Post on 25-Jun-2020
C. A. Bouman: Digital Image Processing - January 7, 2020 1
Continuous Space Fourier Transform (CSFT)
Forward CSFT:
F (u, v) =
∫ ∞
−∞
∫ ∞
−∞f (x, y)e−j2π(ux+vy)dxdy
Inverse CSFT:
f (x, y) =
∫ ∞
−∞
∫ ∞
−∞F (u, v)ej2π(ux+vy)dudv
• Space coordinates:
1. Usually, x is horizontal and y is vertical coordinate
2. Usually, y points down
3. Raster order - Television scans rapidly from left to
right and more slowly from top to bottom.
• Frequency coordinates:
1. u corresponds to horizontal frequency components (ver-
tical strips).
2. v corresponds to vertical frequency components (hori-
zontal strips).
C. A. Bouman: Digital Image Processing - January 7, 2020 2
Useful Continuous Space Signal Definitions
δ(x, y)△= δ(x) δ(y)
rect(x, y)△= rect(x) rect(y)
sinc(x, y)△= sinc(x) sinc(y)
circ(x, y)△= rect(
√
x2 + y2)
• A 2-D function f (x, y) is said to be separable if it is
formed by the product of two 1-D functions.
f (x, y) = g(x)h(y)
rect(x, y), sinc(x, y), and δ(x, y) are separable functions.
• Is circ(x, y) a separable function?
C. A. Bouman: Digital Image Processing - January 7, 2020 3
CSFT Properties Inherited from CTFT
• Some properties of the CSFT are very similar to corre-
sponding CTFT properties.
Property Space Domain Function CSFT
Linearity af (x, y) + bg(x, y) aF (u, v) + bG(u, v)Conjugation f ∗(x, y) F ∗(−u,−v)Scaling f (ax, by) 1
|ab|F (u/a, v/b)
Shifting f (x− x0, y − y0) e−j2π(ux0+vy0)F (u, v)Modulation ej2π(u0x+v0y)f (x, y) F (u− u0, v − v0)Convolution f (x, y) ∗ g(x, y) F (u, v)G(u, v)Multiplication f (x, y)g(x, y) F (u, v) ∗G(u, v)Duality F (x, y) f (−u,−v)
• Inner product property∫ ∞
−∞
∫ ∞
−∞f (x, y)g∗(x, y)dx dy
=
∫ ∞
−∞
∫ ∞
−∞F (u, v)G∗(u, v)du dv
C. A. Bouman: Digital Image Processing - January 7, 2020 4
Properties Specific to CSFT
• But some properties of the CSFT are quite unique to the
2-dimensional problem.
Property Space Domain Function CSFT
Separability f (x)g(y) F (u)G(v)
Rotation f
(
A
[
xy
])
|A|−1F(
[u, v]A−1)
C. A. Bouman: Digital Image Processing - January 7, 2020 5
Separability of CSFT
F (u, v) =
∫ ∞
−∞
∫ ∞
−∞f (x, y)e−j2π(ux+vy)dxdy
=
∫ ∞
−∞
[∫ ∞
−∞f (x, y)e−j2πuxdx
]
e−j2πvydy
Define the CTFT of f (x, y) in the variable x
F̃ (u, y) =
∫ ∞
−∞f (x, y)e−j2πuxdx
Then the CSFT may be computed as the CTFT of F̃ (u, y) in
y
F (u, v) =
∫ ∞
−∞F̃ (u, y)e−j2πvydy
• Comment: 2-D CSFT can be computed as two 1-D CTFT’s.
C. A. Bouman: Digital Image Processing - January 7, 2020 6
CSFT of Separable Functions
Let
g(t)CTFT⇔ G(f )
h(t)CTFT⇔ H(f )
Then
g(x)h(y)CSFT⇔ G(u)H(v)
Proof:
F (u, v) = CSFT {g(x)h(y)}=
∫ ∞
−∞
∫ ∞
−∞g(x)h(y) e−j2π(ux+vy)dxdy
=
∫ ∞
−∞
∫ ∞
−∞g(x)h(y) e−j2πuxe−j2πvydxdy
=
[∫ ∞
−∞g(x)e−j2πuxdx
] [∫ ∞
−∞h(y)e−j2πvydy
]
= G(u)H(v)
C. A. Bouman: Digital Image Processing - January 7, 2020 7
Useful CSFT Transform Pairs
• 2-D delta function:
CSFT {δ(x, y)} = CSFT {δ(x)δ(y)}= CTFT {δ(x)} · CTFT {δ(y)}= 1 · 1 = 1
−4
−2
0
2
4
−4
−2
0
2
40
0.2
0.4
0.6
0.8
1
X Axis
delta(x,y)
Y Axis
⇒ 1
• 2-D rect function:
CSFT {rect(x, y)} = CSFT {rect(x)rect(y)}= CTFT {rect(x)} · CTFT {rect(y)}= sinc(u) sinc(v)
= sinc(u, v)
−4
−2
0
2
4
−4
−2
0
2
40
0.2
0.4
0.6
0.8
1
X Axis
rect(x,y)
Y Axis
⇒
−4
−2
0
2
4
−4
−2
0
2
4−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
X Axis
sinc(fx,fy)
Y Axis
C. A. Bouman: Digital Image Processing - January 7, 2020 8
Rotated Functions
• Let the matrix A be an orthonormal rotation of angle θ
