ℱuv=∫−∞∞∫−∞∞fxye−(iux)e−(ivy)dxdy
ℱ
u
v
y
x
f
x
y
u
x
v
y
where ℱℱis 2DFT, u and
v is freq 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
β
α
h
x
α
y
β
u0
α
v0
β
change of variables
α′
=x−α
α′
x
α
,
β′
=y−β
β′
y
β
∫−∞∞∫−∞∞hx−αy−βei
u0
αei
v0
βdαdβ=∫−∞∞∫−∞∞h
α′
β′
ei
u0
(x−
α′
)ei
v0
(y−
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′
where
∫−∞∞∫−∞∞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
u0
α′
)e−(i
v0
β′
)d
α′
d
β′
β′
α′
h
α′
β′
u0
x
α′
v0
y
β′
u0
x
v0
y
β′
α′
h
α′
β′
u0
α′
v0
β′
and
∫−∞∞∫−∞∞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 2D Fourier transform of
hxy
h
x
y
evaluated at freq
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
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 inverse 2DFT