Segmentation of Striatal Brain Structures from High Resolution PET Images

Algorithm 1

Graph partition algorithm (Weighted Kernel 𝑘-means). This algorithm assigns the voxel 𝑎_𝑖 to the cluster 𝜋_𝑐 if the cluster 𝜋_𝑐 minimizes the distance function from point 2 [7].

Function [{𝜋_𝑐}^𝑘_𝑐=1]=Kernel_𝐾means(𝐾,𝑘,𝑊,𝑡_max,{𝜋_𝑐⁽⁰⁾}^𝑘_𝑐=1)

Input →

𝐾:Kernel matrix;

𝑘:number of clusters;

𝑊:sum of weights for each point;

𝑡_max=maximumnumberofiterations;

{𝜋_𝑐⁽⁰⁾}^𝑘_𝑐=1:intial cluster.

Output →

{𝜋_𝑐}^𝑘_𝑐=1:final clusters.

1. If no intial clustering is given, initialize the 𝑘 clusters (i.e., randomly).

Set the iteration counter 𝑡=0.

2. For each point 𝑎_𝑖 and every cluster 𝑐, compute:

𝑑(𝑎_𝑖,𝑚_𝑐)=𝐾_𝑖𝑖∑−2^{𝑎𝑗∈𝜋𝑐(𝑡)}𝑊_𝑗𝑘_𝑖𝑗∑_{𝑎𝑗∈𝜋𝑐}^(𝑡)𝑊_𝑗+∑_{𝑎𝑗,𝑎𝑙∈𝜋𝑐}^(𝑡)𝑊_𝑗𝑊_𝑙𝐾_𝑗𝑙(∑_{𝑎𝑗∈𝜋𝑐}^(𝑡)𝑊_𝑗)²

3. Find

𝑐^∗(𝑎_𝑖)=argmin_𝑐𝑑(𝑎_𝑖,𝑚_𝑐),resolvingtiesarbitrarily.

Compute the updated cluster

𝜋_𝑐^(𝑡+1)={𝑎∶𝑐^∗(𝑎_𝑖)=𝑐}.

4. If not converged or 𝑡_max>𝑡, set 𝑡=𝑡+1 and go to step 2;

Otherwise, stop and output the final clusters

{𝜋_𝑐}^𝑘_𝑐=1={𝜋_𝑐^(𝑡+1)}^𝑘_𝑐=1