in the y direction. To convert the continuous function f(x,y) to a
sampled version f(mAx,nAy), multiply f(x,y) with a grid of narrow unit
pulses at intervals of Ax and Ay. This grid of narrow unit pulses is
defined as
s(x,y) = Z
m_- n=-~
6 (x-m x,y-n y)
(3.8)
and the sampled image is
fs(mAx,nAy) = f(x,y) s(x,y).
(3.9)
The sampled version is the product of the continuous image and the
sampling function s(x,y). The spectrum of the sampled version can be
determined using the convolution theorem (equation 2.8).
Fs(u,v)= F(u,v) S(u,v)
where
and
S(u,v)=
(3.10)
F(u,v) is the Fourier transform of f(x,y)
S(u,v) is the Fourier transform of s(x,y)
6 (u-mAu,v-nAv)
where u = /Ax and v = / Ay
Thus Fs(u,v) = ff F(u-uo,v-Vo) Z 6 (uo-mAu,vo-nAv) duo dvo
m= co n= oo
(3.11)
Upon changing the order of summation and integration and invoking the
sampling property of the delta function (equation 2.17), this becomes
0O 00
F(u,v) = E Z F(u-mAu,v-nAv).
m= -o n= .
(3.12)