MAP Estimation of Dense Volume Template from Population Using the EM Algorithm

Shape statistics in CA are local, locally defined relative to template coordinates. Generating templates empirically from populations is a fundamental operation ubiquitous to the discipline. Several methods based on Bayesian statistics have emerged for submanifolds and dense image volumes. For the dense image volume case, given the observable $I^{D_{1}},I^{D_{2}},\dots$ the problem is to estimate the template in the orbit of dense images $I\in {\mathcal {I}}$ . Ma's procedure takes an initial hypertemplate $I_{0}\in {\mathcal {I}}$ as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism $I\doteq \phi ^{0}\cdot I_{0}$ .

The observable are modelled as conditional random fields, $I^{D_{i}}$ a Gaussian random field with mean field $\phi ^{i}\cdot \phi ^{0}\cdot I_{0}$ . The unknown variable to be estimated explicitly by MAP is the mapping of the hyper-template $\phi ^{0}$ , with the other mappings considered as nuisance or hidden variables which are integrated out via the Bayes procedure. This is accomplished using the expectation-maximization EM algorithm.

The orbit-model is exploited by associating the unknown flows to their log-coordinates, the initial vector field in the tangent space at the identy so that $Exp_{id}(v^{i})\doteq \phi ^{i}$ , with $Exp_{id}(v^{0})$ the mapping of the hyper-template. The MAP estimation problem becomes

\max _{v^{0}}p(I^{D},\theta =v^{0})=\int p(I^{D},\theta =v^{0})\pi (v^{1},v^{2},\dots )dv

The EM algorithm takes as complte data the vector-field coordinates parameterizing the mapping, $v_{i},i=1,\dots$ and compute iteratively the conditional-expectation

$Q(\theta =v^{0};\theta ^{old})=E(\log p(I^{D},\theta =v^{0}|v^{1},v^{2},\dots )|I^{D},\theta ^{old})$