Tutorial on Fourier Theory University of Otago（傅里叶理论奥塔哥大学教程）.pdf

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 fr equency. In other words, any space or time varying data can be transformed into a different domain called thefr equency 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 fr equency 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 spatialfr equency. 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, which shows frequency graphs of 4 different images. If you have trouble inter- preting the frequency graphs on the top low; The low frequency terms are on the 1 Figure 1: Images in the spatial domain are in the middle row, and their frequency space are shown on the top row. The bottom row shows t