Bayesian model of computational anatomy

The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, including medical imaging, neuroscience, physics, probability, and statistics.

The central focus of the sub-field of computational anatomy within medical imaging is mapping information across anatomical coordinate systems most often dense information measured within a magnetic resonance image (MRI).

The introduction of flows into CA, which are akin to the equations of motion used in fluid dynamics, exploit the notion that dense coordinates in image analysis follow the Lagrangian and Eulerian equations of motion.

In models based on Lagrangian and Eulerian flows of diffeomorphisms, the constraint is associated to topological properties, such as open sets being preserved, coordinates not crossing implying uniqueness and existence of the inverse mapping, and connected sets remaining connected.

The Bayes theory dictates that the model is characterized by the prior on the source,

, written as densities the log-posterior takes the form The random orbit model which follows specifies how to generate the group elements and therefore the random spray of objects which form the prior distribution.

The random orbit model of Computational Anatomy first appeared in[4][5][6] modelling the change in coordinates associated to the randomness of the group acting on the templates, which induces the randomness on the source of images in the anatomical orbit of shapes and forms and resulting observations through the medical imaging devices.

For the study of deformable shape in CA, the high-dimensional diffeomorphism groups used in computational anatomy are generated via smooth flows

To ensure smooth flows of diffeomorphisms with inverse, the vector fields

[8][9] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm: where

In the random orbit model of computational anatomy, the entire flow is reduced to the initial condition which forms the coordinates encoding the diffeomorphism.

then geodesic positioning with respect to the Riemannian metric of Computational anatomy solves for the flow of the Euler-Lagrange equation.

The random orbit model induces the prior on shapes and images

, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows.

The random orbit model induces the prior on shapes and images

, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows.

take many forms including (i) disease type such as neurodegenerative or neurodevelopmental diseases, (ii) structure type such as cortical or subcortical structures in problems associated to segmentation of images, and (iii) template reconstruction from populations.

, MAP estimation maximizes the posterior: This requires computation of the conditional probabilities

The model on images in the orbit take the form of a multi-modal mixture distribution The conditional Gaussian model has been examined heavily for inexact matching in dense images and for landmark matching.

The anatomical labelling of parcellated structures are manual delineations by neuroanatomists.

is hidden and acts on the background space of coordinates of the randomly selected template image

The MAP segmentation can be iteratively solved via the expectation–maximization algorithm 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.

as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism

In the Bayesian random orbit model of computational anatomy the observed MRI images

Ma's procedure for dense imagery takes an initial hypertemplate

as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism

, with the other mappings considered as nuisance or hidden variables which are integrated out via the Bayes procedure.

The orbit-model is exploited by associating the unknown to be estimated flows to their log-coordinates

via the Riemannian geodesic log and exponential for computational anatomy the initial vector field in the tangent space at the identity so that