![Page 1: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/1.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
1
Digital Image Processing
Image Restoration and Reconstruction
1
• Image Degradation Model (Linear/Additive)
, , , ,
, , , ,
g x y f x y h x y x y
G u v F u v H u v N u v
![Page 2: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/2.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
2
Digital Image Processing
Image Restoration and Reconstruction
2
• Source of noise
– Objects Impurities
– Image acquisition (digitization)
– Image transmission
• Spatial properties of noise
– Statistical behavior of the gray-level values of pixels
– Noise parameters, correlation with the image
• Frequency properties of noise
– Fourier spectrum • White noise (a constant Fourier spectrum)
![Page 3: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/3.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
3
Digital Image Processing
Image Restoration and Reconstruction
3
• Some Noises Distribution:
![Page 4: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/4.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
4
Digital Image Processing
Image Restoration and Reconstruction
4
• Test Pattern
– Histogram has three spikes (Impulse)
– For two independent random variables (x, y)
Z X Yz x y p z p x p y
![Page 5: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/5.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
5
Digital Image Processing
Image Restoration and Reconstruction
5
• Noisy Images(1):
![Page 6: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/6.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
6
Digital Image Processing
Image Restoration and Reconstruction
6
• Noisy Images (2):
![Page 7: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/7.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
7
Digital Image Processing
Image Restoration and Reconstruction
7
• Periodic1 Noise:
– Electronic Devices
1 Disturbance
![Page 8: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/8.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
8
Digital Image Processing
Image Restoration and Reconstruction
8
• Estimation of Noise Parameters
– Periodic noise • Observe the frequency spectrum
– Random noise with PDFs • Case 1: Imaging system is available
– Capture image of “flat” environment
• Case 2: Single noisy image is available
– Take a strip from constant area
– Draw the histogram and observe it
– Estimate PDF parameters or measure the mean and variance
![Page 9: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/9.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
9
Digital Image Processing
Image Restoration and Reconstruction
9
• Noise Estimation:
– Shape: Histogram of a subimage (Background)
![Page 10: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/10.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
10
Digital Image Processing
Image Restoration and Reconstruction
10
• Noise Estimation: – Parameters:
– Momentum
– Power Spectrum Estimation:
1
1
Maximum Likelihood me argtho : ad m x ,SN
N
i iii
z p z
Solve ?
22PDF Parameters
i
i
i i
z S
i i
z S
z p z
z p z
2 22
1
1, , ,
, : Flat (No object) Subimages
N
i
i
i
N u v E x y x yN
x y
![Page 11: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/11.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
11
Digital Image Processing
Image Restoration and Reconstruction
11
• Noise-only spatial filter:
• Mean Filters:
– A m×n mask centered at (x, y): Sxy
– An algebraic operation on the mask pixels!
, , , , , ,g x y f x y x y G u v F u v N u v
![Page 12: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/12.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
12
Digital Image Processing
Image Restoration and Reconstruction
12
• Mean Filter: – Arithmetic mean:
• Noise Reduction (uncorrelated, zero mean) vs. Blurring
– Geometric mean:
• Same smoothing with less detail degradation:
( , )
1ˆ , ,xys t S
f x y g s tmn
1
( , )
ˆ , ,xy
mn
s t S
f x y g s t
![Page 13: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/13.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
13
Digital Image Processing
Image Restoration and Reconstruction
13
• Mean Filter:
– Harmonic mean:
• Salt Reduction (Pepper will increase)/ Good for Gaussian like
– Contra-harmonic mean:
• Q>0 for pepper and Q<0 for salt noise
( , )
ˆ ,1
,xys t S
mnf x y
g s t
1
( , )
( , )
,
ˆ ,,
xy
xy
Q
s t S
Q
s t S
g s t
f x yg s t
![Page 14: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/14.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
14
Digital Image Processing
Image Restoration and Reconstruction
14
Arithmetic Geometric
Noisy
More-Less Blurring
• Example (1):
– Additive Noise
– Mean • Arithmetic
• Geometric
Original
![Page 15: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/15.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
15
Digital Image Processing
Image Restoration and Reconstruction
15 Q=+1.5
• Example (2):
– Impulsive
– Contra-harmonic
– Correct use
Noisy (Pepper) Noisy (Salt)
Q=-1.5
![Page 16: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/16.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
16
Digital Image Processing
Image Restoration and Reconstruction
16 Q=-1.5
• Example (3):
– Impulsive Noise
– Conrtra-harmonic
– Wrong use
Q=+1.5
![Page 17: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/17.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
17
Digital Image Processing
Image Restoration and Reconstruction
17
• Noise-only spatial filter:
• Mean Filters:
– A m×n mask centered at (x, y): Sxy
– An Ordering/Ranking operation on mask members!
