Incorporating Uncertainty

If the score maps given by the classifier are not extremely reliable, we need to balance between the reference structure locations constrained by the atlas and the evidence from score maps.

We do this in a Bayesian setting. Suppose the atlas gives a prior distribution for the structure's location, represented by $P(\theta)$. An observation distribution $P(s | \Omega)$ models classification uncertainty, where $\Omega$ is the set of voxels that belong to a particular structure, and $s$ is the computed score volume.

Let $\Omega(\theta)$ denote this structure's voxel locations transformed by $\theta$, and let $s$ denote the score volume, then the likelihood of the score volume is $P(s | \Omega(\theta))$. The optimal transform $\theta^*$ should maximize the posterior:

$$ \theta^* = \arg\max P(s | \Omega(\theta)) P(\theta).$$

Our formulation corresponds to the following specification of the two distributions:

Observation Distribution $$P(s | \Omega(\theta)) = \frac{1}{Z} \exp(f(\theta)),$$ where $Z$ is the normalization factor that can be ignored in optimization, and $f(\theta)$ is the score function defined as the total score over the relevant voxels, $$f(\theta) = \sum_{x \in \Omega(\theta)} s(x) .$$

Location Prior Distribution $$P(\theta) = \mathcal{N}(\theta; \theta_0, \Sigma ),$$ where $\theta_0$ is the structure location after global transformation of the atlas, and $\Sigma$ is the uncertainty matrix that should be specified according to the following principle: For dimensions in which the detector is uncertain, the prior should be more strongly imposed, and so the corresponding entries in $\Sigma$ should be small. For dimensions that the detector is very certain about, the prior's impact should be reduced, and so the corresponding entries should be large.

Comment: Implicitly, we translate uncertainty in texture into uncertainties in location and shape.

New Objective Function Take the logarithm of the posterior, we see the objective is equivalent to maximizing $$\theta^* = \arg\max f(\theta) - \eta (\theta - \theta_0)^T \Sigma^{-1} (\theta - \theta_0)$$

Incorporating the prior amounts to adding a regularization term to the score function to penalize deviation from the atlas-suggested location. $\eta$ controls the strength of penalty.

Specify Prior Uncertainty Based on Score Hessians One choice for the prior uncertainty matrix is $\Sigma = H$, where $H$ is the average hessian of the score function computed over many instances of the score maps of the structure. The hessians can be computed at the structure's global transformed location, $\theta_0$, or at the maxima of $f$.

The rational behind this choice is that the hessian of score function reflects the certainty of the detector in different dimensions. If the eigenvalue of the hessian corresponding to a particular dimension is consistently small, which means the score function is often flat in that dimension, then the detector is not trustworthy. In this case we assert the correctness of the prior estimate. On the other hand, if the eigenvalue is consistently large for a dimension, which means the score function is very peaked and the detector is confident, then we would like to reduce the impact of prior and increase the reliance of score functions.

Comment: I think it is important for $H$ to be an average computed from many subjects and treated as a priori information rather than some quantity that relies on the score maps of the current particular subject.