New Chapter 5 - Encsusers.encs.concordia.ca/~weiping/DIP W12/New Chapter 5.pdf · Chapter 5: Image Restoration and Reconstruction 3 Adaptive filters3 Adaptive filters Local noise

Chapter 5: Image Restoration and ReconstructionChapter 5: Image Restoration and Reconstruction

Degradation Model

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


Different Noises



Test Pattern



Noise Corrupted Test Pattern



Noise Corrupted Test Pattern (Cont’d)



Sine Noise Corrupted Image



Noise Corrupted Strip Image



N i d ti ti l filt iNoise reduction – spatial filtering

1 fi

),(),(),( yxyxfyxg ),(),(),( vuNvuFvuG 1 Mean filters

Arithmetic mean filter

xySts

tsgmn

yxf),(

),(1),(ˆ

Geometric mean filtermn

Sts sy

tsgyxf

1

),(

),(),(ˆ

Harmonic mean filter

St tsg

mnyxf

)( )(1),(ˆ

xySts tsg),( ),(

Contraharmonic mean filter

xySts

Qtsgyxf ),(

1),()(ˆ


xySts

Qtsgyxf

),(),(

),(


2 Order statistic filters2 Order-statistic filters

Median filter ),(),(ˆ),(

tsgmedianyxfxySts

Max and min filter ),(max),(ˆ),(

tsgyxfxySts

ˆ ),(min),(ˆ),(

tsgyxfxySts

1Midpoint filter

),(min),(max

21),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

Alpha-trimmed filter

xySts

r tsgdmn

yxf),(

),(1),(ˆ



3 Adaptive filters3 Adaptive filtersLocal noise reduction filter

Ad ti di filt

LL

myxgyxgyxf ),(),(),(ˆ2

2

Adaptive median filter

Stage A: A1=zmed-zminA2=zmed-zmaxA2 zmed zmaxIf A1>0 and A2<0, go to stage BElse increase the window sizeIf window size < S repeat stage AIf window size < Smax, repeat stage AElse output zmed

Stage B: B1=zxy-zminB2=zxy-zmaxIf B1>0 and B2<0, output zxy


, p xyElse output zmed


Arithmetic and Geometric Filtering

GuassianGuassian noise added


3X3 Averaging Geometric mean filtering


Contraharmonic Filtering

(a) Pepper noise (b) Salt noise


Q=1.5 Q=-1.5


Wrong Selection of Parameter

Q=1.5 Q=-1.5



O d S i i Fil iOrder Statistic Filtering

(a) Pepper & salt noise

(b) 3X3 Median


2nd Median Filtering 3rd Median Filtering


Results of Max & Min Filtering

3X3 Max filtering 3X3 Min filtering

Filtering of pepper noise corrupted image



il i d i i il iMean Filtering & Order Statistic Filtering

U if lUniform noise Uniform + salt & pepper noise

5X5 Arithmetic Geometric mean5X5 Arithmetic

Median filtering Trimmed mean filtering


filtering


Fil i f G i N i i h 0Filtering of Gaussian Noise with m=0

Gaussian noise added Arithmetic mean filtering


Geometric mean filtering Adaptive median filtering


Median and Adapti e Median Filtering

Salt & pepper t d Median filtering Adaptive

di fil i

Median and Adaptive Median Filtering

corrupted Median filtering median filtering



Frequency Selective Filters

Ideal bandstop filter

Butterworth bandstop filter

Gaussian bandstop filter



Removal of Sine Noise



Sine Noise Image



Notch Filters



N t h Filt iNotch Filtering









Filtering process:

)} ,,({),(1

yxgFvuG

),()],(1{[),( 1 vuGvuHFyx N R

),(),(),(),(ˆ yxyxwyxgyxf

ba 2

2 ˆˆ1

To determine w(x,y), consider

btas

yxftysxfba

yx2 ),(),()12)(12(

1),(

Where is average of in the neighborhood)(f̂)(f̂

ba

tysxfyxf ),(ˆ1),(ˆ

Where is average of in the neighborhood.),( yxf),( yxf


btas

yfba

yf ),()12)(12(

),(


2

2

2

])()()([

)],(),(),([

)12)(12(1),(

b

bt

a

asyxyxwyxg

tysxtysxwtysxg

bayx

]),(),(),([ yxyxwyxg

),(),(thatAssume yxwtysxw

),(),(),(),( yxyxwyxyxw 2

)([ tysxg2

)],(),(),([

)],(),(),([

)12)(12(1),(

b

bt

a

asyxyxwyxg

tysxyxwtysxg

bayx

)]()()([ yyyg

By setting 0),(),(2

yxwyx

)()()()(

we obtain


),(),(

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

yxyxgyxyxgyxw




Modeled by PSF

)( vuG G )(

Estimation degradation function

),(ˆ),(),(

vuFvuGvuH

s

ss

AvuGvuH ),(),(

6/522 )()( vukH


)(),( vukevuH


Air Turbulence Model

K=0.0025


K=0.001 K=0.00025


Motion Blur Model



T

dttyytxxfyxg 00 )](),([),(

Motion blurring model

0

)](),([),( )(200

0

dtdxdyetyytxxfvuG vyuxjT

),(),(),(0

)]()([2

0

00 vuFvuHdtevuFT

tvytuxj

T

dtevuHT

tvytuxj 0

)]()([2 00),(

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

)( bjT


)()](sin[)(

),( vbuajevbuavbua

TvuH


Inverse filtering

),(),(ˆ vuNvuG),(),(

),(),(),(

),(vuH

vuFvuH

vuF

For exampleFor example

6/522 ])2/()2/[(),( NvMukevuH ),( evuH



Inverse FilteringInverse Filtering



ˆWiener (MMSE) filtering 22 )ˆ( ffEe

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

),(ˆ2

*

2

*

vuGvuHvuGvuSvuH

vuF f

)(),(1

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

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

),(

2

22

vuGvuH

vuSvuSvuHvuSvuHvuS ff

),(),(/),(),(),( 2 vuG

vuSvuSvuHvuHf

1 1

21 1

2 )(ˆ)(M NM N

fF

1 1

2

0 0

2

1 12

0 0

)],(ˆ),([

),(

),(

),(

M Nu v

M Nu v

yxfyxf

yxf

vuN

vuFSNR

0 00 0 x yu v

)(),(1)(ˆ

2

vuGvuH

vuF

1 1

2)](ˆ)([1 M N

yxfyxfMSE


),(),(),(

),( 2 vuGKvuHvuH

vuF

0 0

)],(),([x y

yxfyxfMN

MSE


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* vuH

),(),(),(

),(),( 22 vuGvuPvuH

vuHvuF

where P(u v) is FT ofwhere P(u,v) is FT of

010141

010),( yxp


010


Constrained LS Filtering



Constrained LS Filtering with iteration of

Using correct noise parameters

Using wrong noise parameters



Projection, back projection, and superimpose



Reconstruction using back projections



Reconstruction using different projections



Reconstruction using back projections





dxdyyxyxfg jkkkj )sincos(),(),(

dxdyyxyxfg )sincos(),(),(

1 1M N


1

0

1

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

M

x

N

ykj yxyxfgg


222

otherwise 0

x),(

222 ryAyxf

dyyfdxdyxyxfg

),( )(),(),(

rr

2222

dyAdyyfgrr

2222

),(),(

r 2)()(

22 rAgg

otherwise 0)(),( gg



),(),( dyxfyxf 0

),(),( yfyf

)()( ff

0

),(),( yxfyxf

),ysing(xcos),(),( kkk kgyxfk


),sincos(),( yxgyxf


Reconstruction of back projected image using sinogramsReconstruction of back-projected image using sinograms

j 2

degG j 2),(),(

dxdydeyxyxfG j2)sincos(),(),(

FT of projection:

dxdyeyxf yxj )sincos(2),(

)()( FG


)sincos(

),(),( sin;cos

wFvuFG vu

~ Fourier slice


R t ti i ll l b filt d b k j tiReconstruction using parallel-beam filtered backprojection

)(

dudvevuFyxf vyuxj )(2),(),(

2

)sincos(2)()( ddGf yxj


0 0

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


0

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

ddeGyxf

yx

j

sincos0

2),(),(



T t l t b k j t d iTo get complete, back-projected image:

1) Compute 1-D FT of each projection2) Multiply each FT by the filter function which has

already multiplied by a suitable window3) Obtain the inverse 1-D FT of each resulting filtered

||

3) Obtain the inverse 1 D FT of each resulting filtered transform

4) Integrate (sum) all the 1-D inverse transforms from step 3)step 3).

Frequency-domain windowing function:

1)-(M0

12cos)1(

)(

Mcc

h


otherwise 0






R t ti i f b filt d b k j tiReconstruction using fan-beam filtered backprojection



sinD

ddfT

2

)i()(1)( ddyxsgyxfT 0

)sincos(),(21),(

sinsincoscossincos rryx)cos(

sinsincoscossincos

r

rryx

ddrsgyxfT

T

2

0

)cos(),(21),(

DgrfDT

DT

),sin(21),(

2 )/(sin

)/(sin

1

1


ddDDrsDT

cossin)cos()/(sin

Chapter 5: Image Restoration and ReconstructionChapter 5: Image Restoration and Reconstruction m1 2

ddDDrsprfm

cossin)cos(),(21),(

)sin(sin)cos( ' RDr )()(

ddDRsprfm

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

2

m

)()sin

()sin( 2

sR

Rs

dhqR

rfm

m

)'(),(1),(2

02

)(sin2

1)(2

sh

cos),(),( Dpq

),sin(),(),( Dggp

)(i)( D If


)(,sin),( nmnDgmnp If









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New Chapter 5 - Encsusers.encs.concordia.ca/~weiping/DIP W12/New Chapter 5.pdf · Chapter 5: Image Restoration and Reconstruction 3 Adaptive filters3 Adaptive filters Local noise