, , ,g x y f x y x y
![Page 18: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/18.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
18
Digital Image Processing
Image Restoration and Reconstruction
18
• Median: Most Familiar OS filter:
– Little Blurring/Single/Double valued noises
• Max/min:
– Find Brightest/Darkest Points (Pepper/Salt Reducer)
( , )
ˆ , ,xys t S
f x y median g s t
( , )( , )
ˆ ˆ, , , , min ,xyxy s t Ss t S
f x y Max g s t f x y g s t
![Page 19: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/19.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
19
Digital Image Processing
Image Restoration and Reconstruction
19
• Midpoint:
– OS+Mean Properties (Gaussian, Uniform)
( , )( , )
1ˆ , , min ,2 xyxy s t Ss t S
f x y Max g s t g s t
![Page 20: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/20.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
20
Digital Image Processing
Image Restoration and Reconstruction
20
• Alpha Trimmed mean:
– Delete (d/2) lowest and highest samples
– d = 0 → Mean Filter
– d = mn-1 → Median Filter
– Others: Combinational Properties
( , )
1ˆ , ,xy
r
s t S
f x y g s tmn d
![Page 21: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/21.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
21
Digital Image Processing
Image Restoration and Reconstruction
21
• Example (1):
– Impulsive Noise
– Median Filter
– 1-2-3- passes
– Less Blurring
Noisy (Salt & Pepper) One Pass
Two Pass Three Pass
![Page 22: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/22.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
22
Digital Image Processing
Image Restoration and Reconstruction
22
• Example (2):
– Max and min Filters:
3×3 Max Filter 3×3 min Filter
![Page 23: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/23.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
23
Digital Image Processing
Image Restoration and Reconstruction
23
• Example (3):
– Noise: • Uniform (a) and Impulsive (b)
– Mask size: • 5×5
– Filters • Arithmetic mean (c)
• Geometric mean (d)
• Median (e)
• Alpha Trimmed (f)
![Page 24: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/24.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
24
Digital Image Processing
Image Restoration and Reconstruction
24
• Adaptive, local noise reduction:
– If is small, return g(x, y)
– If ,return value close to g(x, y)
– If ,return the arithmetic mean mL
– Non-stationary noise or poor estimation:
2
2ˆ , , , ,
,L
L
f x y g x y g x y m x yx y
2
2 2
L
2 2
L
2
2max 1
,L x y
![Page 25: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/25.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
25
Digital Image Processing
Image Restoration and Reconstruction
25
Original Noisy A. Mean
G- Mean Local
• Example:
– N(0,100)
• Filters:
– Arithmetic Mean
– Geometric Mean
– Local
![Page 26: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/26.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
26
Digital Image Processing
Image Restoration and Reconstruction
26
• Adaptive Median:
1 min 2 max
1 2
1 min 2 max
1 2
max
max
,
if 0 and < 0
,
ˆ ˆif 0 and < 0
increase Mask size:
( ): Check " " condition.
ˆ
t
( ):
then else
hen
else
med med
xy xy
xy xy xy med
xy m
A Z Z A Z Z
A A
B Z Z B Z Z
B B Z Z Z Z
S A
S Z Z
ed
min
max
Minimum intensity in
Maximum intensity in
Median of intensity in
Intensity value at ( , )
Maximum allowed size of
xy
xy
med xy
xy
Max xy
z S
z S
z S
z x y
S S
![Page 27: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/27.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
27
Digital Image Processing
Image Restoration and Reconstruction
27
• Example:
– Noise: Salt and Pepper (Pa=Pb=0.25)
Noisy Median (7×7) Adaptive Median (Smax=7)
![Page 28: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/28.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
28
Digital Image Processing
Image Restoration and Reconstruction
28
• Band Reject Filter (Periodic Noises)
– Ideal, Butterworth, and Gaussian
IBRF BBRF(1) GBRF
![Page 29: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/29.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
29
Digital Image Processing
Image Restoration and Reconstruction
29
• Band Reject Filter (Periodic Noises)
0
0 0
0
2
2 2
0
22 2
0
1 ,2
, 0 ,2 2
1 ,2
1,
,1
,
,1, 1 exp
2 ,
n
I
B n
G
WD u v D
W WH u v D D u v D
WD u v D
H u vD u v W
D u v D
D u v DH u v
D u v W
![Page 30: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/30.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
30
Digital Image Processing
Image Restoration and Reconstruction
30
• Example:
Noisy Spectrum
BBRF(1) DeNoised
![Page 31: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/31.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
31
Digital Image Processing
Image Restoration and Reconstruction
31
• Band Pass Filter (Extract Periodic Noises)
– Analysis of Spectrum in different bands
– Noise Pattern:
, 1 ,bp brH u v H u v
![Page 32: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/32.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
32
Digital Image Processing
Image Restoration and Reconstruction
32
• Notch Reject Filter: INRF
GNRF
BNRF
NPF=1-NRF
![Page 33: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/33.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
33
Digital Image Processing
Image Restoration and Reconstruction
33
• Notch Filter (Single Frequency Removal):
