•Image Degradation Model (Linear/Additive)sbisc.sharif.edu/~miap/Files/DIP4MIAP(ForView... · 2011. 2. 20. · ee.sharif.edu/~miap E. Fatemizadeh, Sharif University of Technology,
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ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20111
Medical Image Analysis and Processing
Image Restoration
• Image Degradation Model (Linear/Additive)
, , , ,
, , , ,
g x y h x y f x y x y
G u v H u v F u v N u v
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20112
Medical Image Analysis and Processing
Image Restoration
• Source of noise– 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– Ex. white noise (a constant Fourier spectrum)
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20113
Medical Image Analysis and Processing
Image Restoration
• Noise Model
2
22
1 exp22
zp z
2
2 expz a
p z z a u z ab b
1
1 !
b baza zp z e u z
b
azp z ae u z
1p z u z a u z bb a
a bp z P z a P z b
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20114
Medical Image Analysis and Processing
Image Restoration
• Test Pattern– Histogram has three Spikes!
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20115
Medical Image Analysis and Processing
Image Restoration
• Noisy Images– Gaussian– Rayleigh– Gamma
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20116
Medical Image Analysis and Processing
Image Restoration
• Noisy Images– Exponential– Uniform– Salt & Pepper
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20117
Medical Image Analysis and Processing
Image Restoration
• Periodic Noise:– Electronic Devices
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20118
Medical Image Analysis and Processing
Image Restoration
• Periodic noise– Observe the frequency spectrum
• Random noise with PDFs– Case 1: imaging system is available
• Capture images of “flat” environment
– Case 2: noisy images available• Take a strip from constant area• Draw the histogram and observe it• Measure the mean and variance
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 20119
Medical Image Analysis and Processing
Image Restoration
• Medical Example:– MRI Artifact:
• Phantom:
Phantom Gibbs Noise
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201110
Medical Image Analysis and Processing
Image Restoration
• Medical Example:– CT Metal Artifact:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201111
Medical Image Analysis and Processing
Image Restoration
• Noise Estimation:– Shape: Histogram of a subimage (Background)
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201112
Medical Image Analysis and Processing
Image Restoration
• Noise‐only spatial filter:– g(x,y)=f(x,y)+η(x,y)
• Adaptive, local noise reduction:– If is small, return g(x,y)– If L>> , return value close to g(x,y)– If L≈ , return the arithmetic mean mL
2
2ˆ , , , L
L
f x y g x y g x y m
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201113
Medical Image Analysis and Processing
Image Restoration
Original Noisy A- Mean
G- Mean Local
• Example:
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E. Fatemizadeh, Sharif University of Technology, 201114
Medical Image Analysis and Processing
Image Restoration
• Linear Degradation:
, , ,
, , ,
, , , ,
,
L-System:
LSI-Syst
, ,
, , , ,, ,
e :, ,
m
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 yG u v F u v H u v N u v
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201115
Medical Image Analysis and Processing
Image Restoration
• Degradation Estimation:– Image Observation:
• Look at the image and
– Experiments:• Acquire image using well defined object (Flat, pinhole, and etc.)
– Modeling:• Introduce certain model for certain degradation using physical knowledge.
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201116
Medical Image Analysis and Processing
Image Restoration
• Degradation (Using Observation/PSF)
, ,, ,ˆ ,s
s sPSFs
H u v G u vH u v H u v
AF u v
Original Object Degraded Object
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201117
Medical Image Analysis and Processing
Image Restoration
• Atmospheric Turbulence:
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E. Fatemizadeh, Sharif University of Technology, 201118
Medical Image Analysis and Processing
Image Restoration
• Modeling of turbulence in atmospheric images:
5 62 2, expH u v k u v
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201119
Medical Image Analysis and Processing
Image Restoration
• Motion Blurring Modeling:
0 0
0 00
20 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
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201120
Medical Image Analysis and Processing
Image Restoration
• Linear one/Two dimensional motion blurring:
0
0 0
, , sin
,
, sin
j uaMax
j ua vb
Tx t at T t T H u v ua eua
x t at T y t bt TTH u v ua vb e
ua vb
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201121
Medical Image Analysis and Processing
Image Restoration
• Motion Blurring Example:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201122
Medical Image Analysis and Processing
Image Restoration
• MR Motion Artifact:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201123
Medical Image Analysis and Processing
Image Restoration
• Motion Blurring Discrete Modeling:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201124
Medical Image Analysis and Processing
Image Restoration
• Inverse Filtering:– Without Noise:
Problem of division by ze
, , ,ˆ , ,ˆ ˆ,o!,
r
G u v F u v H u vF u v F u v
H u v H u v
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201125
Medical Image Analysis and Processing
Image Restoration
• Inverse Filtering:– With Noise:
Problem of division by zeroImpossible 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
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201126
Medical Image Analysis and Processing
Image Restoration
• Pseudo Inverse (Constrained) Filtering:– Set infinite (large) value to zero;– Multiply H(u,v) by a I/G/B LPF
, ˆ ,ˆ ,ˆ ,, ˆ ,
, ˆ ,ˆˆ ,,ˆ ,
THR
THRTHR
THR
THR
G u vH u v H
H u vF u v
G u vH u v H
H
G u vH u v H
H u vF u vT H u v H
0THRH
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201127
Medical Image Analysis and Processing
Image Restoration
Full Band 60 Band
70 Band 85 Band
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E. Fatemizadeh, Sharif University of Technology, 201128
Medical Image Analysis and Processing
Image Restoration
• Phase Problem:– Look at this formulation:
– We preserve the Correct Phase!
