Classification Algorithm

From the Markov PDFs, which are estimated from the initial classification, we reassign voxels based on optimizing the information content of the labels. We observe that the energy in (6.9) can be reduced, based on a steepest-descent strategy, if each voxel $t$ is assigned to the class $k$ that maximizes the probability $P_k ({\bf z}_t)$. This is an iterative process where the Markov PDFs define a classification that, in turn, redefines the PDFs. Because the PDFs get implicitly redefined after every iteration, via the updated classification, the PDF estimates lag, so to speak, the classification. We have found this to be an acceptable approximation, although some recent work [17] introduces an additional term in the update rule to avoid this lag.

Given a classification $\{ \mathcal{T}_k^m = \{ t \in \mathcal{T}: L^m_t = k \} \}_{k=1}^K$ at iteration $m$, the algorithm iterates as follows:

1. 
   For $k= 1, 2, 3, 4$ and $\forall t \in \mathcal{T}$, estimate $P^m_k ({\bf z}_t)$ nonparametrically, as described in Section 6.2.
   
2. 
   Update the classification labels:
   

   $$\forall t \in \mathcal{T}: L^{m+1}_t = \mathop{\mbox{argmax }}_{k} P^m_k ({\bf z}_t).$$ (148)

   
   
   
   
3. 
   Stop upon convergence, i.e., when $\sum_{t \in \mathcal{T}} \delta (L^{m+1}(t) - L^m(t)) < \epsilon$, where $\delta (\cdot)$ is the Kronecker-delta (unit impulse) function and $\epsilon$ is a small threshold.