1 22 2
1 0 0
1 22 2
2 0 0
1 0 2 0
2
0
1 2
1 2
2
0
, 2 2
, 2 2
0 , and ,,
1 O.W.
1,
1, ,
, ,1, 1 exp
2
I
B
G
D u v u M u v N v
D u v u M u v N v
D u v D D u v DH u v
H u vD
D u v D u v
D u v D u vH u v
D
![Page 34: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/34.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
34
Digital Image Processing
Image Restoration and Reconstruction
34
• Example •Horizontal Pattern in space
•Vertical Pattern in Frequency
Spectrum Notch in Vertical lines
Notch Passed Spectrum Vertical
Notch Rejected Spectrum
![Page 35: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/35.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
35
Digital Image Processing
Image Restoration and Reconstruction
35
• Several Single Frequency Noises:
Images Spectrum
![Page 36: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/36.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
36
Digital Image Processing
Image Restoration and Reconstruction
36
• Multiple Notch will disturb the image:
– Extract interference pattern using Notch Pass
– Now Perform Filter Function in Spatial Domain as An Adaptive-Local Point Process:
– Optimization Criteria: minimum variance of filtered images
1( , ) ( , ) ( , ) , ( , ) ( , )NP NPN u v H u v G u v x y H u v G u v
ˆ , , , ,f x y g x y w x y x y
2
ˆminf
![Page 37: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/37.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
37
Digital Image Processing
Image Restoration and Reconstruction
37
22
ˆ 2
2
2
ˆ 2
2
1 ˆ ˆ, , ,2 1
1ˆ ˆ, ,2 1
1, , , ,
2 1
, , ,
Smoothness of ( , ):
, , , ,
d d
fs d t d
d d
s d t d
d d
fs d t d
x y f x s y t f x yd
f x y f x s y td
x y g x s y t w x s y t x s y td
g x y w x y x y
w x y
w x s y t w x y w x
, , ,y x y w x y x y
• Optimum Notch Filter:
![Page 38: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/38.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
38
Digital Image Processing
Image Restoration and Reconstruction
38
• Optimum Notch Filter:
• Computed for each pixel using surrounded mask
2
ˆ 2
2
2
ˆ
2 2
1, , , ,
2 1
, , ,
, , , , ,0 ,
, , ,
d d
fs d t d
f
x y g x s y t w x y x s y td
g x y w x y x y
x y g x y x y g x y x yw x y
w x y x y x y
![Page 39: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/39.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
39
Digital Image Processing
Image Restoration and Reconstruction
39 Former images Spectrum (without fftshift)
• Result:
![Page 40: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/40.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
40
Digital Image Processing
Image Restoration and Reconstruction
40
• Result:
N(u,v) η(x,y)
![Page 41: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/41.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
41
Digital Image Processing
Image Restoration and Reconstruction
41
Filtered Image • Result:
![Page 42: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/42.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
42
Digital Image Processing
Image Restoration and Reconstruction
42
• Linear Degradation:
L-System:
LSI-System:
, , ,
, , ,
, , , ,
, , ,
, , , ,
, , , ,
g x y H f x y x y
H f x y f H x y d d
h x y H x y
H f x y f h x y d d
g x y f x y h x y x y
G u v F u v H u v N u v
![Page 43: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/43.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
43
Digital Image Processing
Image Restoration and Reconstruction
43
• Degradation Estimation:
– Image Observation: • Look at the image and …
– Experiments: • Acquire image using well defined object (pinhole)
– Modeling: • Introduce certain model for certain degradation using physical
knowledge.
![Page 44: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/44.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
44
Digital Image Processing
Image Restoration and Reconstruction
44
• Degradation (Using Experimental PSF)
,
,s
PSF
G u vH u v
A
Original Object Degraded Object
![Page 45: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/45.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
45
Digital Image Processing
Image Restoration and Reconstruction
45
• Atmospheric Turbulence:
NASA
![Page 46: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/46.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
46
Digital Image Processing
Image Restoration and Reconstruction
46
• Modeling of turbulence in atmospheric images:
5 62 2, expH u v k u v
![Page 47: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/47.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
47
Digital Image Processing
Image Restoration and Reconstruction
47
• Motion Blurring:
– Camera Limitation;
– Object movement.
