Transcript Document
Digital Image Processing
Chapter 5: Image Restoration
A Model of the Image
Degradation/Restoration Process
Degradation
Degradation function H
Additive noise ( x, y )
Spatial domain
g ( x, y) h( x, y) * f ( x, y) ( x, y)
Frequency domain
G(u, v) H (u, v) F (u, v) N (u, v)
Restoration
g ( x, y) Restoration Filter fˆ ( x, y)
Noise Models
Sources of noise
White noise
Image acquisition, digitization,
transmission
The Fourier spectrum of noise is
constant
Assuming
Noise is independent of spatial
coordinates
Noise is uncorrelated with respect to
the image itself
Gaussian noise
The PDF of a Gaussian random variable,
z,
1
( z ) 2 / 2 2
p( z )
e
2
Mean:
Standard deviation:
Variance: 2
70% of its values will be in the range
( ), ( )
95% of its values will be in the range
( 2 ), ( 2 )
Rayleigh noise
The PDF of Rayleigh noise,
2
( z a ) 2 / b
( z a )e
p( z ) b
0
Mean:
a b/4
Variance:
b( 4 )
4
2
for z a
for z a
Erlang (Gamma) noise
The PDF of Erlang noise,
is a positive integer,
a b z b1 a z
e
p( z ) (b 1)!
0
b
Mean:
a
b
2
Variance:
2
a
a0
,
for z 0
for z 0
b
Exponential noise
The PDF of exponential noise,
ae a z
p( z )
0
Mean:
1
a
Variance:
1
2
a
2
a0
for z 0
for z 0
,
Uniform noise
The PDF of uniform noise,
1
if a z b
p( z ) b a
otherwise
0
ab
Mean:
2
2
(
b
a
)
2
Variance:
12
Impulse (salt-and-pepper) noise
The PDF of (bipolar) impulse noise,
Pa
p ( z ) Pb
0
for z a
for z b
ot herwise
b a : gray-level b
will appear as a
light dot, while level a will appear like
a dark dot
Unipolar: either Pa or Pb is zero
Usually, for an 8-bit image,
(black) and b =0 (white)
a =0
Modeling
Gaussian
Rayleigh
Electronic circuit noise, sensor noise due
to poor illumination and/or high
temperature
Range imaging
Exponential and gamma
Laser imaging
Impulse
Quick transients, such as faulty switching
Uniform
Least descriptive
Basis for numerous random number
generators
Periodic noise
Arises typically from electrical or
electromechanical interference
Reduced significantly via frequency
domain filtering
Estimation of noise parameters
Inspection of the Fourier spectrum
Small patches of reasonably constant
gray level
For example, 150*20 vertical strips
Calculate ,
, a , b from
zi p( zi )
zi S
( zi ) p( zi )
2
2
zi S
Restoration in the Presence of Noise
Only-Spatial Filtering
Degradation
Spatial domain
g ( x, y) f ( x, y) ( x, y)
Frequency domain
G(u, v) F (u, v) N (u, v)
Mean filters
Arithmetic mean filter
ˆf ( x, y) 1
g ( s, t )
m n ( s ,t )S xy
Geometric mean filter
fˆ ( x, y ) g ( s, t )
( s ,t )S xy
1
mn
Harmonic mean filter
Works well for salt noise, but fails fpr
pepper noise
fˆ ( x, y )
mn
( s ,t )S xy
1
g ( s, t )
Contraharmonic mean filter
Q0
Q0
: eliminates pepper noise
: eliminates salt noise
fˆ ( x, y )
Q 1
g
(
s
,
t
)
( s ,t )S xy
g ( s, t )
( s ,t )S xy
Q
Usage
Arithmetic and geometric mean filters:
suited for Gaussian or uniform noise
