•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,

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

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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)

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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

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Image Restoration

• Test Pattern– Histogram has three Spikes!

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Image Restoration

• Noisy Images– Gaussian– Rayleigh– Gamma

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Image Restoration

• Noisy Images– Exponential– Uniform– Salt & Pepper

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Image Restoration

• Periodic Noise:– Electronic Devices

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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

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Image Restoration

• Medical Example:– MRI Artifact:

• Phantom:

Phantom Gibbs Noise

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Image Restoration

• Medical Example:– CT Metal Artifact:

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Image Restoration

• Noise Estimation:– Shape: Histogram of a subimage (Background)

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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

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Image Restoration

Original Noisy A- Mean

G- Mean Local

• Example:

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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

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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.

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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

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Image Restoration

• Atmospheric Turbulence:

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Image Restoration

• Modeling of turbulence in atmospheric images:

5 62 2, expH u v k u v

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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

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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

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Image Restoration

• Motion Blurring Example:

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Image Restoration

• MR Motion Artifact:

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Image Restoration

• Motion Blurring Discrete Modeling:

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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

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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

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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

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Image Restoration

Full Band 60 Band

70 Band 85 Band

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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

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Image Restoration

• Phase Problem:

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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)

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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

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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

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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

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Image Restoration

• Degradation plus Noise:– White Noise

2

2

Select Interavtively

1 HH H K

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Image Restoration

• Wiener Filter is known as:– Wiener-Hopf– Minimum Mean Square Error– Least Square Error

• Problems with Wiener:– PFF– PNN

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Image Restoration

• Phase in Wiener Filter:

• No Phase compensation!

22

2

1

1

NN

FFNN

FF

HW PHH

PW PH HP W H

H

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Image Restoration

• Wiener Filter vs. Inverse Filter:

202

1 0lim

0 0NNPNN

FF

HH HW W HP HH HP

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Image Restoration

Full Inverse Pseudo Inverse Wiener

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Image Restoration

Motion Blurring +Noise Wiener

Inverse

Noise D

ecrease

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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

=

=

=

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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

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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

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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

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Image Restoration

• Results:

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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

Documents

•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,