chapter5-2 restoration and depredations.ppt

Image processing

1. Digital Image Processing
Chapter 5: Image Restoration

2. A Model of the Image
Degradation/Restoration Process

3. Degradation
Degradation function H
Additive noise
Spatial domain
Frequency domain),(yx ),(),(*),(),( yxyxfyxhyxg  ),(),(),(),( vuNvuFvuHvuG 

4. Restoration),(
ˆ
Filtern Restoratio),( yxfyxg 

5. Noise Models
Sources of noise
Image acquisition, digitization,
transmission
White noise
The Fourier spectrum of noise is
constant
Assuming
Noise is independent of spatial
coordinates
Noise is uncorrelated with respect to
the image itself

6. Gaussian noise
The PDF of a Gaussian random variable,
z,
Mean:
Standard deviation:
Variance:22
2/)(
2
1
)(




z
ezp   2


7. 70% of its values will be in the range
95% of its values will be in the range )(),(    )2(),2(  

8. Rayleigh noise
The PDF of Rayleigh noise,
Mean:
Variance:








az
azeaz
b
zp
baz
for 0
for )(
2
)(
/)(
2 4/ ba 4
)4(
2 



b

10. Erlang (Gamma) noise
The PDF of Erlang noise, ,
is a positive integer,
Mean:
Variance:









0for 0
0for
)!1()(

1
z
ze
b
za
zp
za
bb a
b
 2
2
a
b
 0a b

11. Exponential noise
The PDF of exponential noise, ,
Mean:
Variance:






0for 0
0for
)(

z
zae
zp
za a
1
 2
21
a
 0a

12. Uniform noise
The PDF of uniform noise,
Mean:
Variance:







otherwise0
if
1
)(
bza
ab
zp 2
ba
 12
)(
2
2 ab


13. Impulse (salt-and-pepper) noise
The PDF of (bipolar) impulse noise,
 : gray-level will appear as a
light dot, while level will appear like
a dark dot
Unipolar: either or is zero







otherwise0
for
for
)( bzP
azP
zp
b
a ab b a aP bP

14. Usually, for an 8-bit image, =0
(black) and =0 (white)b a

15. Modeling
Gaussian
Electronic circuit noise, sensor noise due
to poor illumination and/or high
temperature
Rayleigh
Range imaging
Exponential and gamma
Laser imaging

16. Impulse
Quick transients, such as faulty switching
Uniform
Least descriptive
Basis for numerous random number
generators

20. Periodic noise
Arises typically from electrical or
electromechanical interference
Reduced significantly via frequency
domain filtering

22. Estimation of noise parameters
Inspection of the Fourier spectrum
Small patches of reasonably constant
gray level
For example, 150*20 vertical strips
Calculate , , , from  a b 


Sz
ii
i
zpz)( 


Sz
ii
i
zpz )()(
22


24. Restoration in the Presence of Noise
Only-Spatial Filtering
Degradation
Spatial domain
Frequency domain),(),(),( yxyxfyxg  ),(),(),( vuNvuFvuG 

25. Mean filters
Arithmetic mean filter
Geometric mean filter


xy
Sts
tsg
mn
yxf
),(
),(
1
),(
ˆ mn
Sts
xy
tsgyxf
1
),(
),(),(
ˆ











26. Harmonic mean filter
Works well for salt noise, but fails fpr
pepper noise


xySts tsg
mn
yxf
),( ),(
1
),(
ˆ

27. Contraharmonic mean filter
 : eliminates pepper noise
 : eliminates salt noise





xy
xy
Sts
Q
Sts
Q
tsg
tsg
yxf
),(
),(
1
),(
),(
),(
ˆ 0Q 0Q

30. Usage
Arithmetic and geometric mean filters:
suited for Gaussian or uniform noise
Contraharmonic filters: suited for
impulse noise

32. Order-statistics filters
Median filter
Effective in the presence of both bipolar
and unipolar impulse noise)},({median),(
ˆ
),(
tsgyxf
xySts


33. Max and min filters
max filters reduce pepper noise
min filters salt noise)},({max),(
ˆ
),(
tsgyxf
xySts
 )},({min),(
ˆ
),(
tsgyxf
xy
Sts


34. Midpoint filter
Works best for randomly distributed noise,
like Gaussian or uniform noise







)},({min)},({max
2
1
),(
ˆ
),(),(
tsgtsgyxf
xyxy
StsSts

35. 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


xy
Sts
rtsg
dmn
yxf
),(
),(
1
),(
ˆ

39. Adaptive, local noise reduction filter
If is zero, return simply the value
of
If , return a value close to
If , return the arithmetic
mean value2
 ),(yxg 22
L


