Degradation process: Objective of restoration: To reconstruct …users.encs.concordia.ca/~weiping/ELEC6641-W14/Chapter 5.pdf · 2014. 3. 19. · Chapter 5: Image Restoration and Reconstruction

© 1992–2008 R. C. Gonzalez & R. E. Woods

Chapter 5: Image Restoration and Reconstruction

Degradation process:

Objective of restoration: To reconstruct original image as accurately as possible

Degradation and restoration model:

Degradation function H: normally linear and shift invariant system



1. The restoration in general is a complex process. It depends on the distortion model H and the nature of the noise.

2. If there is no distortion, then restoration becomes a denoising problem.

3. Usually, it is assumed that the noise is independent of the image, and noises at different pixels are also independent.

4. The signal to noise ratio is defined as

5. Peak signal to noise ratio is defined as

2

10 10210log =20log dBf f

n n

SNRσ σσ σ

=

10 1025520log =20log dBpeak pixel valueSNR

η ησ σ

=



(a) A digital image with standard deviation 55 , (b) the same image added with a noise of standard deviation 3 (making SNR to 25.26 dB), (c) the same image with SNR around 6 dB.



Different Noises



Gaussian noise:

Rayleigh noise:

Gamma noise:

−−= 22

2 2)(exp

21)(

σµ

πσxxp

4/ba πµ +=4

)4(2 πσ −= b

an=µ 22 an=σ

a > 0, n is a positive integer.



Test Pattern



Noise Corrupted Test Pattern



Noise Corrupted Test Pattern (Cont’d)



Sine Noise Corrupted Image



Noise Corrupted Strip Image



Noise reduction – spatial filtering

1 Mean filters

Arithmetic mean filter

),(),(),( yxyxfyxg η+= ),(),(),( vuNvuFvuG +=

∑∈

=xySts

tsgmn

yxf),(

),(1),(ˆ

Geometric mean filter

Harmonic mean filter

mn

Sts sy

tsgyxf

1

),(

),(),(ˆ

= ∏

∈

∑∈

=

xySts tsg

mnyxf

),( ),(1),(

ˆ

Contraharmonic mean filter

∑

∑

∈

∈

+

=

xy

xy

Sts

QSts

Q

tsg

tsgyxf

),(

),(

1

),(

),(),(ˆ



2 Order-statistic filters

Median filter

Max and Min filters

Midpoint filter

Alpha-trimmed filter

{ }),(),(ˆ),(

tsgmedianyxfxySts ∈

=

{ }),(max),(ˆ),(

tsgyxfxySts ∈

=

{ }),(min),(ˆ),(

tsgyxfxySts ∈

=

{ } { }

+=

∈∈),(min),(max

21),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

∑∈−

=xySts

r tsgdmnyxf

),(),(1),(ˆ

Remove dark pixels

Remove white pixels

Remove salt-pepper noise

Remove outliers by excluding d/2 brightest and darkest pixels



2

2

2

2

2 2

2 2 2 2

( , ) ( , ) ( ( , ) )

Local mean

Noise variance

Local variance

If (less noise), then

If (more noise), then and i.e. when there is lo

NL

L

L

N

L

N L

N L N L L

f x y g x y g x y m

m

f g

f m

σσ

σ

σ

σ σ

σ σ σ σ

∧

∧

∧

= − −

= =

=

=

≈ ≈t of noise, we just average it over pixels.

∑∈ ),(),(

),( 1yxSts

tsgmn

3 Adaptive filters



Local noise reduction filter

Pseudo code for adaptive median filter:

[ ]LL

myxgyxgyxf −−= ),(),(),(ˆ 22

σση

Stage A: A1=zmed-zmin A2=zmed-zmax If A1>0 and A20 and B2



Arithmetic and Geometric Filtering

Guassian noise added

3X3 Averaging Geometric mean filtering



Contraharmonic Filtering

(a) Pepper noise

(b) Salt noise

Q=1.5 Q=-1.5



Q=1.5 Q=-1.5

Wrong Selection of Parameter



(a) Pepper & salt noise

(b) 3X3 Median

Order Statistic Filtering

2nd Median Filtering 3rd Median Filtering



Results of Max & Min Filtering

Filtering of pepper noise corrupted image

3X3 Max filtering 3X3 Min filtering



Mean Filtering & Order Statistic Filtering

Uniform noise

5X5 Arithmetic

Median filtering

Uniform + salt & pepper noise

Geometric mean

Trimmed mean filtering



Filtering of Gaussian Noise with m=0

Gaussian noise added Arithmetic mean filtering

Geometric mean filtering Adaptive median filtering



Salt & pepper corrupted Median filtering

Adaptive median filtering

Median and Adaptive Median Filtering



Frequency Selective Filters

Ideal bandstop filter

Butterworth bandstop filter

Gaussian bandstop filter



Removal of Sine Noise



Sine Noise Image



Notch Filters



Notch Filtering



Filtering process:

