Alias, Schmalias

While the great line from Romeo and Juliet: “a rose by any other name would smell as sweet” rings true, would a digital rose smell as sweet?  We often think of the digital world as a mere “renaming” of the real world.  But some interesting effects emerge from digitizing, and one of them is commonly known as aliasing.

If you’ve danced in a club with strobe lights, you’ve laughed at the slow motion effect that results from your eyes interpolating between positions. Sampling data can be thought of in much the same way. As a young child I was in love with both dance and mathematics, and I recall my uncle describing to me a David Parson’s piece involving a strobe light. The piece, Caught, involves a man appearing to fly through the air as a strobe light flashes on at just the right moments in a progression of jumps.

While I certainly didn’t think of it this way at the time, the choreographer Parsons was sampling his dancing at the same rate that he jumped. In this way he could appear to be at the same height above the ground (i.e. floating/flying) at every moment. If we think of Parsons (who it doesn’t hurt to think of since he was pretty gorgeous) moving up and down over a sine wave as he jumps, then we are simply sampling at a rate of exactly once per period and essentially leaving out the information that he ever touches the ground. So what made me think of all this again?

Mathalicious’s blog! While Mathalicious’s lessons require a subscription to view, the blog (and a few sample lessons) are free, and the most recent lesson entitled “Spinning your Wheels” is about aliasing, the same phenomenon mentioned above, in which we sample less frequently than is necessary to faithfully reproduce the information (movement) that is occurring. Mathalicious explains the “Wagon Wheel effect” which results in a car’s wheels appearing to stop moving or to spin backwards even as the car (or wagon) is moving forwards.

Chris Lusto's illustration of how a wheel that is spinning can appear still if it's movement is captured at a rate that coincides with it's turning through an angle of rotational symmetry.

Chris Lusto’s illustration of how a wheel that is spinning at a rate of 72 degrees per frame may appear still if consecutive frames are as shown above (where the green spoke would not be colored green in the film).  Any object that undergoes a symmetry in the time between successive frames will appear stationary.

A nice point made at the beginning of the post is that this effect is due completely to the nature of digital media in which the image is sampled. In the same way that we interpolate position when strobe lights are flashing, we interpolate the movement of the wheel as pixels on a screen change color. While a car wheel doesn’t have spokes, its hub caps often have the typical five-fold symmetry described in the blog post, and Chris (the blogger) has put in some great animation widgets to help the reader understand this phenomenon without ever mentioning Nyquist, sampling rate, or the word aliasing. While I applaud his explanation, I also think it would be worth plotting the sine wave traced out by one point on the wheel.  Then the typical definition of aliasing would be more directly connected to this wonderful example. Another great post of Chris’s on the Mathalcious blog that is in the same Digital Signal Processing vein is the Siren Song which illuminates the Doppler Effect and even addresses what happens if the siren travels at more than the speed of light!

Likewise, there are spatial examples of aliasing such as the Moire patterns that emerge from digital photographs of objects that have periodic patterns (like brick walls and textured clothes).

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Also aliasing manifests in music as what we commonly call distortion when higher pitches are aliased down to lower ones. Any analysis of a cyclic phenomenon can be colored by the affects of aliasing. I happened upon the arxiv paper “Could sampling make Hares eat Lynxes?” which discusses the potential of aliasing as an explanation for misinterpretations of cyclic behaviors of populations in the context of the Lotke-Volterra predator-Prey model. What examples of aliasing have you experienced or found interesting?

Hares high-fiving after eating a lynx? :)

Hares high-fiving after eating a lynx? 🙂

 

This entry was posted in Applied Math, K-12 Mathematics, Math Education, Mathematics and Computing, Recreational Mathematics and tagged , , , , , , , , , , . Bookmark the permalink.

One Response to Alias, Schmalias

  1. Muhamed says:

    Excellent article, like it. This is first time I am on this blog and I hope that I will read more articles like this one

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