03-4 - Fuzzy techniques.pdf

7/28/2019 03-4 - Fuzzy techniques.pdf

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Fuzzy techniques for intensitytrans ormat ons an spat a

filtering

Spring 2008 ELEN 4304/5365 DIP 1

byGlebV.Tcheslavski:[email protected]://ee.lamar.edu/gleb/dip/index.htm

PreliminariesFuzzy sets are used to incorporate knowledge in the solutions of

problems, whose formulation is based on imprecise concepts

Let Z be a set of elements (objects) with a generic element of Z

denoted as z; that is Z = {z}. This set is called a universe or

discourse.

A fuzzy set A in Z is characterized by a membership function A(z)that associates a real number in [0 1] with each element of Z. The


it is to one, the higher the grade of membership is.

In ordinary (crisp) sets, an element either belongs or does not

belong to a set.


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Preliminaries

In fuzzy sets, however, we say that all members for which A(z) = 1are full members of the set. All members for which A(z) = 0 are notmembers of the set. The members for which A(z) is between 0 and1 are partial members of the set. Therefore, a fuzzy set is an

ordered pair consisting of values of z and a corresponding

membership function:

,A z z z Z=


For continuous variables, the set A can be infinitely large. For

discrete values of z, the elements of A are shown explicitly.

Preliminaries

Membership function for a Crisp set Membership function for a Fuzzy set

Limitin the a e to inte er ears, a fuzz set would be:


{(1,1), (2,1),...,(20,1), (21,0.9), (22,0.8),...,(25,0.5),...,(29,0.1), (30,0),...}A =

Age 21 has a 0.9 degree of membership in the set


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Preliminaries/Definitions

A fuzzy set is empty iff its membership function is identically zero in Z.Two fuzzy sets are equal (A = B) iffA(z) = B(z) for all z Z.

The complement (NOT) of a fuzzy set A denoted by or NOT(A) isa set whose membership function is (z) = 1 - A(z) for all z Z.

A fuzzy set A is a subset of a fuzzy set B iffA(z) B(z) for all z Z.

The union (OR) of two fuzzy sets A and B denoted as AB or A OR B is a fuzzy set U with membership function U(z) = max[A(z), B(z)] forall z Z.


The intersection (AND) of two fuzzy sets A and B denoted as AB orA AND B is a fuzzy set I with membership function I(z) = min[A(z),

B(z)] for all z Z.

Preliminaries



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Preliminaries

Although fuzzy logic and probability may

,

significant difference between them

While the probability states: There is a 50% chance that the person

is young (assuming that the person is either young or not), the

fuzzy statement would be A persons degree of membership among


degree).

Common membership functionsTriangular: 1 ( )

( ) 1 ( )

a z b a b z a

z z a c a z a c


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Common membership functionsS-shape:

2

0

2

z a

z aa z b

c a


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Using fuzzy sets

Assume that we need to categorize fruits (based on their colors) intothree groups: verdant (green), half-mature (yellow), and mature

. ,

yellow, and red are vague

notations of our color sensation and are

needed to be expressed in fuzzy format

(fuzzified). This is achieved by

defining membership as a function of

Spring 2008 ELEN 4304/5365 DIP 11

. ,

notion of color is rather linguistic:

there are regions of wavelengths that

may be associated with each color!

Using fuzzy setsThe logics underlying the classification can be expressed by the

following 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-mature

OR

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

Spring 2008 ELEN 4304/5365 DIP 12

ese ru es orma y sum our now e ge a ou e pro em. e

next step is to determine a procedure utilizing color and the

knowledge base to create the output of the fuzzy system

implication (inference).


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Using fuzzy sets

Fuzzy inputs to the system lead tofuzzy outputs out of it. Unlike the

independent variable, theindependent variable for theoutput is maturity.

Two sets of membership functions together with the rule basecontain all the information needed for classification. For

Spring 2008 ELEN 4304/5365 DIP 13

ns ance, e express on re ma ure represens anintersection (AND) operator. Since the independent variables

for the input and output are different, the result will be 2D.

Using fuzzy sets

Input or re mature

Combinedrepresentation

The resultof AND

Spring 2008 ELEN 4304/5365 DIP 14

output

3( , ) min{ ( ), ( )}red mat z v z v =


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Using fuzzy sets

In practice, we are interested in the output resulting from aspecific input. Let z0 is a specific value of red. The fuzzy

3 ,

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

Assuming that

0( )red z c =

Spring 2008 ELEN 4304/5365 DIP 15

We can find the membership function of maturity (for theclass mature) for the specific input (color)

Using fuzzy setsUsing the same approach, we can find the fuzzy responses fortwo other rules and the specific input z0:

2 0 2 0( ) min ( ), ( , )yellowQ v z z v =

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

These equations are the outputs of specific rules for a giveninput. The complete (aggregated) fuzzy output is:

Spring 2008 ELEN 4304/5365 DIP 16

1 2 3=Or, in general:

{ }{ }0 0( ) max min ( ), ( , )s rsr

Q v z z v =


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Using fuzzy sets

In our situation r= {1, 2, 3}ands = {green, yellow, red}

Therefore, the response Q of afuzzy system is the union ofindividual fuzzy responses for thegiven input.

In the fruit example, we notice that

Spring 2008 ELEN 4304/5365 DIP 17

1 wavelength z0, the fruit is notverdant.

