A New Kernel-Based Fuzzy Level Set Method for Automated Segmentation of Medical Images in the Presence of Intensity Inhomogeneity

Algorithm 1

The algorithm of GKFCM clustering.

Input:

(I) The data set: 𝑋=𝑥₁,𝑥₂,…,𝑥_𝑛, 𝑥_𝑖∈𝑅^𝑠

(II) The number of clusters: 𝑐where2≤𝑐≤𝑛.

(III) The stopping criterion of algorithm: 𝜀>0

(IV) The initials of cluster centers: 𝑎⁽⁰⁾=(𝑎₁⁽⁰⁾,𝑎₂⁽⁰⁾,…,𝑎_𝑐⁽⁰⁾)

Output:

(I) The final cluster centers: 𝑎=𝑎₁,𝑎₂,…,𝑎_𝑐

(II) The final membership function: 𝑈_𝑀𝐹=𝜇_𝑖𝑗∣𝑖=1,…,𝑐;𝑗=1,…,𝑛;

Algorithm:

(a) Initialize 𝑐,𝑚>1, 𝜀>0 for positive constants.

(b) Let 𝑠=1 and estimate 𝜎² by 𝜎²=∑^𝑛_𝑗=1𝑥_𝑗−𝑥²/𝑛 where 𝑥=∑^𝑛_𝑗=1𝑥_𝑗/𝑛

(c) Compute 𝜂_𝑖^(𝑠) using

       𝜂_𝑖=min_{𝑖′≠𝑖}1−𝐾𝑎_𝑖′,𝑎_𝑖max_𝑘1−𝐾𝑎_𝑘,𝑥

(d) Compute 𝜇^(𝑠) with 𝑎^(𝑠−1) and 𝜂_𝑖^(𝑠) using

  𝜇_𝑖𝑗=1−𝐾𝑥_𝑗,𝑎_𝑖+𝜂_𝑖1−𝐾𝑥_𝑗,𝑎_𝑖^{−1/(𝑚−1)}∑^𝑐_𝑘=11−𝐾𝑥_𝑗,𝑎_𝑖+𝜂_𝑖1−𝐾𝑥_𝑗,𝑎_𝑖^{−1/(𝑚−1)},𝑖=1,…,𝑐;𝑗=1,…,𝑛;𝐾𝑥,𝑦=exp(−𝑥−𝑦²/𝜎²)

(e) Update 𝑎^(𝑠) with 𝑎^(𝑠−1), 𝜇^(𝑠) and 𝜂_𝑖^(𝑠) using

       𝑎_𝑖=∑^𝑛_𝑗=1𝜇^𝑚_𝑖𝑗𝐾𝑥_𝑗,𝑎_𝑖𝑥_𝑗+𝜂_𝑖𝐾𝑥_𝑗,𝑎_𝑖𝑥_𝑗∑^𝑛_𝑗=1𝜇^𝑚_𝑖𝑗𝐾𝑥_𝑗,𝑎_𝑖+𝜂_𝑖𝐾𝑥_𝑗,𝑎_𝑖,𝑖=1,…,𝑐

(f) If 𝑎^(𝑠)−𝑎^(𝑠−1)<𝜀 STOP and OUTPUT

 Else 𝑠=𝑠+1 and return to step c in Algorithm.