1

( , ) { ( , )},( , ) {[1 ( , )] ( , )}NR

G u v F g x yx y F H u v G u vη −

=

= −

),(),(),(),(ˆ yxyxwyxgyxf η−=

[ ]∑∑−=−=

−++++

=b

bt

a

as

yxftysxfba

yx2

2 ),(ˆ),(ˆ)12)(12(

1),(σ

To determine w(x,y), consider

∑∑−=−=

++++

=b

bt

a

astysxf

bayxf ),(ˆ

)12)(12(1),(ˆ

Where is average of in the neighborhood.

),(ˆ yxf),(ˆ yxf



2

2

]),(),(),([

)],(),(),([

)12)(12(1),( ∑∑

−=−=

−

−++++−++

++=

b

bt

a

asyxyxwyxg

tysxtysxwtysxg

bayx

η

ησ

),(),(thatAssume yxwtysxw =++

),(),(),(),( yxyxwyxyxw ηη =2

2

)],(),(),([

)],(),(),([

)12)(12(1),( ∑∑

−=−=

−

−++−++

++=

b

bt

a

asyxyxwyxg

tysxyxwtysxg

bayx

η

ησ

By setting 0),(),(2=

∂∂

yxwyxσ

),(),(

),(),(),(),(),( 22 yxyx

yxyxgyxyxgyxwηη

ηη

−

−=

we obtain



Modeled by PSF

),(ˆ),(),(

vuFvuGvuH

s

ss = A

vuGvuH ),(),( =

Estimation degradation function

6/522 )(),( vukevuH +−=



Air Turbulence Model

K=0.0025

K=0.001 K=0.00025



Motion Blur Model



∫ −−=T

dttyytxxfyxg0

00 )](),([),(

Motion blurring model

[ ]

),(),(),(

)](),([),(

0

)]()([2

)(200

0

00 vuFvuHdtevuF

dtdxdyetyytxxfvuG

Ttvytuxj

vyuxjT

==

−−=

∫

∫ ∫∫

+−

+−∞

∞−

∞

∞−

π

π

dtevuHT

tvytuxj∫ +−=0

)]()([2 00),( π

Further assuming linear motion in both x and y directions: TbttyTattx /)(/)( 00 ==

)()](sin[)(

),( vbuajevbuavbua

TvuH +−++

= πππ



Inverse filtering

),(),(

),(),(),(

),(ˆvuHvuN

vuFvuHvuG

vuF +==

For example

6/522 ])2/()2/[(),( NvMukevuH −+−−=



Inverse Filtering



Wiener (MMSE) filtering { }22 )ˆ( ffEe −=

),(),(/),(),(

),(),(

1

),(),(/),(),(

),(),(),(),(),(

),(),(),(ˆ

2

2

2

*

2

*

vuGvuSvuSvuH

vuHvuH

vuGvuSvuSvuH

vuHvuGvuSvuHvuS

vuSvuHvuF

f

ff

f

+=

+=

+=

η

ηη

∑∑

∑∑

∑∑

∑∑−

=

−

=

−

=

−

=−

=

−

=

−

=

−

=

−≈= 1

0

1

0

2

1

0

1

0

2

1

0

1

0

2

1

0

1

0

2

)],(ˆ),([

),(ˆ

),(

),(

M

x

N

y

M

u

N

vM

u

N

v

M

u

N

v

yxfyxf

yxf

vuN

vuFSNR

),(),(

),(),(

1),(ˆ 2

2

vuGKvuH

vuHvuH

vuF

+≈ ∑∑

−

=

−

=

−=1

0

1

0

2)],(ˆ),([1M

x

N

yyxfyxf

MNMSE



Inverse and Wiener Filtering

Full inverse filtering Radially limited filtering Wiener filtering



Motion Blur and Additive Noise

Corrupted image

Inverse and Wiener filtering



Constrained LS Filtering

ηHfg +=Degraded image in matrix form:

