Images and Signals

When I first started learning more about graphics theory in my introduction to graphics class, I was confused beyond belief. I had a vague understanding of what Fourier transforms were but that was through my math class where I only saw the theoretical results. I could not wrap my head around how this had anything to do with graphics. Here’s my attempt at explain some of the confusing parts though this page likely does it more justice (and it has good pictures! You simply cannot learn without pictures).

As with most new ideas, staring at it long enough will crack it open. I blame my inability to grasp the material on my lack of mathematical familiarity with Fourier transforms. Basically, all I learned in this area was that there was a large class of functions (including all bounded, continuous ones) that can be represented as a Fourier series. What does this mean? It means that for any continuous function, you can find a linear combination of and of with different frequencies. Actually finding this linear combination amounts to performing a Fourier transform.

Still, this is probably a lot of mathematical jargon that still may make no sense. The take-away is the following:

Any function can be decomposed into its low and high frequency components.

This is referred to as its spectral domain.

There are a lot of other reasons we study Fourier series because there are actually a lot of basis functions we could have chosen. What makes ’s and ’s so special is that it is translation invariant. Suppose you’re given a sinusoidal signal, . Translational invariance means we can represent as a linear combination of and . We do not need to introduce ’s and ’s with different phases.

Now to actual images. Actually, instead, let’s simplify it a lot. We have no color, just grayscale and our image is 1D instead of 2D. Let’s say our image is just some curve, . Note that is not necessarily an integer. can be any real number. To draw the actual image on the screen, we will just sample at integer values (for now). We can get the Fourier transform of which is another function, that tells us how much of that frequency contributes to the image.

This is all just scratching the surface and leaves out so many details that make all of the theory quite amazing. I know that when I was learning this stuff, I wish someone could’ve have explained these parts to me. Maybe this is of no help to you. It’s probably one of those things you have to pick through on your own and develop your own abstractions to really get it.