![Page 48: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/48.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
48
Digital Image Processing
Image Restoration and Reconstruction
48
• Motion Blurring Modeling:
0 0
0 0
0
2
0 0
0
2
0
, ,
, ,
, , , ,
T
Tj ux vy
Tj ux t vy t
g x y f x x t y y t dt
G u v f x x t y y t dt e dxdy
G u v F u v e dt F u v H u v
![Page 49: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/49.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
49
Digital Image Processing
Image Restoration and Reconstruction
49
• Linear one/Two dimensional motion blurring:
0
0 0
, , sin
, , sin
j ua
Max
j ua vb
at Tx t t T H u v ua e
T ua
at bt Tx t y t H u v ua vb e
T T ua vb
![Page 50: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/50.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
50
Digital Image Processing
Image Restoration and Reconstruction
50
• Motion Blurring Example (1):
– a=b=0.1 and T=1
– Original (Left) and Notion Blurred (Right)
![Page 51: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/51.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
51
Digital Image Processing
Image Restoration and Reconstruction
51
• Motion Blurring Example (2):
![Page 52: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/52.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
52
Digital Image Processing
Image Restoration and Reconstruction
52
• Uniform Motion Blurring (?):
![Page 53: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/53.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
53
Digital Image Processing
Image Restoration and Reconstruction
53
• Out of Focus Blurring:
2 2
2
1,
1 1
0 1
x yh x y circ
R R
rcirc r
r
![Page 54: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/54.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
54
Digital Image Processing
Image Restoration and Reconstruction
54
• Performance Metrics in Image Restoration:
– Definition • 𝑓(𝑥, 𝑦): Clean Image
• 𝑛(𝑥, 𝑦): zero mean uncorrelated noise with variance 𝜎2
• 𝑔(𝑥, 𝑦): Blurred Image, 𝑓(𝑥, 𝑦)(𝑥, 𝑦)
• 𝑓 𝑥, 𝑦 : Restored Image
– SNR (Signal to Noise Ratio), Image Quality:
– BSNR (Blurred SNR)
2
,
10 2
,
10 logx y
f x y
SNR
2
,
10 2
,
10 logx y
g x y
BSNR
![Page 55: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/55.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
55
Digital Image Processing
Image Restoration and Reconstruction
55
• Performance Metrics in Image Restoration:
– ISNR (Improvement in SNR):
2
,
10 2
,
, ,
10 logˆ, ,
x y
x y
f x y g x y
ISNR
f x y f x y
![Page 56: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/56.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
56
Digital Image Processing
Image Restoration and Reconstruction
56
• Inverse Filtering:
– Without Noise:
, , ,ˆ , ,
Problem of division by zero or NaN!
ˆ ˆ, ,
G u v F u v H u vF u v F u v
H u v H u v
![Page 57: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/57.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
57
Digital Image Processing
Image Restoration and Reconstruction
57
• Inverse Filtering:
– With Noise:
Problem of division by zero
Impossible to recover even
, , , , ,ˆ , ,ˆ
if H(.,.) is known!
ˆ ˆ, , ,
!
!
