image degradation model (linear/additive) g u v f u v h u...

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 2012

1

Digital Image Processing

Image Restoration and Reconstruction

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• 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

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• 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)

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• Some Noises Distribution:

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• 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

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• Noisy Images(1):

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• Noisy Images (2):

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• Periodic1 Noise:

– Electronic Devices

1 Disturbance

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• 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

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• Noise Estimation:

– Shape: Histogram of a subimage (Background)

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• 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

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• 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

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• 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

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• 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

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Arithmetic Geometric

Noisy

More-Less Blurring

• Example (1):

– Additive Noise

– Mean • Arithmetic

• Geometric

Original

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15 Q=+1.5

• Example (2):

– Impulsive

– Contra-harmonic

– Correct use

Noisy (Pepper) Noisy (Salt)

Q=-1.5

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16 Q=-1.5

• Example (3):

– Impulsive Noise

– Conrtra-harmonic

– Wrong use

Q=+1.5

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• 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

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• 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

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• 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

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• 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

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• Example (1):

– Impulsive Noise

– Median Filter

– 1-2-3- passes

– Less Blurring

Noisy (Salt & Pepper) One Pass

Two Pass Three Pass

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• Example (2):

– Max and min Filters:

3×3 Max Filter 3×3 min Filter

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• Example (3):

– Noise: • Uniform (a) and Impulsive (b)

– Mask size: • 5×5

– Filters • Arithmetic mean (c)

• Geometric mean (d)

• Median (e)

• Alpha Trimmed (f)

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• 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

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Original Noisy A. Mean

G- Mean Local

• Example:

– N(0,100)

• Filters:

– Arithmetic Mean

– Geometric Mean

– Local

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• 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

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• Example:

– Noise: Salt and Pepper (Pa=Pb=0.25)

Noisy Median (7×7) Adaptive Median (Smax=7)

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• Band Reject Filter (Periodic Noises)

– Ideal, Butterworth, and Gaussian

IBRF BBRF(1) GBRF

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• 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

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• Example:

Noisy Spectrum

BBRF(1) DeNoised

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• Band Pass Filter (Extract Periodic Noises)

– Analysis of Spectrum in different bands

– Noise Pattern:

, 1 ,bp brH u v H u v

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• Notch Reject Filter: INRF

GNRF

BNRF

NPF=1-NRF

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• 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

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• Example •Horizontal Pattern in space

•Vertical Pattern in Frequency

Spectrum Notch in Vertical lines

Notch Passed Spectrum Vertical

Notch Rejected Spectrum

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• Several Single Frequency Noises:

Images Spectrum

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• 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

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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:

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• 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

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39 Former images Spectrum (without fftshift)

• Result:

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• Result:

N(u,v) η(x,y)

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Filtered Image • Result:

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• 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



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• 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.

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• Degradation (Using Experimental PSF)

,

,s

PSF

G u vH u v

A

Original Object Degraded Object

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• Atmospheric Turbulence:

NASA

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• Modeling of turbulence in atmospheric images:

5 62 2, expH u v k u v

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• Motion Blurring:

– Camera Limitation;

– Object movement.

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• 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

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• 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

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• Motion Blurring Example (1):

– a=b=0.1 and T=1

– Original (Left) and Notion Blurred (Right)

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• Motion Blurring Example (2):

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• Uniform Motion Blurring (?):

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• Out of Focus Blurring:

2 2

2

1,

1 1

0 1

x yh x y circ

R R

rcirc r

r

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• 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

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• 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

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• 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

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• 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

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• 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

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• Example:

– Full Band (a)

– Radius 50 (b)

– Radius 70 (c)

– Radius 85 (d)

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Ghost

• Threshold Effect:

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• Phase Problem:

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• 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)

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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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– 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

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• Degradation plus Noise:

22

2 2

12

2

Uncorrelated Noise and Imag

, , , ,

, ,

e:

, ,

,

1 1

FF

NNFF NN

FF

NNFF

FFNN



P H HW u v

PP H P HP

H H

PH H PH H

SN

P

R

P

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• Degradation plus Noise:

– White Noise

2

2

Select Interavtively

1,

HW u v

H H K

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• Wiener Filter is known as:

– Wiener-Hopf

– Minimum Mean Square Error

– Least Square Error

• Problems with Wiener:

– PFF

– PNN

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• Phase in Wiener Filter:

• No Phase compensation!

22

2

1

1

NN

FF

NN

FF

HW

PHH

PWPH

HP W H

H

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• Wiener Filter vs. Inverse Filter:

22

10

lim

0 0NNP

NN

FF

HH HW W H

P HH HP

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• Other Wiener Related Filter:

1

2

0.52

1, 0 1

1 1exp

SNN

FF

NN

FF

HW

PHH

P

W jP H

HP

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• 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

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72 Full Inverse Pseudo Inverse Wiener

• Example:

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Degraded Wiener Inverse

No

ise Decrease

• Example:

– Motion Blurring+Noise

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• 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

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• 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

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– 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

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– Diagonalization of Circular Convolution:

– Extension to 2D case is similar!

DFT

Multiply

IDFT

f g

WDW f W D W f

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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High Noise Mid Noise Low Noise

• Example:

– CLS

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Correct Noise Parameter Incorrect Noise Parameter

• Example:

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• 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

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• 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

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• Example (Uniform Noise): – Blurred by a 7×7 Disc (BSNR=20dB)

– Restored Using Generalized Inverse Filter with T=10-3

(ISNR=-32.9dB)

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• Example (Gaussian Noise): – Blurred by a 5×Gaussian (σ2=1) (BSNR=20dB)


(ISNR=-36.6dB)

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(ISNR=+0.61 dB)

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(ISNR=-1.8 dB)

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– Restored Using CLS Filter with ϒ=1 (ISNR=+2.5 dB)

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– Restored Using CLS Filter with ϒ=1 (ISNR=+1.3 dB)

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– Restored Using CLS Filter with ϒ=10-4 (ISNR=-21.9 dB)

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– Restored Using CLS Filter with ϒ=10-4 (ISNR=-22.1 dB)

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• Example:

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• Wiener Filter Example (input):

– Original (Left) and degraded with SNR =7dB (Right)

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• Wiener Filter Example (output):

– Known Noisy Image (Left) and Know Clear Image (Right)

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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

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– 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.

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– 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):

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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

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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

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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

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• Example (1):

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• Example (1):

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• Results:

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• Results:

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• Spatial Transform:

tiepoints

y

x

y

xr(x,y)

s(x,y)

f(x,y) g(x’,y’)

original distorted

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• An example of Spatial Transform:

8765

4321

cxycycxcy

cxycycxcx

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• Gray Level Interpolation:

– Main Problem: Non-Integer coordination!

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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

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original Tiepoints after distortion

Nearest neighbor

Bilinear

restored

restored

• Example (1):

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original distorted

Difference

restored

• Example (2):

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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

image degradation model (linear/additive) g u v f u v h u...

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