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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
Spring 2008 ELEN 4304/5365 DIP 2
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=
Spring 2008 ELEN 4304/5365 DIP 3
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:
Spring 2008 ELEN 4304/5365 DIP 4
{(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.
Spring 2008 ELEN 4304/5365 DIP 5
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
Spring 2008 ELEN 4304/5365 DIP 6
7/28/2019 03-4 - Fuzzy techniques.pdf
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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
Spring 2008 ELEN 4304/5365 DIP 7
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
IF z A AND z A AND AND z A THEN v B
IF z A AND z A AND AND z A THEN v B
( , )ELSE v B
Spring 2008 ELEN 4304/5365 DIP 21
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
+ + =
+ +
Spring 2008 ELEN 4304/5365 DIP 24
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