Texture Segmentation Using Mutual Information

The problem of texture segmentation is, at a high level, very similar to that of MRI classification that we considered in the previous chapter--essentially, we want to partition the image into mutually-exclusive and collectively-exhaustive sets in such a way that these partitions comprise stationarity Markov PDFs that are as compact as possible. For the current work, the number of partitions remains a free parameter of the system. For MRI classification, we modeled the intensities in each tissue class as an instance of a stationary MRF. For texture segmentation, this model continues to hold by the employed definition of a texture: regularity in Markov statistics. Therefore, we choose to employ the same optimality metric as before, i.e., the mutual information between the labels and the data.

Consider a discrete RV $L : T \rightarrow \Z$ 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.,

$$\forall t \in \mathcal{T}, P ({\bf Z}_t \vert L(t) = k) = P_k ({\bf Z}).$$ (152)

We define the optimal segmentation as the one that maximizes the mutual information between $L$ and $\Z$, i.e.,

$$I (L, {\bf Z}) = h ({\bf Z}) - h ({\bf Z} \vert L)$$

$$= h ({\bf Z}) - \sum_{k=1}^{K} P (L = k) h ({\bf Z} \vert L = k).$$ (153)

Thus, the optimal segmentation is

$$\mathop{\mbox{argmin }}_{ \{ \mathcal{T}_k \}_{k=1}^K } \Bigg( - ... ...l{T}\vert} \sum_{k=1}^K \sum_{t \in \mathcal{T}_k} \log P_k ({\bf z}_t) \Bigg).$$ (154)

This rather-large nonlinear optimization problem potentially has many local minima. Similar to the approach for MRI classification in the previous chapter, we impose smoothness constraints on the Markov PDFs via a suitable choice of the kernel-parameter $\sigma$. For texture segmentation, however, we have found that we need additional smoothness constraints on the boundaries of the segmented regions because of: (a) the higher variability in textures encountered in real images that does not conform very well with the stationary-ergodic Markov model, and (b) we do not use any prior information to obtain a good initial-segmentation estimate like the one for MRI brain tissue classification. To impose such regularizations, we can use standard variational formulations, such as the level-set framework [118,153,117]. Thus, we borrow from a rather extensive body of work on variational methods for image segmentation, in particular the Mumford-Shah model [110], its extensions to motion, depth, and texture, and its implementation via level-set flows [153,168,117].