PRELIMINARIES 2.1 FUNDAMENTALS 21

In visual data, that same transition corresponds to the spectrum between blurred imagery

and images rich in visual detail. Note that high frequencies refer to wild intensity excur-

sions. This tends to correspond to visual detail like edges and texture in high contrast

images. High frequencies that are subjectively determined to add nothing to the informa-

tion content of the signal are usually referred to as noise. Since blurred images have

slowly varying intensity functions, they lack significant high frequency information. In

either case, music and images are time- and spatially-varying functions whose informa-

tion content is embedded in their frequency spectrum. The conversion between the spa-

tial and frequency domains is achieved by means of the Fourier transform.

We are familiar with other instances in which mathematical transforms are used to

simplify a solution to a problem. The logarithm is one instance of such a transform. It

simplifies problems requiring products and quotients by substituting addition for multi-

plication and subtraction for division. The only tradeoff is the accuracy and time neces-

sary to convert the operands into logarithms and then back again. Similar benefits and

drawbacks apply to the Fourier transform, a method introduced by the French physicist

Joseph Fourier nearly two centuries ago. He derived the method to transform signals

between the spatial (or time) domain and the frequency domain. As we shall see later,

using two representations for a signal is useful because some operations that are difficult

to execute in one domain are relatively easy to do in the other domain. In this manner,

the benefits of beth representations are exploited.