Improvement of 2-Partition Entropy Approach Using Type-2 Fuzzy Sets for Image Thresholding

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2. Fuzzy 2-Partition Entropy Technique

In the fuzzy 2-partition entropy approach proposed in (Cheng, 1998), an image is modelled by 2 fuzzy sets which have membership functions and there is no sharp boundary between these sets (Assas, 2012). An image is modelled by two fuzzy sets dark and bright, whose membership functions are defined as follows:

(1)(2)

Where x is the independent variable and a and c are parameters determining the shape of the above two membership functions.

We assume that the images have 256 gray levels ranging from 0 to 255. Then, an exhaustive search is used to find the pair aopt and copt which forms a fuzzy 2-partition that has the maximum entropy as follows:For a = 0 to254 For c = (a+1) to255

• 1.

  For given a and c. new membership functions µd(i) and µb(i) are computed, for i =0, ...,255.

• 2.

  Probabilities of the two fuzzy events dark and bright are defined as:

  
  (3)
  
  (4)

where P(i)is the probability of the occurrence of the gray level i =0, ...,255.

• 3.

  The entropy of this fuzzy 2-partition is given by:

  
  (5)

• 4.

  The selected threshold value Topt which is the mid- point of aopt and copt has to satisfy the following criterion function:

  
  (6)

Where