## Friday, January 25, 2013

### Aliasing

In Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss aliasing.
If a component [in the Fourier spectrum] is present whose frequency is more than half the sampling frequency, it will appear in the analysis at a lower frequency. This is the familiar stroboscopic effect in which the wheels of the stagecoach appear to rotate backward because the samples (movie frames) are not made rapidly enough. In signal analysis, this is called aliasing. It can be seen in Fig. 11.15, which shows a sine wave sampled at regularly spaced intervals that are longer than half a period.
First of all, what is all this business about a stagecoach? Fifty years ago, when westerns were all the rage in movies and on TV, aliasing often occurred if the frame rate (typically 24 frames per second for old movies) was lower than the rotation rate of the wheel (if all the spokes of the wheel are equivalent, then you can take the “period of rotation” as the time it takes for one spoke to rotate to the position of the adjacent one, which may be much shorter than the time for the wheel to make one complete rotation). You can see an example of this in the John Wayne movie Winds of the Wasteland (1936), especially in the climactic scene of the stagecoach race. In this video of the movie, you can see aliasing of the stagecoach wheel briefly at time 55:40. For those of you who are more discriminating in your movie tastes, you can see another example of aliasing 14 minutes and 15 seconds into Stagecoach, a John Wayne classic from 1939 directed by John Ford. In my opinion, the greatest western is the John Ford masterpiece The Man Who Shot Liberty Valance. What more could you ask for than both John Wayne and Jimmy Stewart in the same production? You can see aliasing briefly when Stewart drives his buckboard out of Shinbone to practice his pistol shooting (without much success). Another time when you see a wheel rotate backwards in this movie does not involve aliasing; it is (Spoiler Alert!) after Stewart Wayne kills Valance (Lee Marvin), when Pompey (Woody Strode) takes the drunken Wayne to his ranch house where he backs up the buckboard (that was a joke….).

But I digress. Aliasing can happen in space as well as time, and can therefore affect images. If spatial frequencies in the structure of an object correspond to wavelengths smaller than the twice the pixel size, low spatial frequency artifacts, such as Moire patterns, can appear in the image, shown nicely in this figure. One can minimize aliasing by first filtering (anti-aliasing) before sampling. Some rather extreme cases of aliasing can been seen in Figs. 11.41 and 12.11 of Intermediate Physics for Medicine and Biology.

Stagecoach, with John Wayne.

#### 1 comment:

1. This phenomena is important in the field of imaging and has been a topic involved in some interesting discoveries/debates. Spatial and temporal sampling is critical in interpreting recorded signals. Filtering, in general, has consequences. When you know what you are looking for, it is much easier to apply appropriate filters than when you are using the signals to look for unresolved underlying mechanisms.