User:Mim.cis/sandbox/MAP Estimation of Deformable Template

=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 ) d v $$ The EM algorithm takes as complte data the vector-field coordinates parameterizing the mapping, $$v_i,i=1,\dots$$ and compute iteratively the conditional-expectation

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