1 ˆ ,ˆ ,ˆ , ,1 ˆ ,
THR
THRTHR
H u v HH u v
F u v G u vH u v H
H
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201129
Medical Image Analysis and Processing
Image Restoration
• Phase Problem:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201130
Medical Image Analysis and Processing
Image Restoration
• Wiener Filtering :
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
Degradationf(x,y) g(x,y)
De Degradationg(x,y) f(x,y)
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201131
Medical Image Analysis and Processing
Image Restoration
• 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
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201132
Medical Image Analysis and Processing
Image Restoration
• Wiener Filtering in 2D case:– Special Cases:
• Noise Only:
Uncorrelated Noise and Im, , , , ,
ag,
,, ,
e
,
:
FF
FF NN
g x y f x y x y G u v F u v N u v
P u vW u v
P u v P u v
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201133
Medical Image Analysis and Processing
Image Restoration
• Degradation plus Noise:
2 2
2 2
122
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 HPH H PH H
SN
P
R
P
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201134
Medical Image Analysis and Processing
Image Restoration
• Degradation plus Noise:– White Noise
2
2
Select Interavtively
1 HH H K
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201135
Medical Image Analysis and Processing
Image Restoration
• Wiener Filter is known as:– Wiener-Hopf– Minimum Mean Square Error– Least Square Error
• Problems with Wiener:– PFF– PNN
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201136
Medical Image Analysis and Processing
Image Restoration
• Phase in Wiener Filter:
• No Phase compensation!
22
2
1
1
NN
FFNN
FF
HW PHH
PW PH HP W H
H
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201137
Medical Image Analysis and Processing
Image Restoration
• Wiener Filter vs. Inverse Filter:
202
1 0lim
0 0NNPNN
FF
HH HW W HP HH HP
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201138
Medical Image Analysis and Processing
Image Restoration
Full Inverse Pseudo Inverse Wiener
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201139
Medical Image Analysis and Processing
Image Restoration
Motion Blurring +Noise Wiener
Inverse
Noise D
ecrease
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201140
Medical Image Analysis and Processing
Image Restoration
• Iterative Wiener Filter:– We formulate for noise‐only case:
1
1 1 1
21 1
0. 0
1.
2.
3.
4.
5. Repeat 2,3,4 until convergence.
i
iFF GG
iFF
iFF NN
i i i
i iFF
i
P P
PWP P
F W G
P E F
=
=
=
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201141
Medical Image Analysis and Processing
Image Restoration
• 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:
• Noise 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
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201142
Medical Image Analysis and Processing
Image Restoration
• 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 fa
ff vv f v
fa
f v
f a f
ff 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
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201143
Medical Image Analysis and Processing
Image Restoration
• Parameter Estimation:
2
2
2 2 2
Local Noisy Image Variance
Variance in a smooth image region or background
, ,
g
v
f g vx y x y
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201144
Medical Image Analysis and Processing
Image Restoration
• Results:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201145
Medical Image Analysis and Processing
Image Restoration
• Results:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201146
Medical Image Analysis and Processing
Image Restoration
• Results:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201147
Medical Image Analysis and Processing
Image Restoration
• Results:
ee.sharif.edu/~miap
E. Fatemizadeh, Sharif University of Technology, 201148
Medical Image Analysis and Processing
Image Restoration
• 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
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