Ho Kuni 120412

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Fuzzy techniques in image processing

Sami Hokuni

April 12, 2012

1/ 62 Sami Hokuni Fuzzy techniques in image processing

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In a traditional set theory an element has only two options: iteither belongs totally to some set or it doesn’t belong there atall.

If we have set A such that [0, 1] ∈ A, then

− x = 1 belongs totally to the set A

− y = 2 doesn’t belong at all to the set A

This method is not very flexible if we have some imprecision


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Let us assume that we have set Z of all people.From this set we would want to define a subset A called”young people”.

To define this subset we need to decide some threshold age,

for example 20 years.If you are exactly 20 years old, you belong to the subset”young people”.

However, if you are 20 years and one second old, you dontbelong to this subset at all.


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In this case we would need more flexible system, which wouldhave some kind of gradual transition

Young → relatively young → somehow young → not so young

→ not at all young.Fuzzy logic offers us a way to do this kind of transitions.


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Fuzzy set is characterized by its membership function µ.

Let us assume that we have fuzzy set A in set Z , and z issome element in Z .

Then value µA(z ) represents degree of membership of element

z in A.Degree of membership is some real number from range [0, 1].

Degree of membership tells us, how strongly an elementbelongs to the subset.


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If µA(z )=1, then z belongs totally to the fuzzy set A.

If µA(z )=0, then z doesn’t belong at all to the fuzzy set A.

If µA(z ) ∈ (0, 1), then z belongs partially to the fuzzy set A.


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Fuzzy set A is an ordered pair consisting of values z ∈ Z anda membership function that gives the degree of membershipto each z ∈ Z

A = {z , µA(z )|z ∈ Z }

Previous subset of ”young people” can then be expressed as

A = {(1, 1), · · · , (20, 1), (21, 0.9), · · · , (29, 0.1), (30, 0), · · · }

In this example people who are 30 or older do not belong tothe subset ”young people” at all.


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Membership value is not the same as probability.

For example, in the figure on the last slide we saw that themembership value for z = 25 was 0.5.

Probablilistic statement: There is 50% chance that thisperson is young.

Here person is considered to either be in the set of youngpeople or not in that set.

Fuzzy statement: Person is young to some degree, and herethis degree is 0.5.

Fuzzy logic is characterized by imprecision, not byrandomness.


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Fuzzy set A ∈ Z is empty if and only if µA(z ) = 0 for all

z ∈ Z .Two fuzzy sets A ∈ Z and B ∈ Z are equal (A = B ) if andonly if µA(z ) = µB (z ) for all z ∈ Z .

Fuzzy set A is a subset of fuzzy set B if and only if µA(

z ) ≤

µB (

z ) for all

z ∈

Z .

Complement of a fuzzy set(NOT) A, denoted by A, is the setwhose membership function is

µA(z ) = 1 − µA(z )

for all z ∈ Z .


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The union(OR) of two fuzzy sets A and B , denoted A ∪ B , is

a fuzzy set U with the membership function

µU (z ) = max {µA(z ), µB (z )}

for all z ∈ Z .

The intersection(AND) of two fuzzy sets A and B , denotedA ∩ B , is a fuzzy set I with the membership function

µI (z ) = min {µA(z ), µB (z )}

for all z ∈ Z .


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Membership function can be defined in many different ways

A few next slides will present some of the most common types

of membership functions


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Triangular


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Trapezoidal


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Sigmoidal


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Let us suppose that we have an electric motor, whose health

we would like to measure.To simplify the case we can assume that it is enough tomeasure average vibration frequency.

Three are three ranges of average frequency:

− In low range motor is performing normally

− In mid range motor is performing marginally

− In high range motor is in the near-failure mode

Because these ranges sound imprecise, using fuzzy logic onthem might be a good idea.


