Digital Image Processing - University of Ioanninacnikou/Courses/Digital_Image_Processing/2012-201… · 5 Principles of fuzzy set theory • Let Z= {z} be a set of elements with a

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Digital Image Processing

Fuzzy Techniques for Intensity Transformations and Spatial Filtering

Christophoros Niko

University of Ioannina - Department of Computer Science

Christophoros [email protected]

2 Contents

In this lecture we will look at spatial filtering techniques:q

– General principles of fuzzy set theory– Intensity transformations using fuzzy sets– Spatial filtering using fuzzy sets

C. Nikou – Digital Image Processing (E12)

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3 Introduction

• Fuzzy sets provide a framework to incorporate human logic in problems with imprecise concepts.

• Set membership– Crisp sets: the membership function assigns

values of 0 or 1 (the element belongs to the set or not).


– Fuzzy sets: the membership function has a gradual transition between 0 and 1 (the element has a degree of membership).

4 Introduction (cont.)

• Example: let Z be the set of all people and we want to define a subset A, the set of young people.


Crisp set Fuzzy set

• We may make statements as: young, relatively young, not so young...

• It is not a probability!

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5 Principles of fuzzy set theory

• Let Z= {z} be a set of elements with a generic element denoted by z.

• A fuzzy set A in Z is characterized by a membership function μA(z) that associates to each element of z a real number in [0,1], the grade of membership.

• A fuzzy set is an ordered pair


y p

{ }, ( ) |A z z z ZμΑ= ∈

6 Principles of fuzzy set theory (cont.)

• Empty fuzzy set: ( ) 0,A z z Zμ = ∀ ∈

• Equality:

• Complement (NOT):

• Subset:

( ) 1 ( )AA z zμ μ= −

if andonly if ( ) ( ),A BA B z z z Zμ μ= = ∀ ∈

if and onlyif ( ) ( ),A BA B z z z Zμ μ⊂ ≤ ∀ ∈


• Union (OR):

• Intersection (AND):

A B

[ ]: ( ) max ( ), ( )U A BU A B z z zμ μ μ= ∪ =

[ ]: ( ) min ( ), ( )I A BI A B z z zμ μ μ= ∩ =

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7 Principles of fuzzy set theory (cont.)


8 Common membership functions


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9 Using fuzzy sets

• Example: Use colour to categorize fruit into three groups: verdant, half-mature and mature. Ob i i f i l d• Observations at various stages of maturity led to the conclusions:

– A verdant fruit is green– A half mature fruit is yellow– A mature fruit is red.

• The colour is a vague description and has to be


The colour is a vague description and has to be expressed in fuzzy format.

– Linguistic variable (colour) with a linguistic value (e.g. red) is fuzzified through the membership function.

10 Using fuzzy sets (cont.)


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• The problem specific knowledge may be formalized in the form of fuzzy IF-THEN rules:

– R1: IF the color is green, THEN the fruit is verdant.

OR– R2: IF the color is yellow, THEN the fruit is half-

t


mature.OR

– R3: IF the color is red, THEN the fruit is mature.


• The next step is to perform inference or implication, that is, to use the inputs and the knowledge (IF THEN rules) to obtain the outputknowledge (IF-THEN rules) to obtain the output.

• As the input is fuzzy, the output (maturity) is, in general, also fuzzy.


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• For the sake of clarity, let’s see it through R3:– IF the color is red, THEN the fruit is mature.

• Red AND mature is the intersection (AND) of the membership functions μred (z) and μmat (v).

• Notice that the independent variables are different (z and v) and the result will be two-dimensional (2D).

• The intersection corresponds to the minimum:


The intersection corresponds to the minimum:

{ }3( , ) min ( ), ( )red matz v z vμ μ μ=



{ }3( , ) min ( ), ( )red matz v z vμ μ μ=

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• In general, we are interested in a specific input (e.g. a value of red z0).T fi d h i bl f h AND• To find the output variable, we perform the AND operation between μred (z0)=c and the general 2D result μ3(z,v):

{ }3 0 3 0( ) min ( ), ( , )redQ v z z vμ μ=



• Equivalently, for the other rules:{ }2 0 2 0( ) min ( ), ( , )yellowQ v z z vμ μ= { }2 0 2 0yellow

{ }1 0 1 0( ) min ( ), ( , )greenQ v z z vμ μ=

• The complete fuzzy output is given by:

1 2 3OR ORQ Q Q Q=

which is the union (OR) of the three individual fuzzy


which is the union (OR) of the three individual fuzzy sets. Because OR is defined as the max operator:

{ }{ }0 0( ) max min ( ), ( , )s rsrQ v z z vμ μ=

{1,2,3}, { , , }r s green yellow red= =

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• Input of the membership functions to colour z0.0

• Individual output for each rule

– the clipped cross-sections discussed previously


d scussed p e ous y

• Union of the outputs


• We have the complete output corresponding to a specific input (colour z0).p p ( 0)

• To obtain a crisp value for the maturity of that colour (defuzzification), one way is to compute the center of gravity:

1( )K

vvQ v

=∑


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1( )

vK

v

vQ v

=

= ∑∑

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• We may combine more than one inputs.


20Contrast enhancement using fuzzy

sets• The problem may be stated using the following

rules:– IF a pixel is dark, THEN make it darker– IF a pixel is gray, THEN make it gray– IF a pixel is bright, THEN make it brighter

• Both input and output are fuzzy terms


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sets (cont.)• We are dealing with constants in the output in this

example, membership and the expression is p , p psimplified:

0 0 00

0 0 0

( ) ( ) ( )( ) ( ) ( )

dark d gray g bright b

dark gray bright

z v z v z vv

z z zμ μ μ

μ μ μ× + × + ×

=+ +



sets (cont.)• Notice the difference in the hair and forehead with

respect to histogram equalization.p g q


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sets (cont.)• The histogram expanded but its main characteristics were

kept contrary to histogram equalization.


24 Spatial filtering using fuzzy sets

• A boundary extraction algorithm may have the rulesthe rules

– If a pixel belongs to a uniform region, then make it white

– Else make it black• Uniform region, black and white are fuzzy

sets and we have to define their their


sets and we have to define their their membership functions

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• A simple set of rules:– IF d2 is zero AND d6 is zero THEN z5=white– IF d6 is zero AND d8 is zero THEN z5=white– IF d8 is zero AND d4 is zero THEN z5=white– IF d4 is zero AND d2 is zero THEN z5=white– ELSE z5=black



• Membership functions (for input: zero, output: black and white) and fuzzy rules


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• Membership functions (for input: zero, output: black and white) and fuzzy rules


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Digital Image Processing - University of Ioanninacnikou/Courses/Digital_Image_Processing/2012-201… · 5 Principles of fuzzy set theory • Let Z= {z} be a set of elements with a