Image Restoration / Filtering in Frequency Domain

Image Restoration

Gaussian
Nose Distributions
Uniform
Lognormal
Rayleigh
Exponential
Erlang

Spatial Filters

Example:
Adaptive Spatial Filters
g
2 2
v
n,
m)
f ( n,
m)
(
2
4000
3000
2000
1000
0
0 50 100 150 200 250
5000
4000
3000
2000
1000
0
wiener2.m
0 50 100 150 200 250

Filtering in Frequency Domain
H(u,v) - Filter Transfer Function
Fundamental Concept:
f ( x,
y)*
h(
x,
y)
H(
u,
v)
F(
u,
v)

Lowpass in Frequency Domain
v u
Ideal lowpass filter:
1,
H(
u,
v)
0,
if
if
D(
u,
v)
D
D(
u,
v)
D
Buterworth lowpass filter:
1
n(
u,
v)
1[
D(
u,
v) / D0]
H
2n
0
0
D(u,v)
Gaussian lowpass filter:
2
2
D ( u,
v) / 2
;
0
H( u,
v)
exp
D

Highpass in Frequency Domain
Main Concept:
H
HP
( u,
v)
1
H ( u,
v)
LP
Ideal
Hghipass
Buterworth
Highpass
Gaussian
Highpass

Highpass in Frequency Domain
Main Concept:
H
HP
( u,
v)
1
H ( u,
v)
LP

BandReject in Frequency Domain
Ideal Buterworth Gaussian

Restoration in Frequency Domain
Fundamental Concept:
f
( x,
y (
x,
)
g( x,
y)
H
y

Restoration in Frequency Domain
Linear, spatially invariant process :
g( x,
y)
h(
x,
y)*
f ( x,
y)
(
x,
y)
G( u,
v)
H(
u,
v)
F(
u,
v)
N(
u,
v)
Fundamental Concept (Direct Inverse Filtering):
G(
u,
v)
F'(
u,
v)
H(
u,
v)
N(
u,
v)
F'(
u,
v)
F(
u,
v)
H(
u,
v)

Restoration in Frequency Domain
g
Hf
η
Problem: to inverse H
Constrained Least Squares method
C
2
2
f ( x,
y)
smoothness minimum .
Frequency Domain Solution:
H ( u,
v)*
F' ( u,
v)
G(
u,
v)
2 2
H ( u,
v)
P(
u,
v)
0
p(
x,
y)
1
0
1
4
1
0
1
laplacin mask
0

Degraded Image
PSF Estimation
1
2
3
4
5
Original
6
7
1 2 3 4 5 6 7
0.05
0.04
0.03
0.02
0.01
0
20
15
10
5
0
0
5
10
15
20
Restored Image
OTF
OTF
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
80
60
40
20
0
0
20
40
60
80

Restoration in Frequency Domain
Wiener Filtering: minimum of statistical error function
F'(
u,
v)
1
H ( u,
v)
2
2
2
H ( u,
v)
H ( u,
v)
N(
u,
v)
2
/ F(
u,
v)
R
N(
u,
v)
F(
u,
v)
2
2
f
A
A
const.
A
1
M N
|
N(
u,
v) |
2
averagenoise power