Digital image warping

Fourier transforms are central to the study of signal processing. They offer a

powerful set of analytical tools to analyze and proc$ss singi a0d multidimensional sig-

napd_syse_ms: The great impact thakEom'ier_tra snsfformsha_s_ha_d on sgnal proih

is dug:, in large part, to the fundamental understandiog ginned by exarmmng a sgnalTTom

 entirely different viewpoint7

We had earlier considered an arbitrary input function f (x) to be the sum of an

infinite number of impulses, each scaled and shiftdi'sish[ 'in  gr]lap ofitriagihfi-

ti /ou'i'scovered that an alternfi- u-n'lpo-sible: f (x)earr be tafffi

the sum of'hn 'ififihit'hurlr. o 'mus0idai x.' "f'hi 'rivvi(Jnt ]S'jSfiflable

bce ie 'eln}e o/5-ax linear, spae-invariant system to a complex exponential

(sinusoid) is another cmpllOx exponential of the same frequency but altered amplitude

and phase. DeLe`minrgtheptde-sand-phase-iftsfr`upd-`!s ).e.central

topic of F'ier analgi_S,.. Conversely, the act of adding these scaled and shi8'--

FtiaOiSqOdih-df'i':own a Fourier synthesis. Fourier analS,i and yfithesi'e

made possible by the Fourier transform pair:

where i =',/21-, and

F(u) = f f (x)e-i2WC dx (2.1.13)

f (x) = i F(u)e+ianUX du (2.1.14)

e i2nux = cos2gux _+ isin2gux (2.1.15)

is a succinct expression for a complex exponential at frequency u.

The definition of the Fourier transform, given in Eq. (2.1.13), is valid for any

integrable function f (x). It decomposes f (x) into a sum of complex exponentials. The

complex function F (u) specifies, for each frequency u, the amplitude and phase of each

complex exponential. F(u) is commonly known as the signal's frequency spectrum.

This should not be confused with the Fourier transform of a filter, which is called the fre-

quency response (for 1-D filters) or the modulation transfer function (for 2-D filters).

The frequency response of a filter is computed as the Fourier transform of its impulse

response.

It is important to realize that f (x) and F (u) are two different representations of the

same function. In particular, f(x) is the signal in the spatial domain and F(u) is its

counterpart in the frequency domain. One goes back and forth between these two

representations by means of the Fourier transform pair. The transformation from the fre-

quency domain back to the spatial domain is given by the inverse Fourier transform,

defined in Eq. (2.1.14).

Although f (x) may be any complex signal, we are generally interested in real func-

tions, i.e., standard color images. The Fourier transform of a real function is usually

complex. This is actually a clever encoding of the orthogonal basis set, which consists of

sine and cosine functions. Together, they specify the amplitude and phase of each fre-

quency component, i.e., a sine wave. Thus, we have F(u) defined as a complex function

of the form R (u) + i! (u), where R (u) and 1 (u) are the real and imaginary components,

respectively. The amplitude, or magnitude, of F (u) is defined as

[F (u) l = 5]R2(u) + 12(u) (2.1.16)

It is often referred to as the Fourier spectrum. This should not be mistaken with the

Fourier transform F(u), which is itself commonly known as the spectrum. In order to

avoid confusion, we shall refer to [F(u) l as the magnitude spectrum. The phase spec-

trum is given as

[I(u) ] (2.1.17)

½(u) = tan -1 [ R'J

This specifies the phase shift, or phase angle, for the complex exponential at each fre-

quency u.