Minimize [ ]∑∑−

=

−

=

∇=1

0

1

0

22 ),(M

x

N

yyxfC

Subject to 22ˆ ηfH-g =

),(),(),(

),(),(ˆ 22*

vuGvuPvuH

vuHvuF

+=

γ

where P(u,v) is FT of

−−−

−=

010141

010),( yxp



Constrained LS Filtering



Constrained LS Filtering with iteration of

Using correct noise parameters

Using wrong noise parameters

γ

γ



Projection, back projection, and superimpose



Reconstruction using back projections



Reconstruction using different projections



Reconstruction using back projections



∫ ∫∞

∞−

∞

∞−

−+= dxdyyxyxfg jkkkj )sincos(),(),( ρθθδθρ

∫ ∫∞

∞−

∞

∞−

−+= dxdyyxyxfg )sincos(),(),( ρθθδθρ

∑∑−

=

−

=

−+==1

0

1

0)sincos(),(),(),(

M

x

N

ykj yxyxfgg ρθθδθρθρ



≤+

=otherwise 0

x),(

222 ryAyxf

dyyfdxdyxyxfg ∫∫ ∫∞

∞−

∞

∞−

∞

∞−

=−= ),( )(),(),( ρρδθρ

dyAdyyfgr

r

r

r ∫∫−

−−

−

−−==

22

22

22

22),(),(

ρ

ρ

ρ

ρρθρ

≤−

==otherwise 0

r 2)(),(22 ρρρθρ rAgg



),ysing(xcos),(),( kkk θθθθρθ +== kgyxf k),sincos(),( θθθθ yxgyxf +=

∫=π

θ θ0

),(),( dyxfyxf

∑=

=π

θθ

0),(),( yxfyxf



∫∞

∞−

−= ρθρθω πωρdegG j2),(),(

dxdyeyxf

dxdydeyxyxfG

yxj

j

)sincos(2

2

),(

)sincos(),(),(

θθπω

πωρ ρρθθδθω

+−∞

∞−

∞

∞−

∞

∞−

∞

−∞−

∞

∞

−

∫∫

∫ ∫ ∫

=

−+=

[ ] cos ; sin( , ) ( , )( cos sin )

u vG F u v

Fω θ ω θ

ω θ

ω θ ω θ= =

=

= +

FT of projection:

Reconstruction of back-projected image using sinograms

~ Fourier slice



Reconstruction using parallel-beam filtered backprojection

∫ ∫∞

−∞−

∞

∞

+= dudvevuFyxf vyuxj )(2),(),( π

∫ ∫∞

+=π

θθπω θωωθω2

0 0

)sincos(2),(),( ddeGyxf yxj



∫ ∫∞

∞−

+=π

θθπω θωθωω0

)sincos(2),(),( ddeGyxf yxj

θωθωωθθρ

ππωρ ddeGyxf

yx

j

sincos0

2),(),(+=

∞

∞−∫ ∫

=



To get complete and back-projected image: 1) Compute 1-D FT of each projection 2) Multiply each FT by the filter function which has

already multiplied by a suitable window 3) Obtain the inverse 1-D FT of each resulting filtered

transform 4) Integrate (sum) all the 1-D inverse transforms from

step 3).

||ω

≤≤

−−+

= otherwise 0

1)-(M0 1

2cos)1()(

ωπωω M

cch

Frequency-domain windowing function:



Reconstruction using fan-beam filtered backprojection



αβθ += αρ sinD=

θρρθθθρπ

ddyxsgyxfT

T∫ ∫ −+=−

2

0

)sincos(),(21),(

)cos(sinsincoscossincos

ϕθθϕθϕθθ

−=+=+

rrryx

[ ] θρραθθρπ

ddrsgyxfT

T∫ ∫ −−=−

2

0

)cos(),(21),(

[ ] βαααϕαβ

βααϕαπ

α

ddDDrs

DgrfDT

DT

cossin)cos(

),sin(21),(

2 )/(sin

)/(sin

1

1

−−+

+= ∫ ∫−

− −

−

−

Let be the IFT of ( )s ρ | |ω



[ ] βαααϕαββαϕαπ

α

α

α

ddDDrsprfm

m

cossin)cos(),(21),(

2

−−+= ∫ ∫−

− −

)sin(sin)cos( ' αααϕαβ −=−−+ RDr

[ ] βααααβαϕαπ

α

α

α

ddDRsprfm

m

cos)'sin(),(21),(

2

−= ∫ ∫−

− −

)()sin

()sin( 2 αα

αα sR

Rs = βααβαϕα

α

π

dhqR

rfm

m

−= ∫∫

−

)'(),(1),(2

02

)(sin2

1)(2

αα

αα sh

= αβαβα cos),(),( Dpq =

),sin(),(),( βααθρβα +== Dggp

γαβ =∆=∆ [ ]γγγγ )(,sin),( nmnDgmnp += If

The n-th ray in the m-th radial projection is equal to the n-th ray in the (m+n)-th parallel projection!



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Degradation process: Objective of restoration: To reconstruct …users.encs.concordia.ca/~weiping/ELEC6641-W14/Chapter 5.pdf · 2014. 3. 19. · Chapter 5: Image Restoration and Reconstruction