Resolving Eye Shapes

Greg Ingersoll

Thu 31 January 2013 - 12:25:00 CST


If you have ever looked closely at the eyes of certain animals, you may have noticed something about the shapes of their pupils. Some animals, say humans, have round pupils while other animals, cats for instance, have tall thin pupils. And animals like deer and goats have pupils that are wider than they are tall. What are the advantages of each? One way of looking at the answer involves a quick diversion into spatial frequency analysis and properties of Fourier transforms.

Domestic cats have tall narrow pupils.

Consider light coming from two distant stars—pinpricks in the sky. The wavefront of the light expands spherically from each star, and by the time it reaches your eye, the sphere is so big that it seems basically flat, or planar, instead of curved. So what we have is a plane wave from each star, and because the stars are in different places, the waves are traveling at slightly different angles. Having different angles is equivalent to having different spatial frequencies (the reason is straightforward, but is a whole post by itself).

The pupils of deer are wider than they are tall.

When these two plane waves pass through an aperture like the anatomical pupil of your eye, the wavefronts are clipped. Immediately after the aperture, instead of being infinitely wide plane waves, they now stop at the edges of the aperture. These hard edges cause the light to diffract which results in additional frequency content after the aperture. Looking at this in the (spatial) frequency domain can shed some additional light.

In the plane of the aperture, a plane wave arriving at some angle appears as a sine wave when plotting energy vs. location. Consulting a table of Fourier transforms, a sine wave transforms into two delta functions—infinitely thin spikes sitting at particular frequencies. One delta is at the positive frequency and one is at the negative frequency. So having two plane waves at different angles leads to four delta functions in the frequency domain.

Left: two sinusoids of different frequencies (5 and 8 cycles per unit length) in the spatial domain. Right: the resulting delta functions in the spatial frequency domain.

The simplified aperture is just a rectangle function. This function equals 1 over the width of the aperture and equals zero everywhere else. In the frequency domain this becomes a sinc function

$$sinc(x)=\frac{\sin(\pi x)}{\pi x}$$

which is infinitely wide in frequency. And importantly, the horizontal scaling of the sinc function in frequency is inversely proportional to the width of the rectangle function: wide rect, narrow sinc lobes and vice versa.

Left: the rectangular aperture. Right: Its frequency content, the sinc function.

Now the plane waves passing through the aperture is equivalent mathematically to multiplying the plane waves by the aperture. While addition in space is also addition in frequency, multiplication in the spatial domain leads to convolution in frequency and vice versa. (Convolution: so that’s where that name came from.) So what do we get when we convolve a sinc function with two delta functions?

As it happens, convolution of anything with a delta function simply gives you a copy of the something shifted to the location of the delta. So convolving the sinc with four deltas gives us four copies of the sinc in the frequency domain. Instead of two discrete frequencies (four if you count positive and negative), we now still have a fair amount of energy at those original frequencies—the sinc functions are centered there—but we also have energy splashing into infinitely many other frequencies. This is the result of diffraction from the hard aperture.

The frequency domain result of the aperture clipping the two plane waves. Red and blue are the individual waves; black is the sum.

Now let’s move the original distant points of light closer together. The difference between the angles of the plane waves and thus the frequency separation of the delta functions decreases. But now after the convolution, the sinc functions’ main lobes have collided and blended together to where you can no longer tell there are two contributing peaks. You can no longer distinguish, or resolve, the sources as separate points. This is essentially the definition of resolution in an optical system. (Often the points are considered to be minimally resolved when the peak of one sinc function coincides with the first zero of the other.)

Frequencies (angles) are too close together to be resolved. (Frequencies are now 5 and 6 cycles per unit length.)

But wait! What if the aperture gets wider? This causes the sinc functions’ lobes to get narrower meaning that the light sources can be this close together while still being distinguishable. A larger aperture provides higher resolution.

The same frequencies as the previous plot, but increasing the aperture width has narrowed the sinc function, and the spots are resolvable again. (Aperture width increased from 0.5 units to 1.5 units.)

So back to the animals. A cat’s eye is tall and narrow. The vertical dimension of the pupil is larger giving the cat higher-resolution vision vertically than it has horizontally. This is useful for a cat climbing trees and looking for prey above and below. A goat or deer has the opposite orientation. Its pupil is wider than it is tall giving it higher-resolution vision horizontally, useful for a grazing animal watching for predators on open plains. And humans—and many other animals—have round pupils giving essential equivalent resolution vertically and horizontally. (Note that light exclusion also factors strongly into the function of animals’ eyes. For instance, cats often operate in the dark, so their retinas are quite sensitive. Their slit pupils can close down quite far during the day significantly reducing the amount of light getting in.)

The fundamental mathematics here—space/time vs. frequency domains, Fourier transforms, convolution—can be applied to countless problems in science and engineering. Be sure to subscribe to the blog to get new items as they come out, and leave comments and questions.

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