G u v F u v H u v N u v N u vF u v F u v
H u v H u v H u v
![Page 58: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/58.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
58
Digital Image Processing
Image Restoration and Reconstruction
58
• Pseudo Inverse (Constrained) Filtering:
– Set infinite (large) value to zero;
, ˆ ,ˆˆ ,,
ˆ0 ,
THR
THR
G u vH u v H
H u vF u v
H u v H
![Page 59: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/59.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
59
Digital Image Processing
Image Restoration and Reconstruction
59
• Example:
– Full Band (a)
– Radius 50 (b)
– Radius 70 (c)
– Radius 85 (d)
![Page 60: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/60.jpg)
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60
Digital Image Processing
Image Restoration and Reconstruction
60
Ghost
• Threshold Effect:
![Page 61: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/61.jpg)
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61
Digital Image Processing
Image Restoration and Reconstruction
61
• Phase Problem:
![Page 62: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/62.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
62
Digital Image Processing
Image Restoration and Reconstruction
62
• Wiener Filtering in 2D case:
2
2 2
, , , , , ,
ˆ , , . ,
ˆ, , , , , . ,
,
g x y s x y x y G u v S u v N u v
F u v W u v G u v
E u v F u v F u v F u v W u v G u v
E E u v E F WG F WG
E FF WGG W WGF FW G
E F W G W WGF W G F
Degradation f(x,y) g(x,y)
Restoration g(x,y) f(x,y)
![Page 63: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/63.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
63
Digital Image Processing
Image Restoration and Reconstruction
63
• Wiener Filtering in 2D case:
2
2
2
,
, ,,
,
, :Spectral Estimation
, : Cross Spectral Estimation
, ,
FF GG FG GF
FG
GG
XX
XY
XY YX
E E u v P WP W W P WP
E E u v P u vW u v
P u v
P u v E X
P u v E XY
P u v P u v
W
0
![Page 64: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/64.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
64
Digital Image Processing
Image Restoration and Reconstruction
64
• Wiener Filtering in 2D case:
– Special Cases:
• Noise Only:
2* *
2 2* * *
Uncorrelated "zero mean noise" and "im
, , , , , ,
,
age":
,
,,
, ,
FG FF
GG FF NN
FF
FF NN
g x y f x y x y G u v F u v N u v
P u v E F F N E F E F E N P
P u v E F G F N E F E N E F E N E F E N P P
P u vW u v
P u v P u v
![Page 65: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/65.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
65
Digital Image Processing
Image Restoration and Reconstruction
65
• Degradation plus Noise:
22
2 2
12
2
Uncorrelated Noise and Imag
, , , ,
, ,
e:
, ,
,
1 1
FF
NNFF NN
FF
NNFF
FFNN
g x y f x y h x y x y
G u v F u v H u v N u v
P H HW u v
PP H P HP
H H
PH H PH H
SN
P
R
P
![Page 66: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/66.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
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Digital Image Processing
Image Restoration and Reconstruction
66
• Degradation plus Noise:
– White Noise
2
2
Select Interavtively
1,
HW u v
H H K
![Page 67: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/67.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
67
Digital Image Processing
Image Restoration and Reconstruction
67
• Wiener Filter is known as:
– Wiener-Hopf
– Minimum Mean Square Error
– Least Square Error
• Problems with Wiener:
– PFF
– PNN
![Page 68: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/68.jpg)
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Digital Image Processing
Image Restoration and Reconstruction
68
• Phase in Wiener Filter:
• No Phase compensation!
22
2
1
1
NN
FF
NN
FF
HW
PHH
PWPH
HP W H
H
![Page 69: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/69.jpg)
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Digital Image Processing
Image Restoration and Reconstruction
69
• Wiener Filter vs. Inverse Filter:
22
10
lim
0 0NNP
NN
FF
HH HW W H
P HH HP
![Page 70: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/70.jpg)
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70
Digital Image Processing
Image Restoration and Reconstruction
70
• Other Wiener Related Filter:
1
2
0.52
1, 0 1
1 1exp
SNN
FF
NN
FF
HW
PHH
P
W jP H
HP
![Page 71: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/71.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
71
Digital Image Processing
Image Restoration and Reconstruction
71
• Nonlinear Filter:
1,
,
min
ˆ , , , ,
1 HP like
1 LP like
log , ,ˆ ,
0 O.W.
j G u v
j G u v
F u v G u v e G u v G u v
G u v e G u v GF u v
![Page 72: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/72.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
72
Digital Image Processing
Image Restoration and Reconstruction
72 Full Inverse Pseudo Inverse Wiener
• Example:
![Page 73: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/73.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
73
Digital Image Processing
Image Restoration and Reconstruction
73
Degraded Wiener Inverse
No
ise Decrease
• Example:
– Motion Blurring+Noise
![Page 74: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/74.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
74
Digital Image Processing
Image Restoration and Reconstruction
74
• Algebraic Approach
– A Matrix Review: • Matrix Diagonalization1: Each square matrix A (with n linearly
independent eigenvectors) can be decomposed into the very special form:
• P: Matrix composed of the eigenvectors of .
• D: Diagonal matrix constructed from the corresponding eigenvalues of A.
1A PDP
1 http://mathworld.wolfram.com/MatrixDiagonalization.html
![Page 75: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/75.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
75
Digital Image Processing
Image Restoration and Reconstruction
75
• Algebraic Approach:
– Matrix Representation of Circular Convolution:
1
0
0 0 1 2 1 0
1 1 0 1 2 1
2 2 1 0 3 2
1 1 2 3 0 1
C
, Periodicity:
In Matrix Form:
:
N
j
N N
N
N N N N N
y n f n g n f j g n j g i g i N
y g g g g f
y g g g g f
y g g g g f
y g g g g f
C
A Circulant Matrix!
