Navigation

• Main Page
  

• All Pages
• Categories
• Random Page
• Navigation Paths
  

Technical Doc

• 바로씨(BaorSee)
• 고고피알
• Inply
• MWSF
• MWSM
• MWTC
• MWMC
• MWPC
• mQR
• mMap
• mGPIN
• mSiren
• mNotice
• μSearch
• MUST Ucc
  

• Create a new Page
• File Management
• Administration
• Create Account
  

Search the wiki
»

www.must.or.kr

Fuzzy set approaches

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
  

Introduce

• 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) μ_A(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 μ_A(z)는 X로부터 폐구간 [ 0, 1 ]의 한 사상(Mapping)

μ_A : 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 μ_A(z) = 0 (no membership)으로 하며, z가 A에 소속된 정도가 부분적일 때 0 < μ_A(z) < 1 (Partial membership) 값을 갖도록 나타낸다.

정의 2. 퍼지집합

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



Operation

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

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{μ_red(z), μ_mat(v)}
  

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

• 집계 퍼지 출력 집계
  - Q = Q₁ OR Q₂ OR Q₃
  - Q(v) = max_r{min_s{μ_s(z₀),μ_r(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”
  Q(1),Q(2)……Q(K)
  

• 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 μ_green(z), μ_verd(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