ℱuv=∫−∞∞∫−∞∞fxye−(iux)e−(ivy)dxdy
ℱ
u
v
y
x
f
x
y
u
x
v
y
where ℱℱ is the 2D FT
and uu and
vv are frequency variables in
xu
x
u
and
yv
y
v
.

2D complex exponentials are eigenfunctions for 2D LSI systems:

∫−∞∞∫−∞∞hx−αy−βei
u0
αei
v0
βdαdβ=∫−∞∞∫−∞∞h
α
′
β
′
ei
u0
(x−
α
′
)ei
v0
(y−
β
′
)d
α′
d
β′
=ei
u0
xei
v0
y∫−∞∞∫−∞∞h
α
′
β
′
e−(i
u
0
α
′
)e−(i
v
0
β
′
)d
α
′
d
β′
β
α
h
x
α
y
β
u0
α
v0
β
β′
α′
h
α
′
β
′
u0
x
α
′
v0
y
β
′
u0
x
v0
y
β′
α
′
h
α
′
β
′
u
0
α
′
v
0
β
′

(1)
where

∫−∞∞∫−∞∞h
α′
β′
e−(i
u0
α
′
)e−(i
v0
β
′
)d
α′
d
β′
≡H
u0
v0
β′
α′
h
α′
β′
u0
α
′
v0
β
′
H
u0
v0
H
u0
v0
H
u0
v0
is the 2D Fourier transform of

hxy
h
x
y
evaluated at frequency

u0
,
v0
,
u0
v0

gxy=hxy*fxy=∫−∞∞∫−∞∞hx−αy−βfαβdαdβ
g
x
y
h
x
y
f
x
y
β
α
h
x
α
y
β
f
α
β

(2)
Guv=Huvℱuv
G
u
v
H
u
v
ℱ
u
v
gxy=12π2∫−∞∞∫−∞∞Guveiuxeivydudv
g
x
y
1
2
2
v
u
G
u
v
u
x
v
y
The above equation is the inverse 2D FT.