Aliasing and the brain

I am a long-time subscriber of Wired magazine, which I obviously think is a fine publication otherwise I would no longer be a subscriber, but every now and then there will be some “sciency” article that is misleading. The latest one is here.

As it is now common, the article comes with a nicely produced video in which various experiments are conducted to illustrate, for the most part of the video, the phenomenon known as aliasing. What upsets me the most about this one may be its title: “The wagon wheel effect shows the limit of the human brain.” The problem with the title? The wagon wheel effect does not show any limit of the human brain! In fact, aliasing has little to do with the brain, but lots to do with how signals are processed, all of which happens before even hitting your eye. The narrator at some point alerts that in order to “see the illusion” you will need a camera!

Wikipedia has a nice entry on aliasing, or an “effect that causes different signals to become indistinguishable (or aliases of one another).” One way in which aliases are produced is through sampling.

Sampling is a very useful “trick,” which enables you to talk at your cell phone, watch a video on TV, and much more. But once signals are sampled, by a camera or other type of sampling device, information is necessarily lost. By the way, that we can reconstruct signals from samples with enough quality to please our ears and eyes is a small “miracle” of science!

How about solving the following puzzle: say that you were given samples of a signal, for example an audio signal, and that the samples collected over time are as in the following figure:

Now try to answer the following question: what continuous signal might have generated those samples? If you are scratching your head thinking that there are multiples continuous signals that could have generated those samples then you’re on the right track: all such possible solutions are aliases! From the samples, they are indistinguishable!

But perhaps one could ask what would the be “simplest” signal that could have generated such samples? The answer would vary greatly depending on what one would regard as simple but, for instance, here’s the answer that is very close to what some algorithms and circuits on your cell phone would produce. The simplest signal that interpolates those samples is the following sinusoidal signal:

But even in the realm of “simple” sinusoidal signals there would be ambiguity. For example, the two sinusoidals in the next figure equally interpolate the samples:

The red and blue curves are both sinusoidals and they both interpolate every given sample. They are aliases!

But how do you know which one really generated the samples? The answer is simple: you do not! At least not without additional information. If you are in the business of interpreting the samples, as your brain is when you look at the gorgeous water flowing from a hose through the samples of a video camera, you are forced to pick one solution among all other possible solutions. That water seems to be flowing upwards is indistinguishable from other possible explanations, that is the other aliases. By the way, having the water flow upward is the simplest explanation according to the brain and many other signal processing devices that one could build to reconstruct such signal!

That the brain does that is not a limitation, but rather a prowess, given that the problem it is trying to solve comes already with its own limitations and multiple possible answers! Compare that with one alternative, which is that of been confused and not able to come to a clear answer, or shall we say a blurry answer. The brain does that all the time too, and for the most part we are satisfied. For example, when you look directly at a spinning wheel wagon and cannot decide where the spokes are and settle for a blur instead of a clear image. Or when you listen to your favorite musical instrument and hear a tone instead of the rhythmic beating of a drum.

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