The amazing camera Part 2: Perspective!

How large is large?

In the picture below, how can a 9-inch human head appear to be the same size as an 18-foot sphinx head? Of course, the sphinx is farther away. What's happening in the eye or in any other camera?

Woman kissing the Giza sphinx

A camera lens projects objects in the three-dimensional scene onto a two dimensional image sensor. Information about depth is lost. The size of something in a camera image is actually an angle. If the woman's head and the sphinx head both subtend 2° of angle from the camera (or from the eye of an observer), they will be the same size in the projected image.

Side view of woman and sphinx showing that their heads subtend the same 2-degree angle

So what determines the size of the image? Take a point on the object, such as the top of the sphinx's head, and draw a straight line from that object point through the center of the lens and then continuing to the image sensor. (The center of the lens is where the lens surfaces are parallel, so the light ray is not bent by refraction.)

Lens forming an image of a sphinx head

By doing this, we've constructed similar right triangles (not drawn to scale above). This means that the image is smaller than the object by the ratio of their distances from the lens. This also means that when an object is moved farther and farther away from the camera, the image size gets smaller and smaller. If the object could be moved infinitely far away, the image would shrink to a point. This is known as the vanishing point.

Images of sphinx head as it moves farther and farther from the camera

Getting depth perception

There are a number of ways we extract depth information from the two-dimensional images on our retinas. I'll mention only a few, but you can find more discussion on Wikipedia.

Motion parallax is an example of a monocular depth cue, which means we can get it using one eye. When we drive along a highway, close objects appear to pass by faster than distant objects. Again, this is because we see angles rather than locations, and the closer objects have faster angular motion across our field of view.

Buildings moving by as observed from a moving car

Another monocular depth cue is the perspective effect shown in the vanishing point figure above. When we look down a street, the sides of the street appear to converge toward a vanishing point. This helps us judge the distances to cars parked along the street. But the eye can be fooled! Here is the Ames room illusion. The twin sisters are in a trapezoidal room with trapezoidal floor tiling. Perspective makes the floor tiles appear rectangular, so we judge the sisters to be equally far away and therefore of different sizes.

The Ames room illusion Alex valavanis, CC0, via Wikimedia Commons

Stereopsis is a binocular depth cue, that is, one that exploits the fact that we have two eyes. They see the scene from different points of view. The closer an object, the greater the difference in viewing angle between the eyes. The brain can "merge" the two images and get a sense of depth. Stereograms fool the eyes into seeing depth in a flat picture. Look at the picture below and then cross your eyes just enough to overlay neighboring repetitions and then relax. If you succeed, you will see a 3D scene.

Stereogram by Fred Hsu.. Share Alike 3.0 Unported License. https://creativecommons.org/licenses/by-sa/3.0/deed.en

How far away is too far to see?

Sirius is the brightest star in the night sky. Yet it is 8.6 light years away, or about 50 trillion miles. (See my post on Speeding Through the Galaxy to find out how we know this.) How is it possible for the naked eye to see this star? It depends on what is meant by "see." Our eye gets enough light from Sirius to detect it, but we don't have enough visual acuity to see any details.

How much light is enough for our eyes to detect a star? The light from a star or any light source spreads out more and more thinly as one gets farther away. In order for the eye to detect visible light, it needs at least 0.1 nanowatts per square meter (a nanowatt is a billionth of a watt). A human being living on a planet in the Big Dipper system (80 light years away) would not be able to see our sun. The light has spread out too much, making the sun too faint to see. Most of the individual stars in the Milky Way are not visible from Earth for this reason.

So assuming there's enough light, how close does something need to be for a person with perfect vision to see its details? The answer, once again, involves an angle. Any two points in the scene that are closer together than about 1 minute of arc (1/60th of a degree) will be blurred together. This visual acuity limitation is due to the limited density of photoreceptors on the retina and the fact that no lens can focus light to a perfectly sharp point.

Imaging of Sirius by the eys

Without atmospheric blurring, Sirius would subtend an extremely small angle in the sky, about 0.0001 minutes of arc. With atmospheric blurring, this angle would be more like 0.01 minutes of arc. In either case, the optics of the eye will blur the star so that its apparent size is about 1 minute of arc.

Debunking nonsense

There are a lot of misconceptions about this topic and Samuel Rowbotham (1816-1884) is the source of some of them.

Let's do a thought experiment (but similar real experiments are possible in a science classroom). Put an eye chart on a train car and move it farther and farther away along the tracks while viewing or photographing it. We can still predict the locations of features on our camera's image sensor by drawing straight lines through the center of the lens. As with the sphinx face, this will lead to a common vanishing point at infinity. But now we need to blur those features according to the acuity of the camera. For the human eye, that's a blur of about 1 minute of arc.

Eye chart moving farther and farther from the camera, correct picture and zetetic perspective

The picture on the left above shows what happens. As the eye chart moves farther away, the apparent size of everything shrinks and the features all blur. When a letter in the bottom row subtends less than 1 minute of arc, it will blur to gray. The edges of the large E at the top are also blurred. At a very great distance the entire chart would appear as a light gray dot.

Rowbotham proposed an alternative idea that he called zetetic perspective, illustrated on the right above. He claimed that if you focus your line of sight on the bottom row, it vanishes before the rows above it because the smallest letters are the first things to become too small to see. His real agenda was to prove that the Earth is flat. He claimed that the apparent disappearance of the bottom of a ship sailing over the horizon is due to this perspective effect rather than the Earth's curvature. This is not how perspective works! The letters don't vanish; they shrink and blur. But to demonstrate the absurdity of his idea, imagine turning the eye chart upside down. By his logic, the top row would be the first thing to disappear. And by his logic, if you focus your line of sight on the crow's nest at the top of the sailing ship's mast, that should vanish before the rest of the ship by virtue of being so much smaller! So, the next time you see only the top half of the sun during an ocean sunset, you can be confident that the bottom half is blocked from view -- by the Earth.