Sunday, June 28th, 2015
I've been lecturing to university students about video compression. The slides are available here, but one section in particular seems to stand out to the audience: an abstract but approachable introduction to the Fourier Transform. This mathematical operation is fundamental to video compression, but it can be a bit tricky to understand (and teach!).
So let's assume we own a bar, and it's full of rowdy patrons who are all demanding piña coladas. As it turns out, we have a large pre-made batch of the drink that's ready to serve, but it unfortunately contains a very expensive rum. We'd like to serve the drink, but without the expensive rum in it.
Fortunately, we have a magical device, called the DCT-1000, which can separate the piña colada mixture into its constituent ingredients.
Once we've separated the ingredients, we can remove the expensive rum. We can then recombine the remaining ingredients (using a regular blender!), for our patrons.
Our DCT-1000 is a fictional device, but let's take a slightly closer look at how we might perform a similar task using mathematics. In the early 1800s, French mathematician Joseph Fourier, proved the following theorem:
Using this theorem, we can decompose a continuous mathematical signal (our piña colada) into a set of cosine terms (its ingredients). The algorithm that performs this decomopsition is known as the Fourier Transform.
The math behind this operation is a bit complex (we go over it in greater detail in the lecture), but what it's fundamentally doing is separating a signal into its component frequencies.
It turns out, this is an incredibly useful process across many domains including compression, audio and image processing, encryption, artificial intelligence, and more. Anytime you need to analyze, filter, or otherwise manipulate data in the frequency domain, the Fourier Transform (or its many derivatives) are surely involved.
If you'd like a more in depth discussion of the Fourier Transform, and it's popular sibling, the Discrete Cosine Transform (DCT), check out my video compression lecture.
