Aliasing in Movies: Levitating Helicopters and Wheels Rotating Contrary to Motion

by Barry Van Veen

Movies consist of individual images that are recorded or played back at a certain rate. Thus each pixel in the movie is sampled.  A common frame rate for movies is 30 frames per second, which means that each pixel is sampled at 30 Hz. This is sufficiently fast that humans do not perceive flicker. The playback appears smooth, as if we are observing the scene in the real, physical world.  However, this sampling can result in aliasing.

It is impractical to apply a continuous-time low pass filter at each pixel to prevent aliasing when recording video. Thus, you sometimes see unexpected effects in video that are associated with aliasing. If you have watched old western movies, you have probably seen stagecoach wheels rotating "backwards" relative to the direction of movement.  Similar effects are sometimes seen with spoked wheels on cars.

In a previous blog post you listened to the effect of aliasing in a musical piece.   This post features a video depicting aliasing of helicopter blades.  View it for yourself:

You know that helicopters don't levitate.  Why do the blades appear stationary, and how is this related to the video sampling frequency?

Consider the time course of intensity for a pixel within the sweep of the blades.  It might look something like illustrated in Figure 1.  This illustration assumes that the sky is lighter than the blade and thus that the intensity decreases suddenly when a blade passes through the pixel. It also assumes that the rotor is turning at a constant 360 revolutions per minute and that there are five blades.  This corresponds to a blade passing through the pixel 30 times per second.

Sampling this pixel at 30 Hz results in two different situations.  First suppose a pixel is in the sky region at the start of sampling.  The synchrony between the blade-passing frequency and sampling rate results in that pixel always being sky.  This scenario is illustrated in Figure 2.

Second, suppose a pixel is in the blade region at the start of sampling. Again, synchrony between the blade-passing frequency and sampling rate results in this pixel always being blade as illustrated in Figure 3.

Hence, if the sampling frequency is identical to the blade-passing frequency, then the helicopter blades appear stationary in the movie.  Sky pixels always remain sky pixels, and blade pixels always remain blade pixels.  This property holds anytime the ratio of blade-passing frequency to sampling frequency is an integer.  If the ratio is close to, but not exactly an integer, then the blades will appear to rotate one way or the other, just more slowly than their actual rate.

This phenomenon can also be explained in terms of sinusoids and the frequency domain. Any particular pixel intensity within the sweep of the helicopter blades is a periodic signal.  Periodic continuous-time signals have Fourier series representations. We may express the pixel intensity $x(t)$ using the trigonometric form of the Fourier series as

$x(t) = \sum_{k=0}^\infty A_k \cos(2\pi k f_0 t+ \phi_k)$

where $f_0$ is the blade-passing frequency, that is, the fundamental frequency of the pixel intensity illustrated in Figure 1. All of the sinusoids involved in representation of $x(t)$ have frequencies that are integer multiples of the fundamental frequency $f_0$. For the example of Figure 1 we have $f_0 = 30$ Hz.

Now suppose we sample $x(t)$ at $f_s$ Hz. The sampled signal is expressed as $x[n] = x(n/f_s)$, so

$x[n] = \sum_{k=0}^\infty A_k \cos(2\pi knf_0/f_s+ \phi_k)$

The sampling theorem states that to avoid aliasing we must sample at a rate greater than twice the highest frequency in $x(t)$.  While the highest frequency depends on the $A_k$, at a minimum we require $f_s>2f_0$.  If the ratio of blade-passing frequency $f_0$ to sampling frequency $f_s$ is an integer, say $f_0/f_s = m$, then the sampling theorem is clearly being violated.  In this case we have

$x[n] = \sum_{k=0}^\infty A_k \cos(2\pi knm+ \phi_k)$

Now $\cos(2\pi k n m+ \phi_k) = \cos(\phi_k)$ because $knm$ is integer.  That is, a cosine repeats every $2\pi$ radians.  Thus, we see that $x[n]$ is actually a constant with value

$x[n] = \sum_{k=0}^\infty A_k \cos(\phi_k)$

This implies that $x[n] = x(0)$.  The value of the sampled pixel is forever equal to the value at time $t=0$.  A pixel that is a sky or blade pixel always remains a sky or blade pixel.

All of the sinusoids in the Fourier series representation for $x(t)$ alias to zero Hz when the ratio of fundamental frequency to sampling frequency is an integer.

Aliasing leads to some surprising effects in video containing objects rotating at constant rates as it is not practical to apply an anti-aliasing filter to each pixel.  So don't be surprised when you see a levitating helicopter or wheels rotating contrary to the direction of motion.  After all, it is a movie.