Digital image warping

Property

Lineafity

Spatial Scaling

Frequency Scaling

Spatial Shifting

Frequency Shifting

(Modulation)

Convolution

Multiplication

Spatial Domain, f (x)

(Zlf 1 (X)+Ov2f:(x)

f(ax)

f (x-a)

g(x)=f (x)* h(x)

g(x)=f (x)h(x)

Frequency Domain, F (u)

oqFl(u)+o2F2(u)

al-F()

F (u) e -i2,,,,

F(u-a)

G (u) = F (u) H (u)

G (u) = F (u) * H (u)

Table 2.3: Fourier transform properties.

The Fourier transform can be easily extended to multidimensional signals and sys-

tems. For 2-D images f (x,y) that are integrable, the following Fourier transform pair

exists:

F(u,v) : f f f (x,y)e-i2n(ta+vY' dxdy (2.1.21)

f(x,y) = f f V(u,v)e+i2(m+Y) dudv (2.1.22)

where u and v are frequency vaxiables. Extensions to higher dimensions are possible by

simply adding exponent terms to the complex exponential, and integrating over the addi-

tional space and frequency vaxiables.

2.1.5.3. Discrete Fourier Transforms

The discussion thus far has focused on continuous signals. In practice, we deal with

discrete images that are both limited i_n._extent _and sampl a_[is9rete points. The results

d'evetoped-sfrstee mdified to beeful in this domain. We thus come to define

the discrete Fourier tradofln pair:

F (u ) = ' x=O f '.x ) e  2ux/l (2.1.23)

f (x) =  F (u)e 12r"xl (2.1.24)

u=0

in front of the forward transform serves to normalize the spectrum with respect to the

length of the input. There is no strict rule which requires the normalization to be applied

to F(u). In some sources, the lIN factor appears in front of the inverse transform

instead. For reasons of symmetry, other common formulations have the forward and