 ),(yxg 22
L


 Lm  
L
L
myxgyxgyxf  ),(),(),(
ˆ
2
2




41. Adaptive median filter
 = minimum gray level value in
 = maximum gray level value in
 = median of gray levels in
 = gray level at coordinates
 = maximum allowed size ofminz maxz medz xy
z maxS xy
S xy
S xy
S xy
S ),(yx

42. Algorithm:
Level A: A1=
 A2=
 If A1>0 AND A2<0, Go to
 level B
 Else increase the window size
 If window size
 repeat level A
 Else outputminzz
med maxzz
med maxS medz

43. Level B: B1=
 B2=
 If B1>0 AND B2<0, output
 Else outputmin
zz
xy
 max
zz
xy
 xy
z medz

44. Purposes of the algorithm
Remove salt-and-pepper (impulse) noise
Provide smoothing
Reduce distortion, such as excessive
thinning or thickening of object
boundaries

46. Periodic Noise Reduction by Frequency
Domain Filtering
Bandreject filters
Ideal bandreject filter












2
Dv)D(u, if1
2
Dv)D(u,
2
D if0
2
Dv)D(u, if1
),(
0
00
0
W
WW
W
vuH  
2/1
22
)2/()2/(),( NvMuvuD 

47. Butterworth bandreject filter of order n
Gaussian bandreject filtern
DvuD
WvuD
vuH
2
2
0
2
),(
),(
1
1
),(








 2
2
0
2
),(
),(
2
1
1),(







 


WvuD
DvuD
evuH

50. Bandpass filters),(1),( vuHvuH
brbp


52. Notch filters
Ideal notch reject filter

 

otherwise1
Dv)(u,Dor Dv)(u,D if0
),(
0201
vuH  
2/1
2
0
2
01 )2/()2/(),( vNvuMuvuD   
2/1
2
0
2
02 )2/()2/(),( vNvuMuvuD 

53. Butterworth notch reject filter of
order nn
vuDvuD
D
vuH








),(),(
1
1
),(
21
2
0

54. Gaussian notch reject filter









2
0
21 ),(),(
2
1
1),(
D
vuDvuD
evuH

56. Notch pass filter),(1),( vuHvuH
nrnp


58. Optimum notch filtering

59. Interference noise pattern
Interference noise pattern in the spatial
domain
Subtract from a weighted
portion of to obtain an
estimate of ),(),(),( vuGvuHvuN  )},(),({),(
1
vuGvuHyx

 ),(),(),(),(
ˆ
yxyxwyxgyxf  ),(yxg ),(yx ),(yxf

60. Minimize the local variance of
The detailed steps are listed in Page
251
Result),(
ˆ
yxf ),(),(
),(),(),(),(
),(
22
yxyx
yxyxgyxyxg
yxw






64. Linear, Position-Invariant Degradations
Input-output relationship),()],([),( yxyxfHyxg  )],([),( yxfHyxg  0),(yx

65. H is linear if
Additivity)],([)],([
)],(),([
21
21
yxfbHyxfaH
yxbfyxafH

 )],([)],([
)],(),([
21
21
yxfHyxfH
yxfyxfH



66. Homogeneity
Position (or space) invariant)],([ )],([
11 yxfaHyxafH  )],( )],([   yxgyxfH

67. In terms of a continuous impulse
function




  ddyxfyxf ),(),(),( 










 ddyxfH
yxfHyxg
),(),(
)],([),(

68.  
 





















ddyxhf
ddyxHf
ddyxfH
yxg
),,,(),(
),(),(
),(),(
),(

69. Impulse response of H
In optics, the impulse becomes a point
of light
Point spread function (PSF)
All physical optical systems blur
(spread) a point of light to some
degree)],([),,,(   yxHyxh ),,,( yxh

70. Superposition (or Fredholm) integral of
the first kind





 ddyxhf
yxg
),,,(),(
),(

71. If H is position invariant
Convolution integral),()],([   yxhyxH 






 ddyxhf
yxg
),(),(
),(

72. In the presence of additive noise
If H is position invariant),( ),,,(),(
),(
yxddyxhf
yxg
 





 ),( ),(),(
),(
yxddyxhf
yxg
 







73. If H is position invariant
Restoration approach
Image deconvolution
Deconvolution filter),(),(),(),( yxyxfyxhyxg  ),(),(),(),( vuNvuFvuHvuG 

