Digital image warping

f (x) IF(u)[

(a) (b)

VII 

.5 L5 2

(c)

Figure 2.6: Fourier transform. (a) aperiodic signal; (b) spectrum; (c) partial sums.

have continuously varying Fourier components and are described by a Fourier integral.

There am several other symmetries that apply between the spatial and frequency

domains. Table 2.2 lists just a few of them. They refer to functions being real, ima-

ginary, even, and odd. A function is real ifJhe imaginary component is set to zero.

Simlariy, a function is imaginary if its real component is zero. A function f (x) is even

iff(-x):f(x). Iff(-x)=-f(x), then f(x) is said to be an odd function. Finally,

F*(u) refers to the complex conjugate of F(u). That is, if F(u)=R(u)+il(u), then

F* (u)=R(u)-il(u).

Spatial Domain, f (x)

Real

Imaginary

Odd

Real and Odd

Imaginary and Even

Imaginary and Odd

Periodic

Periodic Sampling

Frequency Domain, F (u)

F(-u) =F*(u)

F(-u) = -F*(u)

Even

Real and Even

Imaginary and Odd

Imaginary and Even

Real and Odd

Discrete

Periodic Copies

Table 2.2: Symmetries between the spatial and frequency domains.

The last two symmetries listed above are particularly notable. By means of the

Fourier series, we have already seen periodic signals produce discrete Fourier spectra

(Fig. 2.5). We shall see later that periodic spectra correspond to sampled signals in the

spatial domain. The significance of this symmetry will become apparent when we dis-

cuss discrete Fourier transforms and sampling theory.

In addition to the symmetries given above, there are many properties that apply to

the Fourier transform. Some of them are listed in Table 2.3. Among the most important

of these properties is linearity because it reflects the applicability of the Fourier transform

to linear system analysis. Spatial scaling is of significance, particularly in the context of

simple warps that consist only of scale changes. This property establishes a reciprocal

relationship between the spatial domain and the frequency domain. Therefore, expansion

(compression) in the spatial domain corresponds to compression (expansion) in the fre-

quency domain. Furthermore, the frequency scale not only contracts (expands), but the

amplitude increases (decreases) vertically in such a way as to keep the area constant.

Finally, the property that establishes a correspondence between convolution in one

domain and multiplication in the other is of gmat practical significance. The proofs of

these properties are left as an exercise for the reader. A detailed exposition can be found

in [Bracewell 86], [Brigham 88], and most signal processing textbooks.