,C k j g k j
![Page 76: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/76.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
76
Digital Image Processing
Image Restoration and Reconstruction
76
• Algebraic Approach:
– Diagonalization of Circular Convolution:
1 1
2
,
,
0 0 0
0 1 0
0 0 1
jm nN
K
m n e
G
G
G N
G k DFT g n
C = WDW WDW W W
W
D
![Page 77: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/77.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
77
Digital Image Processing
Image Restoration and Reconstruction
77
• Algebraic Approach:
– Diagonalization of Circular Convolution:
– Extension to 2D case is similar!
DFT
Multiply
IDFT
f g
WDW f W D W f
![Page 78: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/78.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
78
Digital Image Processing
Image Restoration and Reconstruction
78
• Algebraic Approach to Noise Reduction:
– Least Square Approach: • Matrix Representation of Degradation Model:
2 2
2
1
Pseudo Inverse (Left)
1
ˆ arg min arg min
0
2 0
ˆ
T
T
T
T T
T T
f
g Hf n n g Hf
f n g Hf
g Hf g Hf g Hf
f f
g Hf g HfH g Hf
H
f
f H H H g
H H H = I
![Page 79: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/79.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
79
Digital Image Processing
Image Restoration and Reconstruction
79
• Least Square Approach:
– If H is square and H-1 exists then:
– This is Inverse Filter!
1
1ˆ T T
H gH HHf g
ˆ 1f H g
1 1
2
* , ,
,
,
,,
H u
G u v
H u v
H u v G u vx
vf y
![Page 80: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/80.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
80
Digital Image Processing
Image Restoration and Reconstruction
80
• Constrained Least Square (CLS) Approach:
• Q is a linear operator (High pass version of image, Laplacian and etc.)
– Using Lagrange Multiplier:
2 2 2Const. ˆ arg mi o tn
f
f Qf g - Hf n
2 2 2
11
2 2
2 2
1ˆ
: Lagrange Multiplier which should be be satisfy:
T T
T T T T T T
J f Qf g - Hf n
J fQ Qf - H g - Hf
f
f H H Q Q H g H H Q Q H g
g - Hf n
![Page 81: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/81.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
81
Digital Image Processing
Image Restoration and Reconstruction
81
• CLS in frequency domain:
– Diagonalization:
21
21
22
2 2 22 *
1Multiply by :
, : Fourier Transform of Q (Linear
1
ˆ, , , , ,
T
HH H
TH H Q
H Q H
H Q H H u v C u v F u v H u v G u v
W
C u v
**
T * * * *
T T T * * *
* * *
H H = W D WH = WD W = WD W
H = WD W = WD W Q Q = W D W
H H Q Q f H g W D D W f WD W g
D D W f D W g
*
2 2
,ˆ , ,, ,
Operator!)
H u vF u v G u v
H u v C u v
![Page 82: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/82.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
82
Digital Image Processing
Image Restoration and Reconstruction
82
• CLS in frequency domain:
*
1
22ˆ , ,
,
1
,
,
ˆ
F uH
v G u vCH u v v
u v
u
T T Tf Q QH HH g
![Page 83: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/83.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
83
Digital Image Processing
Image Restoration and Reconstruction
83
• Wiener Filter vc. CLS filter:
*
2 2
1
1
,ˆ , ,, ,
1ˆ
0 Inverse Filter
1 Wiener Filter (General form)
T
T
n
T
f n
H u vE F u v G u v
H u v C u vE
f
T T T
R ff
R nn
Q Q R R f H H Q Q H g
![Page 84: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/84.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
84
Digital Image Processing
Image Restoration and Reconstruction
84
• For Laplacian as a measure of smoothness:
0 1 2 1
1 0 1 2
2 1 0 3
1 2 3 0
,0 , 1 ,1
, 1 ,0
M M
M
M M M
e e e
j
e e
C C C C
C C C C
C C C C
C C C C
P j P j N P j
C
P j N P j
Q
0 1 0
, 1 4 1
0 1 0
P x y
![Page 85: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/85.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
85
Digital Image Processing
Image Restoration and Reconstruction
85
• How to estimate γ:
2
2 2
22
2 2
ˆResidual Vector:
It can be shown that is monotically increasing with .
We seek for such that , is our accuracy
1. Initial Guess on
ˆ2. Calculate
3. If criteria
T
r = g - Hf
r r r r
r n
r g - Hf
r n
2 2
2 2
is not reached :
3.a if increase .
3.b if decrease .