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0 10 20 30 40 50 60 70 80

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Average vibration frequency : z

D e g r e e o f m

e m b e r s h i p : µ ( z )

µ low

(z)

µ mid

(z)

µ high

(z)


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Based on previous definitions we can come up with followingfuzzy IF-THEN rules:

IF the frequency is low , THEN motor operation is normal (R 1)

OR

IF the frequency is mid , THEN motor operation is marginal (R 2)

OR

IF the frequency is high, THEN motor operation is near failure (R 3)


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Abnormality in percents : v

D e g r e e o f m e m b e r s h i p

µ norm

(v) µ marg

(v) µ fail

(v)


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IF the frequency is low , THEN motor operation is normal

(R 1).

This rule relates low AND normal .

There is nothing more than intersection operation AND,which we defined earlier.

As a result we get following membership function

µ1(z , v ) = µlow (z ) AND µnorm(v ) = min {µlow (z ), µnorm(v )}

This function is a general result.


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However, we are interested in outputs due to some specificinput z 0.

The degree of membership of z 0 in terms of the lowmembership function is µlow (z 0).

To obtain the output corresponding to the rule R 1 and inputz 0, we use AND operator to the specific value µlow (z 0) andµ1(z 0, v ) (general solution evaluated at z 0).


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Now we need to obtain the overall response in this fuzzysystem.

In other words we need to combine Q 1, Q 2 and Q 3 into one Q .

Originally: R 1 OR R 2 OR R 3.

Therefore complete fuzzy output is given by

Q (v ) = Q 1(v ) OR Q 2(v ) OR Q 3(v )


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OR was previously defines as a max operation

Therefore

Q (v ) = max {Q 1(v ), Q 2(v ), Q 3(v )}

= maxr

mins ,t

{µs (z 0), µt (v )}

where r = {1, 2, 3},s = {low , mid , high} andt = {norm, marg , fail }. Here s and t are paired combinations.

Although Q (v ) was developed specifically for this example, itwould be generalized to n rules.

Then r = {1, 2, · · · , n}.


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D e g r e e o f

m e m b e r s h i p

z0 is some specific input value

µ norm

(v) µ marg

(v) µ fail

(v)

µlow

(z0)

µmid

(z0)

µhigh

(z0)


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D e g r e e

o f m e m b e r s h i p

Q1(v)

Q2(v)

Q3(v)


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D e g r e e

o f m e m b e r s h i p

Q(v)


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For example, let us assume that z 0 = 0.7.

By counting center of gravity for this z 0 we obtain v 0 = 0.76.

If an engine has frequency value of 0.7 it is operating with a76% degree of abnormality.


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In this example rules were rather simple and had only onepart(IF...THEN)

If rules have more than one part, all parts have to be taken

into account in fuzzification.


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For example, let us assume that we have another variablecalled temperature .

Now we would have to define a membership function also fortemperature.

First rule could then go in a following way

IF frequency is low AND temperature is moderate

THEN motor is normal


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Membership function for the first rule would then be

µ1(z , t , v ) = µlow (z ) AND µmod (t ) AND µnorm(v )

= min {µlow (z ), µmod (t ), µnorm(v )}

Next we would select some specific frequency z 0 andtemperature t 0.


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What phases we just did?1. Fuzzify the inputs (µlow , µmid , µhigh)

2. Perform all required fuzzy logical operations (µ1, µ2, µ3).

3. Apply an implication method (Q 1, Q 2, Q 3).

4. Apply an aggregation method for fuzzy sets acquired inphase 3 (Q ).

5. Defuzzify the final output set (center of gravity).


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Nested functions are defined inside another function.

Nested functions are a relatively new feature and therefore

older versions of MATLAB dont support this.