Contraharmonic filters: suited for
impulse noise
Order-statistics filters
Median filter
Effective in the presence of both bipolar
and unipolar impulse noise
fˆ ( x, y) median{g (s, t )}
( s ,t )S xy
Max and min filters
max filters reduce pepper noise
fˆ ( x, y) max {g (s, t )}
( s ,t )S xy
min filters salt noise
fˆ ( x, y) min {g (s, t )}
( s ,t )S xy
Midpoint filter
Works best for randomly distributed noise,
like Gaussian or uniform noise
ˆf ( x, y) 1 max {g ( s, t )} min {g ( s, t )}
( s ,t )S xy
2 ( s ,t )S xy
Alpha-trimmed mean filter
Delete the d/2 lowest and the d/2 highest
gray-level values
Useful in situations involving multiple
types of noise, such as a combination of
salt-and-pepper and Gaussian noise
fˆ ( x, y)
1
g r (s, t )
mn d ( s ,t )S xy
Adaptive, local noise reduction filter
If
of
If
2
is zero, return simply the value
g ( x, y )
2 L2 , return a value close to
g ( x, y )
2
2
If , return the arithmetic
L
mean value mL
ˆf ( x, y) g ( x, y) g ( x, y) m
L
2
2
L
Adaptive median filter
zmin
S xy
zmax
S xy
zmed
z xy
Smax
= minimum gray level value in
= maximum gray level value in
= median of gray levels in S xy
= gray level at coordinates ( x, y )
= maximum allowed size of S xy
Algorithm:
Level A: A1= zmed zmin
A2= zmed zmax
If A1>0 AND A2<0, Go to
level B
Else increase the window size
If window size Smax
repeat level A
Else output zmed
Level B: B1= z xy zmin
B2= z xy zmax
If B1>0 AND B2<0, output
Else output zmed
z xy
Purposes of the algorithm
Remove salt-and-pepper (impulse) noise
Provide smoothing
Reduce distortion, such as excessive
thinning or thickening of object
boundaries
Periodic Noise Reduction by Frequency
Domain Filtering
Bandreject filters
Ideal bandreject filter
W
1
if
D(u,
v)
D
0
2
W
W
H (u , v) 0 if D 0 D(u, v) D 0
2
2
W
1
if D(u, v) D 0
2
2
2 1/ 2
D(u, v) (u M / 2) (v N / 2)
Butterworth bandreject filter of order n
H (u, v)
1
D(u, v)W
1 2
2
D (u, v) D0
Gaussian bandreject filter
H (u, v) 1 e
1 D 2 ( u ,v ) D02
2 D ( u ,v )W
2
2n
Bandpass filters
Hbp (u, v) 1 Hbr (u, v)
Notch filters
Ideal notch reject filter
0 if D1 (u, v) D0 or D2 (u, v) D0
H (u, v)
otherwise
1
D (u , v) (u M / 2 u )
(v N / 2 v )
D1 (u , v) (u M / 2 u0 ) (v N / 2 v0 )
2
0
2
2
2 1/ 2
2 1/ 2
0
Butterworth notch reject filter of
order n
H (u, v)
1
D
1
D1 (u, v) D2 (u, v)
2
0
n
Gaussian notch reject filter
H (u, v) 1 e
1 D1 ( u ,v ) D2 ( u ,v )
2
2
D0
Notch pass filter
Hnp (u, v) 1 Hnr (u, v)
Optimum notch filtering
Interference noise pattern
N (u, v) H (u, v)G(u, v)
Interference noise pattern in the spatial
domain
( x, y) {H (u, v)G(u, v)}
1
Subtract from g ( x, y ) a weighted
portion of ( x, y ) to obtain an
estimate of f ( x, y)
fˆ ( x, y) g ( x, y) w( x, y) ( x, y)
Minimize the local variance of fˆ ( x, y)
The detailed steps are listed in Page
251
Result
w( x, y )
g ( x, y ) ( x, y ) g ( x, y) ( x, y)
( x, y ) ( x, y)
2
2