]>Segmentation of Striatal Brain Structures from High Resolution PET Images : Algorithm 1
Function [ { 𝜋 𝑐 } 𝑘 𝑐 = 1 ] = K e r n e l 𝐾 m e a n s ( 𝐾 , 𝑘 , 𝑊 , 𝑡 m a x , { 𝜋 𝑐 ( 0 ) } 𝑘 𝑐 = 1 )
Input →
𝐾 :Kernel matrix;
𝑘 :number of clusters;
𝑊 :sum of weights for each point;
𝑡 m a x = m a x i m u m n u m b e r o f i t e r a t i o n s ;
{ 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 𝑖 𝑖 ∑ − 2 𝑎 𝑗 ∈ 𝜋 𝑐 ( 𝑡 ) 𝑊 𝑗 𝑘 𝑖 𝑗 ∑ 𝑎 𝑗 ∈ 𝜋 𝑐 ( 𝑡 ) 𝑊 𝑗 + ∑ 𝑎 𝑗 , 𝑎 𝑙 ∈ 𝜋 𝑐 ( 𝑡 ) 𝑊 𝑗 𝑊 𝑙 𝐾 𝑗 𝑙 ( ∑ 𝑎 𝑗 ∈ 𝜋 𝑐 ( 𝑡 ) 𝑊 𝑗 ) 2
3. Find
𝑐 ∗ ( 𝑎 𝑖 ) = a r g m i n 𝑐 𝑑 ( 𝑎 𝑖 , 𝑚 𝑐 ) , r e s o l v i n g t i e s a r b i t r a r i l y .
Compute the updated cluster
𝜋 𝑐 ( 𝑡 + 1 ) = { 𝑎 ∶ 𝑐 ∗ ( 𝑎 𝑖 ) = 𝑐 } .
4. If not converged or 𝑡 m a x > 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
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].