these receptors has two modes: “‍black‍” if it hasn’t been hit by a photon or “‍white‍” if it has. When about half of the receptors in an area are black and half white the results is a nice 50% grey. In an eye the white receptors return to black after a fraction of a second but at any given moment some of them are still “‍exposed‍” (white) and some not.

Now, suppose when the light is at level x 50% of your receptors are exposed. Then any additional photons have a 50% chance of hitting an already-exposed receptor instead of one that can react to the light. Thus, over-simplifying the math, at light level 2x 75% of the receptors are exposed; at 8x 94% are exposed; and so on.

One of the consequences of this model of exposure is that a well-lit picture will have the dark areas of the image more-or-less match the actual brightness of the original scene while the bright areas are much more flattened. The impact of this is to make the image appear brighter overall: a little light goes a long way in the dark regions.

So why things like the spotlight effect I noticed in the lecture? Because of what the eye does with what it sees. The optic center of the brain is very good at detecting edges and lines, places where the pattern of light changes. It is good at doing this in spite of naturally occurring changes in lighting due to shadows and distance from a light source. But in a photograph, light doesn’t change the way it does naturally. The boundary region between where light was dim enough to behave naturally and where it was bright enough to be flattened was detected by my eye as a line, and thus stood out in a way the real world did not.

So why don’t we make cameras that take pictures at the level of real-world light? With digital cameras we can kind of make this happen if we want to. But the results come out looking very dark. For example, consider the two images:

The light coming from the left image is approximately proportional to the light coming from the actual scene. The light coming from the right image is a photograph. The reason the one on the right looks more “‍real‍” is because of the eye’s exposure function.

Recall that the eye, like any other camera, reacts less rapidly to increases of light if the light is already bright. But the eye also has adjustable pupils that pick a level of exposure so that the retna is sending signals scaled like a photograph is scaled to the optic center of the brain, which then processes them, detects colors and edges and so on. A picture has the interesting job of trying to emulate the entire visual experience of being at the scene without being able to control the entirety of your visual experience. It is very rare for the brightest part of the image to be the brightest thing you can see.

A photograph—and most realistic paintings and other artwork—makes up for the fact that the picture is just a piece of what you see by applying the exposure function your eye would have applied if you were there. Your eye then applies another exposure function on top of that, but generally the photo is not very close to the brightest your eye can see so the the photons from the photo are all pretty dim to your eye and make it to the optical center more-or-less unmodified.

If an image is properly exposed for the level of light around it, it will be interpreted by my brain as if I were where the camera was. But if it is a bit too exposed or not exposed enough for the particular lighting my eye experiences when looking at it then artifacts like the spotlight “‍edge‍” standing out occur.

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