74. Estimating the Degradation Function
Estimation by image observation
In order to reduce the effect of noise in
our observation, we would look for
areas of strong signal content),(
ˆ
),(
),(
vuF
vuG
vuH
s
s
s


75. Estimation by experimentation
Obtain the impulse response of the
degradation by imaging an impulse
(small dot of light) using the same
system settings
Observed image
The strength of the impulseA
vuG
vuH
),(
),( ),(vuG A

77. Estimation by modeling
Hufnagel and Stanley
Physical characteristic of atmospheric
turbulence6
5
22
)(
),(
vuk
evuH



79. Image motiondttyytxxfyxg
T
])(),([),(
0
00


80. dtdydxe
tyytxxf
dydxe
dttyytxxf
dydxeyxgvuG
yvxuj
T
yvxuj
T
yvxuj

)](),([

])(),([
),(),(
) (2
0
00
) (2
0
00
) (2































81. Where),(),(
),(
),(),(
0
)]( )( [2
0
)]( )( [2
00
00
vuHvuF
dtevuF
dtevuFvuG
T
tyvtxuj
T
tyvtxuj








 


T
tyvtxuj
dtevuH
0
)]( )( [2
00
),(


82. If andTattx /)(
0 0)(
0ty uaj
T
Tatuj
T
txuj
eua
ua
T
dte
dtevuH

0
]/ [2
0
)]( [2
) sin(



),(
0














83. If andTattx /)(
0 Tbtty /)(
0 )(
)]( sin[
)(
),(
vbuaj
evbua
vbua
T
vuH








85. Inverse Filtering
Direct inverse filtering
Limiting the analysis to frequencies
near the origin),(
),(
),(
),(
),(
),(
ˆ
vuH
vuN
vuF
vuH
vuG
vuF



87. Minimum Mean Square Error (Wiener)
Filtering
Minimize
Terms
 = degradation function

 = complex conjugate of

 =
})
ˆ
{(
22
ffEe  ),(vuH ),(vuH ),(vuH 2
),(vuH ),(),( vuHvuH

88.  = power spectrum
of the noise
 = power spectrum
of the undegraded image2
),(),( vuNvuS 
 2
),(),( vuFvuS
f


89. Wiener filter),(
),(/),(),(
),(
),(
1
),(
),(/),(),(
),(*
),(
),(),(),(
),(),(*
),(
ˆ
2
2
2
2
vuG
vuSvuSvuH
vuH
vuH
vuG
vuSvuSvuH
vuH
vuG
vuSvuHvuS
vuSvuH
vuF
f
f
f
f


































90. White noise),(
),(
),(
),(
1
),(
ˆ
2
2
vuG
KvuH
vuH
vuH
vuF











93. Constrained Least Squares Filtering
Vector-matrix form

 , , :
 :g ),(),(*),(),( yxyxfyxhyxg  1MN MNMN H η f ηHfg 

94. Minimize
Subject to 





1
0
1
0
2
2
),(
M
x
N
y
yxfC 2
2
ˆ
ηfHg 

95. The solution
Where is the Fourier transform
of the function),(
),(),(
),(*
),(
ˆ
22
vuG
vuPvuH
vuH
vuF










 ),(vuP 













010
141
010
),(yxP

97. Computing by iteration
Adjust so that fHgr
ˆ
  a
22
ηr

98. Computation





1
0
1
0
2
2
),(
M
x
N
y
yxrr  
2
1
0
1
0
2
),(
1






M
x
N
y
myx
MN
  





1
0
1
0
),(
1
M
x
N
y
yx
MN
m 
 ][
22
2

mMN η

99. Algorithm
1: Specify an initial value of
2: Compute
3: Stop if is satisfied;
otherwise return to Step 2 after
increasing if or
decreasing if .a
22
ηr   a
22
ηr a
22
ηr

101. Geometric Mean FIlter),(
),(
),(
),(
),(*
),(
),(*
),(
ˆ
1
2
2
vuG
vuS
vuS
vuH
vuH
vuH
vuH
vuF
f






































102. Geometric Transformations
Spatial transformations
Tiepoints),(' yxrx ),(' yxsy

104. Bilinear equations4321),(' cxycycxcyxrx  8765),(' cxycycxcyxsy 

105. Gray-level interpolationdycxbyaxyxv  '''')','(