Weighted Fuzzy Mean(WFM) filter

• For executing the filtering task, the WFM filter adopts a 3×3 sample window.

)1,1(),1()1,1(

)1,(),()1,(

)1,1(),1()1,1(

,

jijiji

jijiji

jijiji

ji

Knowledge base supported image noise removal process-Dynamic

• The image transmission process when applying the WFM filter with a dynamic knowledge base.

Sender:S Hist( ． )Dynamic

Knowledge Base

Channel

Receiver: X WFM( ． ) Y

(source)

(noise)(filtered)

(13’sa parameters)

Knowledge base supported image noise removal process-Static

• The image transmission process when applying the WFM filter with a static knowledge base.

Sender:S

Channel

Receiver: X WFM( ． ) Y

StaticKnowledge Base

Knowledge base supported image noise removal process-Definition

• Definition

- The WFM adopts LR fuzzy sets which can be characterized by the following equation

Let LR(y)=L(y)=R(y) for each y in real, F(x) can be represented by bounded

differences(the symbol▽) .

mxfor

mxR

mxforxm

L

)f(

,

,

mxxm

LRxf )(

Knowledge base supported image noise removal process-Example

• Example of membership functions for the fuzzy sets DK, MD, and BR.

1

membership grade

255gray level

160

00

DK MD BR

Construction algorithm of fuzzy sets-Graph example

Number of pixels

MDendMDbegin

Fuzzy inference rules of WFM filter• Rule 1:if

1

1

1

1

1

1

1

11

)1,(

)1,())1,((),(

)1,1(,),1(,)1,1(

,)1,(,),(,)1,(

,)1,1(,),1(,)1,1(

k lDK

k lDK

jkixf

jkixjkixfjiythen

DKisjixDKisjixDKisjix

DKisjixDKisjixDKisjix

DKisjixDKisjixDKisjix

1

1

1

1

1

1

1

12

),(

),()),((

),(

)1,1(,),1(,)1,1(

,)1,(,),(,)1,(

,)1,1(,),1(,)1,1(

k lMD

k lMD

ljkixf

ljkixljkixf

jiythen

MDisjixMDisjiXMDisjix

MDisjixMDisjiXMDisjix

MDisjixMDisjiXMDisjix

1

1

1

1

1

1

1

13

),(

),()),((

),(

)1,1(,),1(,)1,1(

,)1,(,),(,)1,(

,)1,1(,),1(,)1,1(

k lBR

k lBR

ljkixf

ljkixljkixf

jiythen

BRisjixBRisjiXBRisjix

BRisjixBRisjiXBRisjix

BRisjixBRisjiXBRisjix

• Rule 2:if

• Rule 3:if

Definition- Fuzzy interval• A fuzzy interval I is of LR-type if there exists two shape

functions L and R and four parameter

α, and β to constitute the membership function of I

,,( ), Rmm rl2

rr

r

mxformx

R

mxmfor

mxforxm

L

xIfLR

),(

,1

),1

(

)(_ 1

1

• The fuzzy interval is then denoted by .,,, LRrl mmI

α β

ml mr

f

Definition- Fuzzy estimator

If I is the fuzzy interval stored in the knowledge base, then a fuzzy estimator can be produced by the following formula

)(_ ELRf

2

1

2

)1(

2

1

2

)1(

_

2

1

2

)1(

2

1

2

)1(

_

_1

1

2

2

1

1

2

2

)),((

),()),((

)),(( n

n

n

nljkixf

n

n

n

nljkixljkixf

jiXf

k l

ILR

k l

ILR

ELR

where is a n1×n2 sample matrix centered at the input pixel x(i,j) ．

),( jiX

Fuzzy inference result

where each weight wr is 1 if the 2-norm of associatedintermediate inference result

and the fuzzy estimator is minimum; otherwise it is zero.

),( jiy r

3

1

3

1

),(),(

rr

r

r

w

jiywjiy

r

)),((_ jiXf ELR

Experimental results• The experimental results of test image”Lenna”.

Fig.19.(a) Noise image ”Lenna” with p=0.9, (b) result of WFM filter, (c) result or median filter, (d) Noise image ”Boat” with p=0.9, (e) result of WFM filter, (f) result of median filter.

Experimental results• The experimental results of test image.