Depth Perception, part 1

This post will take a look at how depth perception works. The next post will build on this background to answer yesterday's question.

First experiment:

1. Close your eyelids.
2. Now open them.

If you did the first part of that experiment right away, you shouldn't have known what to do next, because you wouldn't be able to read the instructions. Close your eyes and you can't see any more. Conclusion: you see with your eyes. (Actually, this just shows that eyes are "necessary but not sufficient". But hey, let's not split corneas here.)

Second experiment: Cut open a cow's eye and diagram it. You should get this:


The salient point is that your eye is designed to give your brain the information "lots of light is coming from this direction, not so much from that direction, and none at all from a third direction". Overall, there are lots and lots of directions. When you put them all together, you get an image.

That's what one eye does. The way it does it is optics. First, light comes into the eye and passes through the lens. All light that comes in parallel, regardless of whether it enters the left or right side of the eye, then gets focused to the same part of the retina. That's the glory of the lens. So all the light falling on a certain part of the retina came from the same direction in space, not from entering the same part of your eye.

This directional knowledge lets your build an image. Here is a picture you can look at with your eyes:

To see each pixel, you must look in a slightly different direction. Putting each pixel with its appropriate direction allows you to build the image back up in your mind.

So eyes only care about the direction light comes in. However, you have two eyes, and they're in different places. That means light from the same source will enter from different angles. Here's a gratuitous graphic:


Okay, so light from an object, like a baseball you want to catch (actually, depth perception is not the tool baseball outfielders use to flag down a fly ball, but that's another story), enters your left eye from a direction (&theta_l, &phi_l), and enters your right eye from a different direction (&theta_r, &phi_r). How far away is the object? Here, &theta is the polar angle, the angle between the line from the top of your head to your feet and the path of the light. &phi is the azimuthal angle, telling how far to the right or left the direction is. Some people use a different convention. Mine is like this:

You stand with your head pointed towards z and your face towards x.

Your eyes are separated (obviously) by a distance d. Also, we'll make the approximation that the object is far enough away from your face that it's essentially the same distance from each eye. In that case, the polar angle &theta will be the same for each eye.

The three lines, one connecting your eyes, the other two from each eye to the object, form a triangle like this:

Boom! The math is right in there! I actually just spent ten minutes searching for and installing this thing that should let me start 'ing up my blog, and then I didn't even use it (except to make that sexy logo).

Anyway, yup. where d is the distance to the object, and w is the width of your eyes. &Delta is the difference between the azimuthal angles, and &phi is that azimuthal angle (either one. the difference is small). I made two simplifying assumptions. The first is that the difference in the angles is small, which just means the thing isn't right up in your face. The second is that the object is close to right in front of you, which is necessary to be able to use depth perception on it.

Nice.

Now there's a problem, which is that you only get so much accuracy in determining the angles. As &delta becomes smaller and smaller, small errors result in big problems for d. So depth perception only works for objects close to you, or a medium distance away. For very far objects, you need to use other clues. The same problem occurs for astronomers attempting to find the distance to a star by parallax.