Tutorial on Fourier Theory

Tutorial on Fourier Theory
Yerin Yoo
March 2001
1 Introduction: Why Fourier?
During the preparation of this tutorial, I found that almost all the textbooks on dig-
ital image processing have a section devoted to the Fourier Theory. Most of those
describe some formulas and algorithms, but one can easily be lost in seemingly
incomprehensible mathematics.
The basic idea behind all those horrible looking formulas is rather simple, even
fascinating: it is possible to form any function as a summation of a series
of sine and cosine terms of increasing frequency. In other words, any space or
time varying data can be transformed into a different domain called the frequency
space. A fellow called Joseph Fourier first came up with the idea in the 19th
century, and it was proven to be useful in various applications, mainly in signal
processing.
1.1 Frequency Space
Let us talk about this frequency space before going any further into the details.
The term frequency comes up a lot in physics, as some variation in time, describ-
ing the characteristics of some periodic motion or behavior. The term frequency
that we talk about in computer vision usually is to do with variation in brightness
or color across the image, i.e. it is a function of spatial coordinates, rather than
time. Some books even call it spatial frequency.
For example, if an image represented in frequency space has high frequencies
then it means that the image has sharp edges or details. Let’s look at figure 1 ...