4. Re-calculate new filter and repeat until criteria is meeted
r n
r n
![Page 86: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/86.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
86
Digital Image Processing
Image Restoration and Reconstruction
86
• How to estimate γ:
– Computational Tips:
2 2
1 12 2
0 0
21 12
0 0
1 1
0 0
1 12 2
0 0
2 2 2
2 2
We need and .
ˆ, , , , ,
1,
1,
,
Need only and
M N
x y
M N
n n
x y
M N
n
x y
M N
x y
n n
n n
R u v G u v H u v F u v r x y
n x y mMN
m n x yMN
n x y
MN m
m
r n
r
n
n
![Page 87: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/87.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
87
Digital Image Processing
Image Restoration and Reconstruction
87
High Noise Mid Noise Low Noise
• Example:
– CLS
![Page 88: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/88.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
88
Digital Image Processing
Image Restoration and Reconstruction
88
Correct Noise Parameter Incorrect Noise Parameter
• Example:
![Page 89: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/89.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
89
Digital Image Processing
Image Restoration and Reconstruction
89
• Computation Efforts:
• Gradient Descent Methods of function minimization:
1ˆ
H is size of MN MN (65536 65536)!!!!
T T
g Hf n n g Hf
f H H H g
( 1)min
2Convergence Condition: 0
max
: Eignvalue of
GD
k k k k
k
TE
xx x x x x
x
xx
![Page 90: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/90.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
90
Digital Image Processing
Image Restoration and Reconstruction
90
• Iterative LS or CLS (Tikhonov-Miller):
2
2
0
1 0
2 2 2
0
1
1
2
1.
2.
1
2
1.
2.
T
T
T T T T
k k k k k
T
T T
k k k k k
Φ
T T
T T T
g Hf n n g Hf
g HfΦ f g Hf H g Hf
f
f H g
f f H g Hf H g I H H f f I H H f
J fJ f Qf g - Hf n Q Qf - H g - Hf
f
f H g
f f H g - Hf Q Qf H g I H
L
H Q
:
CLS :
Q f
S
![Page 91: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/91.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
91
Digital Image Processing
Image Restoration and Reconstruction
91
• Example (Uniform Noise): – Blurred by a 7×7 Disc (BSNR=20dB)
– Restored Using Generalized Inverse Filter with T=10-3
(ISNR=-32.9dB)
![Page 92: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/92.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
92
Digital Image Processing
Image Restoration and Reconstruction
92
• Example (Gaussian Noise): – Blurred by a 5×Gaussian (σ2=1) (BSNR=20dB)
– Restored Using Generalized Inverse Filter with T=10-3
(ISNR=-36.6dB)
![Page 93: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/93.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
93
Digital Image Processing
Image Restoration and Reconstruction
93
• Example (Uniform Noise): – Blurred by a 7×7 Disc (BSNR=20dB)
– Restored Using Generalized Inverse Filter with T=10-1
(ISNR=+0.61 dB)
![Page 94: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/94.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
94
Digital Image Processing
Image Restoration and Reconstruction
94
• Example (Gaussian Noise): – Blurred by a 5×Gaussian (σ2=1) (BSNR=20dB)
– Restored Using Generalized Inverse Filter with T=10-1
(ISNR=-1.8 dB)
![Page 95: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/95.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
95
Digital Image Processing
Image Restoration and Reconstruction
95
• Example (Uniform Noise): – Blurred by a 7×7 Disc (BSNR=20dB)
– Restored Using CLS Filter with ϒ=1 (ISNR=+2.5 dB)
![Page 96: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/96.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
96
Digital Image Processing
Image Restoration and Reconstruction
96
• Example (Gaussian Noise): – Blurred by a 5×Gaussian (σ2=1) (BSNR=20dB)
– Restored Using CLS Filter with ϒ=1 (ISNR=+1.3 dB)
![Page 97: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/97.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
97
Digital Image Processing
Image Restoration and Reconstruction
97
• Example (Uniform Noise): – Blurred by a 7×7 Disc (BSNR=20dB)
– Restored Using CLS Filter with ϒ=10-4 (ISNR=-21.9 dB)
![Page 98: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/98.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
98
Digital Image Processing
Image Restoration and Reconstruction
98
• Example (Gaussian Noise): – Blurred by a 5×Gaussian (σ2=1) (BSNR=20dB)
– Restored Using CLS Filter with ϒ=10-4 (ISNR=-22.1 dB)
![Page 99: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/99.jpg)
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E. Fatemizadeh, Sharif University of Technology, 2012
99
Digital Image Processing
Image Restoration and Reconstruction
99
• Example:
![Page 100: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/100.jpg)
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100
Digital Image Processing
Image Restoration and Reconstruction
100
• Wiener Filter Example (input):
– Original (Left) and degraded with SNR =7dB (Right)
![Page 101: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/101.jpg)
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101
Digital Image Processing
Image Restoration and Reconstruction
101
• Wiener Filter Example (output):
– Known Noisy Image (Left) and Know Clear Image (Right)
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Digital Image Processing
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• Iterative Wiener Filter:
– Wiener in Matrix form (Spatial Filter):
1
is Hermitian Transfor,
ˆ
: Autocorrelation of
: Autocorrelation of
!