A =
[
cos(θ) sin(θ)− sin(θ) cos(θ)
]
• Because A is an orthonormal transform
|A| = 1
A−1 = A
t
• Then the CSFT of the function g
(
A
[
xy
])
is given by
CSFT
{
g
(
A
[
xy
])}
= |A|−1G(
[u, v]A−1)
= |A|−1G(
[u, v]At)
= G
(
A
[
uv
])
• So we have
g
(
A
[
xy
])
CSFT⇔ G
(
A
[
uv
])
C. A. Bouman: Digital Image Processing - January 7, 2020 9
Rotated Rect Function
• Rotated 2-D rect function:
rect
(
y + x√2,y − x√
2
)
= rect
(
A
[
xy
])
where
A =
[
1√2
1√2
−1√2
1√2
]
• A is a 45◦ rotation, so it is and orthonormal transform
CSFT
{
rect
(
y + x√2,y − x√
2
)}
= CSFT
{
rect
(
A
[
xy
])}
= sinc
(
A
[
uv
])
= sinc
(
v + u√2,v − u√
2
)
C. A. Bouman: Digital Image Processing - January 7, 2020 10
Rotated 2-D Rect and Sinc Transform Pairs
• Mesh plot
−4
−2
0
2
4
−4
−2
0
2
40
0.2
0.4
0.6
0.8
1
X Axis
rect((y+x)/sqrt(2),(y−x)/sqrt(2))
Y Axis
⇒
−4
−2
0
2
4
−4
−2
0
2
4−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
X Axis
sinc(fx,fy)
Y Axis
• Contour plot
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4rect((y+x)/sqrt(2),(y−x)/sqrt(2))
X Axis
Y A
xis ⇒
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4sinc(fx,fy)
X Axis
Y A
xis
C. A. Bouman: Digital Image Processing - January 7, 2020 11
More Useful CSFT Transform Pairs
• Circ function:
CSFT {circ(x, y)} = jinc(u, v)
where
jinc(u, v) =J1
(
π√u2 + v2
)
2√u2 + v2
and J1(r) is the Bessel function of the first kind order 1.
−4
−2
0
2
4
−4
−2
0
2
40
0.2
0.4
0.6
0.8
1
X Axis
circ(x,y)
Y Axis
⇒
−4
−2
0
2
4
−4
−2
0
2
4−0.2
0
0.2
0.4
0.6
0.8
fx Axis
jinc(fx,fy)
fy Axis
• Notice that both functions are circularly symmetric
C. A. Bouman: Digital Image Processing - January 7, 2020 12
CSFT of a Plane Wave
• Consider an impulse in the 2-D frequency domain.
F (u, v) = δ(u− uo, v − vo)
• Its inverse transform is a 2-D plane wave.
f (x, y) =
∫ ∞
−∞
∫ ∞
−∞δ(u− uo, v − vo)e
j2π(ux+vy)dudv
= ej2π(uox+voy)
• We know that
cos(2π(uox + voy)) =1
2
[
ej2π(uox+voy) + e−j2π(uox+voy)]
• So we have that
cos(2π(uox + voy))CSFT⇔ 1
2[δ(u− uo, v − vo) + δ(u + uo, v + vo)]
C. A. Bouman: Digital Image Processing - January 7, 2020 13
2-D Plane Wave Example
• Example transform pair computed with Matlab 1
x axis
y ax
is
Cosine: U0=2 and V
0=−4
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
u axis
v ax
is
Frequency Response
−10 −5 0 5
−10
−8
−6
−4
−2
0
2
4
6
8
• Graphical representation of space-frequency domain
1/Vo
1/Uo
Po
Plane Wave in Space Domain Impulse in Frequency Domain
φ
φ
Vo
Uo
1/Po
– 1/P0 =√
V 20 + U 2
0
– Rotations in space and frequency
domains are the same.
1f(x, y) = cos (u0 x+ v0 y) + 0.5
C. A. Bouman: Digital Image Processing - January 7, 2020 14
More Examples 2-D Plane Waves
x axis
y ax
is
Cosine: U0=3 and V
0=1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
u axis
v ax
is
Frequency Response
−10 −5 0 5
−10
−8
−6
−4
−2
0
2
4
6
8
x axis
y ax
is
Cosine: U0=1 and V
0=2
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
u axis
v ax
is
Frequency Response
−10 −5 0 5
−10
−8
−6
−4
−2
0
2
4
6
8
C. A. Bouman: Digital Image Processing - January 7, 2020 15
More Examples 2-D Plane Waves
x axis
y ax
is
Cosine: U0=0.5 and V
0=1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
u axis
v ax
is
Frequency Response
−10 −5 0 5
−10
−8
−6
−4
−2
0
2
4
6
8
x axis
y ax
is
Cosine: U0=2 and V
0=4
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
u axis
v ax
is
Frequency Response
−10 −5 0 5
−10
−8
−6
−4
−2
0
2
4
6
8