Using fuzzy setsThe next step is to obtain a crisp outputv0 from fuzzy setQby the process called defuzzification. One common

10

1

( )

( )

k

v

k

v

vQ v

v

Q v

=

=

=

Spring 2008 ELEN 4304/5365 DIP 18

.In our example, v0 = 72.3, which indicates that a fruit isapproximately 72% mature.


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Using fuzzy sets

So far, weconsidered IF-THEN

conditions (IF thecolor is red). Rulescontaining more thanone part are alsopossible:

Spring 2008 ELEN 4304/5365 DIP 19

OR consistency issoft, THEN

mature.

Using fuzzy setsThe following steps are assumed to implement fuzzy logic:1. Fuzzify the inputs: map each scalar input to the [0 1]

2. Perform the required fuzzy logical operations: theoutputs of all condition operators must be combined toyield a single value using the appropriate logic.

3. Apply an implication method: clip each fuzzy outputaccording to the result of corresponding condition rule.

4. A l an a re ation method to the cli ed out ut

Spring 2008 ELEN 4304/5365 DIP 20

fuzzy sets: combine the output of each fuzzy rule in asingle fuzzy set.

5. Defuzzify the final output fuzzy set: obtain a crisp scalaroutput (for instance, as a center of gravity).


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Using fuzzy sets

In general, for M IF-THEN rules, N input variables z1, z2, zN,and one output variable v, assuming that conditions are linked,

1 11 2 12 1 1

1 21 2 22 2 2

1 1 2 2

( , ) ( , ) ... ( , ) ( , )

( , ) ( , ) ... ( , ) ( , )

...

( , ) ( , ) ... ( , ) ( , )

N N

N N

M M N MN M

IF z A AND z A AND AND z A THEN v B



( , )ELSE v B

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whereAij is the fuzzy set associated with the ith rule and thejth

input variable and Bi is the fuzzy set associated with the

output of the ith rule.

Using fuzzy setsEvaluation of conditions for the ith rule produces a scalar output(strength level or firing level of the ith rule) as:

{ }min ( ); 1, 2,...iji A j

z j N = =

A membership function of a fuzzy setAijevaluated at the value of thejth input.

i = 1, 2, M

For the ELSE portion:

Spring 2008 ELEN 4304/5365 DIP 22

m n 1 ; 1, 2,...,E i= =

When instead of AND, the OR logic is used, min should bereplaced by max in the expression for i but not for E.


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Using Fuzzy sets for intensity

transformationsThe process of contrast enhancement can be stated as

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.

Then, the input

Spring 2008 ELEN 4304/5365 DIP 23

membershipfunctions are:

Using Fuzzy sets for intensitytransformationsIn this example, we are interested in constant intensities,whose stren th is modified. Therefore theout ut membershifunctions are singletons (constant). The various degrees ofintensity in [0 1] occur when singletons are clipped by thecorresponding rules. In this situation, for the input z0 the output:

0 0 0

0

0 0 0

( ) ( ) ( )

( ) ( ) ( )

dark d gray g bright b

dark gray bright

z v z v z vv

z z z

+ + =

+ +

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Fuzzy image processing is computationally intensive, since thefuzzification, processing conditions for all rules, implication,aggregation, and defuzzification must be applied to every pixelin the input image!


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Using Fuzzy sets for intensity

transformations

Spring 2008 ELEN 4304/5365 DIP 25

Original low-contrastimage (intensities ina narrow range)

Result of histogramequalization contrast is increased

but there are areaswith overexposedappearance

Result of a rule-basedcontrast modificationapproach

Using Fuzzy sets for intensitytransformations

Histogram of Histogram ofthe originalimage

the equalizedimage

The outputsingletons were

=

Outputhistogram

Spring 2008 ELEN 4304/5365 DIP 26

vg =127 (midgray);vb =255 (white)

A faster technique, such as histogram specification, could be used instead.


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Using Fuzzy sets for spatial

filteringThe basic approach in to define neighborhood propertiescontaining the essence of what the filters need to detect.

For example, to detect boundaries: if a pixel belongs to auniform region, make it white; else, make it black (white andblack are fuzzy sets). A uniform region can be expressed inthrough intensity differences between the pixel at the center ofa neighborhood and its neighbors.

For a 3x3 nei hborhood

Spring 2008 ELEN 4304/5365 DIP 27

denoting the intensity differencesbetween the ith neighbor and the

center point as di, a boundaryextraction can be described as

Using Fuzzy sets for spatialfilteringIF d2 is zero AND d6 is zeroTHEN d5 is whiteIF d6 is zero AND d8 is zeroTHEN d5 is whiteIF d8 is zero AND d4 is zeroTHEN d5 is whiteIF d4 is zero AND d2 is zeroTHEN d5 is white

ELSE d5 is blackNotice, no diagonal member were considered for simplicity andzero is a fuzzy set.

Spring 2008 ELEN 4304/5365 DIP 28

range of intensitylevels for intensitydifferences is[-L+1 L-1]. For a fuzzy set zero For fuzzy sets black

and white


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Using Fuzzy sets for spatial

filtering

,rule sets can berepresented as:

Spring 2008 ELEN 4304/5365 DIP 29

Using Fuzzy sets for spatialfiltering

Spring 2008 ELEN 4304/5365 DIP 30

A CT scan of ahuman head.

Result of fuzzyspatial filteringextracting edges

Result afterintensity scaling

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03-4 - Fuzzy techniques.pdf