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

Algorithm 2

The algorithm of proposed method.

Phase 1: Initializing Segmentation by the GKFCM clustering

(I) Parameters initialization: data set 𝑋=𝑥₁,𝑥₂,…,𝑥_𝑛; cluster number 𝑐(2≤𝑐≤𝑛); 𝜀>0;𝑚=2;

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

(III) Compute 𝜂_𝑖^(𝑠) using

      𝜂_𝑖=min_{𝑖′≠𝑖}(1−𝐾(𝑎_𝑖′,𝑎_𝑖))max_𝑘1−𝐾𝑎_𝑘,𝑥

(IV) Compute 𝜇^(𝑠) with 𝑎^(𝑠−1) and 𝜂_𝑖^(𝑠) by

  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)

(V) 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

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

(VII) Extract ROI’s fuzzy membership matrix, that is, U_MF.

Phase 2: Localizing the initial level set through defuzzification process

(I) Convert the fuzzy partition matrix U_MF to a crisp partition by assigning the pixel 𝑖 to the class 𝐶 with the highest

membership 𝐶_𝑖=arg_𝑘max𝑈_𝑀𝐹_𝑘𝑖∣𝑘=1,2,…,𝑐.

(II) Select the best matching 𝐶_𝑘 of the ROI.

(III) Compute: 𝜑⁰𝑥,𝑦=−𝑐₀×𝐶_𝑘+(1−𝐶_𝑘)×𝑐₀; (𝑐₀=4)

Phase 3: Curve evolution using LSE_BFE segmentation

(I) Set value: 𝑘=1,𝜏=100, 𝜇=0.2/𝜏,𝜆=1/𝜏 and 𝜈=1.

(II) Update 𝑐₁ and 𝑐₂ by (*is the convolution operation):

  𝑐₁(𝑥)=∫(𝑏∗𝐾_𝜎(𝑥))𝐻𝜑(𝑦)⋅𝐼(𝑥)𝑑𝑦∫(𝑏²∗𝐾_𝜎(𝑥))𝐻𝜑(𝑦)𝑑𝑦,𝑐₂(𝑥)=∫(𝑏∗𝐾_𝜎(𝑥))1−𝐻𝜑𝑦⋅𝐼(𝑥)𝑑𝑦∫𝑏²∗𝐾_𝜎(𝑥)1−𝐻𝜑𝑦𝑑𝑦

(III) Update level set function by (11) and

  𝜑^𝑘+1𝑥,𝑦=𝜑^𝑘𝑥,𝑦+𝜏𝜆𝛿𝜑⋅div∇𝜑∇𝜑+𝜇Δ𝜑−div∇𝜑𝛿𝜑⋅𝑒∇𝜑+𝜈₁(𝑥)−𝑒₂(𝑥)

(IV) Update the bias field by

   𝐼(𝑥)⋅𝑐𝑏(𝑥)=¹⋅𝐻𝜑(𝑥)+𝑐₂⋅1−𝐻𝜑(𝑥)∗𝐾_𝜎(𝑐₁²⋅𝐻𝜑(𝑥)+𝑐₂²⋅1−𝐻𝜑(𝑥)∗𝐾_𝜎 where 𝐻(𝜑₀)=1,𝜑₀≥00,otherwise

 is Heaviside function.

(V) If 𝜑^𝑘+1−𝜑^𝑘<𝜀 then STOP and OUTPUT Else 𝑘=𝑘+1 and return to step II in this phase.