Bayesian estimation of templates in computational anatomy: Difference between revisions

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{{Further|LDDMMLarge |deformation diffeomorphic metric mapping|Bayesian model of computational anatomy}}

{{COI|date=December 2017}}

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In the [[Computational anatomy#The random orbit model of computational anatomy|Bayesian random orbit model of computational anatomy]] the observed MRI images <math>I^{D_i}</math> are modelled as a conditionally Gaussian random field with mean field <math>\phi_i \cdot I</math>, with <math>\phi_i</math> a random unknown transformation of the template. The MAP estimation problem is to estimate the unknown template <math> I \in \mathcal{I}</math> given the observed MRI images.

Ma's procedure for dense imagery takes an initial hypertemplate <math> I_0 \in \mathcal{I} </math> as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism <math> I \doteq \phi_0 \cdot I_0 </math>. The observables are modelled as conditional random fields, <math> I^{D_i} </math> a {{EquationNote|conditional-Gaussian}} random field with mean field <math> \phi_i \cdot I \doteq \phi_i \cdot \phi_0 \cdot I_0 </math>. The unknown variable to be estimated explicitly by MAP is the mapping of the hyper-template <math> \phi_0</math>, 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]]expectation–maximization algorithm|expectation–maximization (EM) algorithm]].

The orbit-model is exploited by associating the unknown to be estimated flows to their log-coordinates <math>v_i,i=1,\dots</math> [[Computational anatomy#Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates)|via the Riemannian geodesic log and exponential]] for [[computational anatomy]] the initial vector field in the tangent space at the identity so that <math> \mathrm{Exp}_\mathrm{id}(v_{i}) \doteq \phi_i </math>, with <math> \mathrm{Exp}_\mathrm{id}(v_{0}) </math> the mapping of the hyper-template.