mT T T
f f n
f
n
g = Hf + n
W R H HR H R
f Wf
R f
R n
![Page 103: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/103.jpg)
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Digital Image Processing
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• Iterative Wiener Filter:
– Conditions:
1. The original image and noise are statistically uncorrelated.
2. The power spectral density (or autocorrelation) of the original image and noise are known.
3. Both original image and the noise are zero mean.
![Page 104: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/104.jpg)
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• Iterative Wiener Filter:
– Iterative Algorithm:
1
1. 0
2. 1
ˆ3. 1 1
ˆ ˆ4. 1 1 1
5. Repeat 2,3,4 until convergence.
T T
T
i i i
i i
i E i i
f g
f f n
f
R = R
W R H HR H R
f W g
R = f f
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• Example Iterative Wiener Filter):
![Page 106: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/106.jpg)
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Digital Image Processing
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• Adaptive Wiener Filter:
– Image are Non Stationary!
– Need Adaptive WF which is locally optimal.
– Assume small region which image are stationary
• Image Model in each region:
• Noisy Image:
, , ,
: zero-mean white noise with unit variance!
, : Constant over each region.
f f
f f
f x y x y x y
, , , , :Constant over each regionvg x y f x y v x y
![Page 107: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/107.jpg)
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Digital Image Processing
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• Local Wiener Filter in each region:
2
2 2
2
2 2
2
2 2
,
, ,
ˆ , , ,
ˆ , ,
, : Low-pass filtered on noisy image.
, , , Zero mean assumption
, , : Hi-pass filtered on noi
ff f
a
ff vv f v
f
a
f v
f a f
f
f f
f v
f
f g
f
PW u v
P P
w x y x y
f x y g x y w x y
f x y g x y
x y
x y x y
g x y x y
2
2 2
sy image.
,ˆ , , ,
,
f
f v
x yf x y HP x y LP x y
x y
![Page 108: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/108.jpg)
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• Parameter Estimation:
2
2
2 2 2
Local Noisy Image Variance
Variance in a smooth image region or background
, , ,0
g
v
f g vx y Max x y
![Page 109: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/109.jpg)
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• Example (1):
![Page 110: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/110.jpg)
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• Example (1):
![Page 111: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/111.jpg)
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• Results:
![Page 112: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/112.jpg)
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• Results:
![Page 113: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/113.jpg)
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• Spatial Transform:
tiepoints
y
x
y
xr(x,y)
s(x,y)
f(x,y) g(x’,y’)
original distorted
![Page 114: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/114.jpg)
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Digital Image Processing
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• An example of Spatial Transform:
8765
4321
cxycycxcy
cxycycxcx
![Page 115: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/115.jpg)
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• Gray Level Interpolation:
– Main Problem: Non-Integer coordination!
![Page 116: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/116.jpg)
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Digital Image Processing
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• Gray Level Interpolation:
– Nearest neighbor
– Bilinear interpolation
• Use 4 nearest neighbors
– Cubic convolution interpolation
• Fit a surface over a large number of neighbors
![Page 117: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/117.jpg)
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original Tiepoints after distortion
Nearest neighbor
Bilinear
restored
restored
• Example (1):
![Page 118: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/118.jpg)
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original distorted
Difference
restored
• Example (2):
![Page 119: Image Degradation Model (Linear/Additive) G u v F u v H u ...ee.sharif.edu/~dip/Files/ImageRestorationForPrint.pdf · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,](https://reader033.vdocument.in/reader033/viewer/2022042214/5eb9ed744f36a961dc7890bb/html5/thumbnails/119.jpg)
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• Matlab Image Restoration Command:
– deconvblind: Restore image using blind deconvolution
– deconvlucy: Restore image using accelerated Richardson-Lucy algorithm
– deconvreg: Restore image using Regularized filter
– deconvwnr: Restore image using Wiener filter
– wiener2: Perform 2-D adaptive noise-removal filtering
– edgetaper: Taper the discontinuities along the image edges