Bayesian Classification with Probabilistic-Atlas Priors

The registered, probabilistic atlas plays another role in the proposed classification algorithm. Instead of using data-driven probabilities alone for the classification updates, we can employ a Bayesian estimation strategy to compute the probabilities. The likelihood terms are the data-driven probabilities $P_k ({\bf z}_t)$ that we have computed via Parzen-window density estimation. The posterior is therefore the likelihood multiplied by the prior $P^a_k (t)$, which we derive from the probabilistic atlas. The Bayesian label updates are based on the MAP estimate:

$$\mathop{\mbox{argmax }}_{k} P (L = k \vert {\bf z}_t, t) = \mathop{\mbox{argmax }}_{k} \bigg( P ({\bf z}_t \vert L = k, t) P (L = k \vert t) \bigg)$$

$$= \mathop{\mbox{argmax }}_{k} \bigg( P_k ({\bf z}_t) P^a_k (t) \bigg).$$ (149)

For the proposed method, our empirical evidence suggests that using the atlas directly as a prior can strongly dominate the likelihood and introduce systematic biases in the classification. Pohl et al. [130] report similar findings with a direct use of an atlas prior. For instance, for regions where the prior probability is zero, or near zero, the likelihood can have little effect. In such a case, the final segmentation may be very much like the initialization. Such behavior is likely an artifact from either (a) the limited variability in the atlas resulting from a limited-size population, or (b) the degree of misfit that remains after the registration process during atlas construction. In practice, the prior strictly interpreted from the atlas is too strong, and we have investigated two ways of weakening its affect on the final solution. Section 6.5.2 discusses empirical results and the effect of different priors on the proposed method in more detail. Section 6.5.2 shows the performance with both these priors.

One way of weakening the atlas prior is to use the atlas for discriminating only between two tissue types, namely the brain and nonbrain tissue. In this way, the prior does not interfere with the more subtle distinctions between the different brain tissues. For this, we sum the atlas probabilities for the white matter, gray matter, and cerebrospinal fluid to create one composite atlas that only gives the spatial probability for any kind of brain tissue. This is equivalent to redefining $P^a_k (t), \forall t \in \mathcal{T}$ as

$$\mathrm {For } k=1,2,3, \forall t \in \mathcal{T}: P^a_k (t) = 1 - P^a_4 (t)$$ (150)

We call this the 2-class prior.

Another way of reducing the strength of the prior is to voxel-wise rescale the atlas probabilities in such a way that the probabilities continue to add up to one but are less discriminating between the tissue types. We have used the following function for the desired effect.

$$\mathrm {For } k=1,2,3,4, \forall t \in \mathcal{T}: P^a_k (t) = \frac {1 - v} {4} + v P^a_k (t),$$ (151)

where $v \in [ 0,1 ]$ is a free parameter. The redefined prior probabilities continue to add up to unity: $\forall t \in \mathcal{T}: \sum_{k=1}^4 P^a_k (t) = 1$. A value of $v = 1$ makes no change to the atlas probabilities, whereas $v = 0$ makes every class equiprobable. In this chapter we provide experimental results with a moderate value of $v = 0.5$. We call this the scaled-atlas prior.