
2.1
Linear Operations
on
Pictures
15
Fig.
3
Sinc(*,
y).
the Fourier transform of the Dirac δ function is the limit of the sequence of
Fourier transforms s'mc(u/n, v/n). In other words, when
f(x,y) = S(x,y)
then
F(u,v) = lim s'mc(uln,v/n) =1 (13)
By multiplying both sides of Eq. (12) by exp [/2π(wa+ , integrating both
sides with respect to u and v, and making use of Eq. (7), it is easily shown that
f*
QO
Λ
oo
F(u,
v)
exp
Ijln
(u<x
+
vß)~]
du dv =
/(α,
β) (14)
J
—
00
J
—
00
or equivalently
Λ
oo
Λ
oo
f(x,y)= F(u, v) txp\_j2n(ux + vy)~\ du dv (15)
J
—
00
J
— CO
which is called the inverse Fourier transform of F(u,v). By Eqs, (12) and (15),
f(x, ...