Learning Per-Class Markov Statistics Nonparametrically

The proposed method constructs a segmentation strategy based on a Markov statistical image model [99] that it learns automatically from the input data. It formulates the segmentation problem as an optimization problem to maximize the dependency or mutual information [34] between the segmentation labels and the Markov image statistics.

The proposed approach models brain MRI images as derived from piecewise stationary-ergodic MRFs. For brain MR images, the Markov PDFs at voxels in individual parts of the brain, such as white matter or gray matter, are similar and, hence, the piecewise-stationary model holds to some degree. Indeed, the successful high-quality classifications produced by the proposed method corroborate this claim.

Consider a discrete RV $L : T \rightarrow \Z$, where $\Z$ is the set of integers, that maps each voxel $t \in \mathcal{T}$ to the class it belongs to, i.e., $L(t) = k$ if voxel $t$ is in class $k$. Let $\{\mathcal{T}_k\}_{k=1}^K$ denote a mutually-exclusive and collectively-exhaustive decomposition of the image domain $\mathcal{T}$ into $K$ regions--assumed stationary--such that $\mathcal{T}_k = \{ t \in \mathcal{T} : L(t) = k \}$. The stationarity assumption implies that for each class $k$ the Markov PDFs are exactly the same, i.e.,

$$\forall k = 1, 2, \ldots, K, \forall t \in \mathcal{T}, P ({\bf Z}_t \vert L(t) = k) = P_k ({\bf Z}).$$ (136)

Based on the piecewise stationary-ergodic assumption, the Parzen-window density estimate gives the PDF for class $k$ as

$$P_k ({\bf z}) \approx \frac {1} {\vert\mathcal{A}\vert} \sum_{s \in \mathcal{A}} G_d ({\bf z} - {\bf z}_s, \Psi_d),$$ (137)

where the set $\mathcal{A}$ is a small subset of $\mathcal{T}_k$ chosen at random for each voxel $t_k$.