An Introduction to the Fourier Transform

What is it about?

The Fourier Transform is a fundamental tool in many domains. In this talk, I introduce the ideas that are critical to computer graphics, and in particular the phenomenon of aliasing and its relationship to proper sampling. The course is about the ideas, so there is no math beyond a couple of simple expressions. I focus on the concepts, not the mechanics. After viewing the course, you'll be able to learn the mathematical version of the ideas if you wish - the process will be easier than it would have been otherwise, and the equations will make more sense to you.

Featured Image

Photo by Elisabeth Arnold on Unsplash

Photo by Elisabeth Arnold on Unsplash

Why is it important?

The Fourier Transform is a fundamental tool in computer graphics. Among its many applications, it explains where aliasing comes from, how to efficiently reduce it, and why noise can be a good thing. It tells us how to filter textures, blur images, and choose sampling patterns. The Fourier transform is the basis ofthe JPEG image compression method, and reveals to us why wagons wheels start to rotate backwards as they spin faster. The Fourier Transform not only describes the source of many problems in graphics, it also often tells us how to avoid or suppress them in the first place. Unfortunately, this terrific tool is unfamiliar to many people, and opaque to many others, often because they are put off by its technical language and complicated looking mathematics. Though the usual notation can be daunting at first contact, it's really just a terse way of expressing specific sequences of multiplications and additions. In this course we only assume you remember some high-school algebra. From there, we'll carefully build up to the full Discrete-Time Fourier Transform (and its inverse) that we use every day, by taking small steps and illustrating the process with pictures. By the time we're done, you'll be comfortable with the Fourier Transform, what it does, and how it works. You'll find it a helpful, everyday tool for solving today's problems, and creating tomorrow's innovations.