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function y = tax(income)

adjusted income = income - 6000;

y = compute tax

function y = compute taxy = 0.28*adjusted income;

end

end


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ulow=@(z)1-sigmamf(z,0.27,0.47);umid=@(z)triangmf(z,0.24,0.50,0.74);

uhigh=@(z)sigmamf(z,0.53,0.73);

unorm=@(z)1-sigmamf(z,0.18,0.33);

umarg=@(z)trapezmf(z,0.23,0.35,0.53,0.69);ufail=@(z)sigmamf(z,0.59,0.78);

rules = {ulow; umid; uhigh; }

L=lambdafcns(rules);


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udark=@(z)1-sigmamf(z,0.35,0.5);

ugray=@(z)triangmf(z,0.35,0.5,0.65);

ubright=@(z)sigmamf(z,0.5,0.65);

udarker=@(z)bellmf(z,0.0,0.1);

umidgray=@(z)bellmf(z,0.4,0.5);

ubrighter=@(z)bellmf(z,0.8,0.9);


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Output membership functions(darker, midgray, brighter)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

Intensity of output figure(v)

D e g r e e o f m

e m b e r s h i p

µdarker

(v)

µgrayer(v)µ

brighter(v)


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% System response functions

rules={udark;ugray;ubright};

outmf={udarker,umidgray,ubrighter};

F=fuzzysysfcn(rules,outmf,[0 1]);

% Read image(will be of class uint8)

f image=imread(’saturnus.JPG’);

z=linspace(0,1,256);

% Construct the intensity transformation function

T=F(z);

% Transform the intensities of f image by using T

g image=intrans(f image,’specified’,T);

figure;

imshow(g image);


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In the following tables z i are marked as intensity values of a

pixel and d i = z i − z 5 are marked as the intensity differencebetween center pixel z 5.

z 1 z 2 z 3z 4 z 5 z 6z

7 z

8 z

9

d 1 d 2 d 3d 4 0 d 6d 7 d 8 d 9


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From the previous tables we can form following rules forboundary recognition

IF d 2 is zero AND d 6 is zero THEN make z 5 white




ELSE make z 5 black

To simplify the case only four of the neighbor pixels are used.


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We have four rules(+else) and four inputs.

Each row in the following matrix corresponds to one rule.

Next we need to define the membership functions for

zero(input), white(output) and black(output).


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Let us assume that our image has L different intensity levels.Then the intensity differences can range between −(L − 1)and L − 1.

It is clear that our output picture also has L different intensity

levels.In this example we fix L = 256

In other words we are using an image of class uint8


Input membership function(zero)

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−255 0 255

0

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Intensity differences(z)

D e g r e e o f m

e m b e r s h i p

µzero

(z)


Output membership functions(black and white)

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0

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Intensity of the output image(v)

D e g r e e o f m

e m b e r s h i p

µblack

(v)

µwhite(v)


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% MAKEFUZZYEDGES Script

% Input membership functions

zero=@(z)bellmf(z,-0.3,0);

not used=@(z)onemf(z);

% Output membership functions

black=@(z)triangmf(z,0,0,0.75);

white=@(z)triangmf(z,0.25,1,1);

% There are four rules and four inputs

inmf={zero, not used, zero, not used

not used, not used, zero, zeronot used, zero, not used, zero

zero, zero, not used, not used}


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function g=fuzzyfilt(f)

% Cast f onto floating point

[f,revertclass]=tofloat(f)

% Count the intensity differences

z1=imfilter(f,[0 -1 1],’conv’,’replicate’);

z2=imfilter(f,[0; -1; 1],’conv’,’replicate’);

z3=imfilter(f,[1;-1;0],’conv’,’replicate’);

z4=imfilter(f,[1 -1 0], ’conv’,’replicate’);

% Load the previously made script and use it

s=load(’fuzzyedgesys’);

g=s.G(z1,z2,z3,z4);

g=revertclass(g);


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In the previous pictures we saw that constant regions of theimage appeared gray.

When intensity differences are close to zero we apply theTHEN rules.

Output is then some constant between L−1

2 and L − 1, whatproduces the grayish tone to the image

This can be ”fixed” by using function mat2gray.


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