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Fuzzy set approaches

Modified on 2010/06/12 14:44 by Administrator Categorized as Data Mining, Digital Image Processing

Fuzzy set approaches

  • Fuzzy logic uses truth values between 0.0 and 1.0 to represent the degree of membership (such as using fuzzy membership graph)
  • Attribute values are converted to fuzzy values
    - e.g., income is mapped into the discrete categories {low, medium, high} with fuzzy values calculated
  • For a given new sample, more than one fuzzy value may apply
  • Each applicable rule contributes a vote for membership in the categories
  • Typically, the truth values for each predicted category are summed


  • 컴퓨터를 인간에 가깝게 하는 일의 어려움
    - 퍼지 이론: 애매함을 처리하는 수리 이론

  • Fuzzy logic
    “X”가 “A”라는 집합 A(X)에 속하는 정도를 0과 1 사이의 숫자로 표현 예) μA(X)=0.7

  • Crisp logic
    - 전체 집합 X를 두 개의 Group, 즉 부분집합 A⊆X에 속하고 있는 요소와 속하고 있지 않는 요소에 이분하는 특성함수(characteristics function)에 의해 정의된다


Principles of Fuzzy Set Theory

정의 1. 소속함수

전체 집합 Z의 부분집합 A에 대한 소속함수 μA(z)는 X로부터 폐구간 [ 0, 1 ]의 한 사상(Mapping)

μA : Z → [ 0, 1 ]

으로서, z가 A에 소속된 정도가 0 ≤μA(z) ≤1 값을 나타낸다. 이 때, z가 A에 완전히 소속된 경우 μA(z) = 1 (full membership)로 하고, 소속되지 않은 경우 μA(z) = 0 (no membership)으로 하며, z가 A에 소속된 정도가 부분적일 때 0 < μA(z) < 1 (Partial membership) 값을 갖도록 나타낸다.

정의 2. 퍼지집합

Z가 속한 임의의 원소 각각에 대해 어떤 특정한 성질을 갖는 정도를 나타내는 소속함수 μA(z), 즉 μA : Z → [0, 1]가 정의된다고 하자. 이 경우, 순서쌍의 집합 A = {(z, μA(x))|z∈ Z } 를 소속함수 μA(z)를 갖는 fuzzy set 이라고 한다.



  • Empty : Membership function is identically zero in Z
  • Equality: Two fuzzy set A, B are equal (μA(z) = μB(z) for al z ∈ Z )
  • Subset : A fuzzy set A is subset of a fuzzy set B (μA(z) ≤ μB(z))
  • Complement, Union, Intersection


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Using Fuzzy Sets

  • 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




  • General result involving two membership functions.
    - μ3(z,v) = min{μred(z), μmat(v)}

  • Fuzzy output due to rule R3 and specific input
    - Q3(v) = min{μred(z0), μ3(z0,v)}
    - Q2(v) = min{μyellow(z0), μ2(z0,v)}
    - Q1(v) = min{μgreen(z0), μ1(z0,v)}


  • 집계 퍼지 출력 집계
    - Q = Q1 OR Q2 OR Q3
    - Q(v) = maxr{minss(z0),μr(z0,v)}}
    r = {1,2,3} , s={green, yellow, red}

  • Defuzzification
    - Obtain a crisp output v0 , from fuzzy set Q
    - Way to defuzzify Q to obtain a crisp output is “center of gravity”


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  • Rule-based fuzzy logic step
    - Fuzzify the inputs
    - Perform any required fuzzy logical operations
    - Apply an implication method
    - Apply an aggregation method
    - Defuzzify the final output fuzzy set

  • Rule’s Short hand notation (variable, fuzzy set)
    - Ex) IF the color is green THEN the fruit is verdant
    → IF(z, green) THEN (v, verdant)
    ㆍ v , z color and degree of maturity
    ㆍ Green , verdant is fuzzy set (defined by membership function μgreen(z), μverd(v)
    - M IF-THEN rules, N input variables, one output variable v


Using Fuzzy Sets for Intensity Transformations

  • Singletons
    - membership functions are constant
    - Significantly reduces computational requirement



(a)Low-contrast image (b) Result of using fuzzy, rule-based contrast enhancement

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