Bayesian model of computational anatomy

Computational anatomy (CA) is a discipline within medical imaging focusing on the study of anatomical shape and form at the visible or gross anatomical scale of morphology. 
The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, including medical imaging, neuroscience, physics, probability, and statistics. It focuses on the anatomical structures being imaged, rather than the medical imaging devices. 
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 use of diffeomorphic methods grew quickly to dominate the field of mapping methods post Christensen's
original paper, with fast and symmetric methods becoming available.
The main statistical model.
The central statistical model of Computational Anatomy in the context of medical imaging has been the source-channel model of Shannon theory; the source is the deformable template of images formula_1, the channel outputs are the imaging sensors with observables formula_2 (see Figure). The importance of the source-channel model is that the variation in the anatomical configuration are modelled separated from the sensor variations of the Medical imagery. The Bayes theory dictates that the model is characterized by the prior on the source, formula_3 on formula_4, and the conditional density on the observable
conditioned on formula_6.
In deformable template theory, the images are linked to the templates, with the deformations a group which acts on the template;
see group action in computational anatomy
For image action formula_7, then the prior on the group formula_8 induces the prior on images formula_9, 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.
The random orbit model of Computational Anatomy first appeared in 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. Such a random orbit model in which randomness on the group induces randomness on the images was examined for the Special Euclidean Group for object recognition in which the group element
formula_11 was the special Euclidean group in.
For the study of deformable shape in CA, the high-dimensional diffeomorphism groups used in computational anatomy are generated via smooth flows formula_12 which satisfy the Lagrangian and Eulerian specification of the flow fields satisfying the ordinary differential equation: 
with formula_13 the vector fields on formula_14 termed the Eulerian velocity of the particles at position formula_15 of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert space with the vector fields having 1-continuous derivative . For formula_16, the inverse of the flow is given by
and the formula_17 Jacobian matrix for flows in formula_18 given as formula_19
To ensure smooth flows of diffeomorphisms with inverse, the vector fields formula_14 must be at least 1-time continuously differentiable in space which are modelled as elements of the Hilbert space formula_21 using the Sobolev embedding theorems so that each element formula_22 has 3-square-integrable derivatives. Thus formula_21 embed smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:
where formula_24 with formula_25 a linear operator formula_26 defining the norm of the RKHS. The integral is calculated by integration by parts when formula_27 is a generalized function in the dual space formula_28.
Riemannian exponential.
In the random orbit model of computational anatomy, the entire flow is reduced to the initial condition which forms the coordinates encoding the diffeomorphism. From the initial condition formula_29 then geodesic positioning with respect to the Riemannian metric of Computational anatomy solves for the flow of the Euler-Lagrange equation.
Solving the geodesic from the initial condition formula_29 is termed the Riemannian-exponential, a mapping formula_31 at identity to the group.
The Riemannian exponential satisfies formula_32 for initial condition formula_33, vector field dynamics formula_34,
It is extended to the entire group, formula_41
Depicted in the accompanying figure is a depiction of the random orbits around each exemplar, formula_42, generated by randomizing the flow by generating the initial tangent space vector field at the identity formula_43, and then generating random object formula_44.
Shown in the Figure on the right the cartoon orbit, are a random spray of the subcortical manifolds generated by randomizing the vector fields formula_45 supported over the submanifolds. The random orbit model induces the prior on shapes and images formula_46 conditioned on a particular atlas formula_47. For this the generative model generates the mean field formula_48 as a random change in coordinates of the template according to formula_49, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows.
MAP estimation in the multiple-atlas orbit model.
The random orbit model induces the prior on shapes and images formula_46 conditioned on a particular atlas formula_51. For this the generative model generates the mean field formula_52 as a random change in coordinates of the template according to formula_53, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows. The prior on random transformations formula_54 on formula_55 is induced by the flow formula_56, with formula_57 constructed as a Gaussian random field prior formula_58. The density on the random observables at the output of the sensor formula_59 are given by
Maximum a posteriori estimation (MAP) estimation is central to modern statistical theory. Parameters of interest formula_61 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. Given the observed image formula_62, MAP estimation maximizes the posterior:
This requires computation of the conditional probabilities formula_64. The multiple atlas orbit model randomizes over the denumerable set of atlases formula_65. 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.
Dense emage matching.
Model formula_67 as a conditionally Gaussian random field conditioned, mean field, formula_68. For uniform variance the endpoint error terms plays the role of the log-conditional (only a function of the mean field) giving the endpoint term:
Landmark matching.
Model formula_69 as conditionally Gaussian with mean field formula_70, constant noise variance independent of landmarks. The log-conditional (only a function of the mean field) can be viewed as the endpoint term:
MAP segmentation based on multiple atlases.
The random orbit model for multiple atlases models the orbit of shapes as the union over multiple anatomical orbits generated from the group action of diffeomorphisms, formula_72, with each atlas having a template and predefined segmentation field formula_73. incorporating the parcellation into anatomical structures of the coordinate of the MRI.. The pairs are indexed over the voxel lattice formula_74 with an MRI image and a dense labelling of every voxel coordinate. The anatomical labelling of parcellated structures are manual delineations by neuroanatomists.
The Bayes segmentation problem is given measurement formula_75 with mean field and parcellation formula_76, the anatomical labelling formula_77. mustg be estimated for the measured MRI image. The mean-field of the observable formula_62 image is modelled as a random deformation from one of the templates formula_79, which is also randomly selected, formula_80. The optimal diffeomorphism formula_81 is hidden and acts on the background space of coordinates of the randomly selected template image formula_82. Given a single atlas formula_83, the likelihood model for inference is determined by the joint probability formula_84; with multiple atlases, the fusion of the likelihood functions yields the multi-modal mixture model with the prior averaging over models.
The MAP estimator of segmentation formula_85 is the maximizer formula_86 given formula_62, which involves the mixture over all atlases.
The quantity formula_89 is computed via a fusion of likelihoods from multiple deformable atlases, with formula_90 being the prior probability that the observed image evolves from the specific template image formula_91.
The MAP segmentation can be iteratively solved via the expectation-maximization algorithm
MAP estimation of volume templates from populations and the EM 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.
For the dense image volume case, given the observable formula_93 the problem is to estimate the template in the orbit of dense images formula_6. Ma's procedure takes an initial hypertemplate formula_95 as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism formula_96, with the parameters to be estimated the log-coordinates formula_97 determining the geodesic mapping of the hyper-template formula_98.
In the Bayesian random orbit model of computational anatomy the observed MRI images formula_99 are modelled as a conditionally Gaussian random field with mean field formula_100, with formula_101 a random unknown transformation of the template. The MAP estimation problem is to estimate the unknown template formula_102 given the observed MRI images.
Ma's procedure for dense imagery takes an initial hypertemplate formula_95 as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism formula_96. The observables are modelled as conditional random fields, formula_105 a random field with mean field formula_106. The unknown variable to be estimated explicitly by MAP is the mapping of the hyper-template formula_107, 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 algorithm.
The orbit-model is exploited by associating the unknown to be estimated flows to their log-coordinates formula_108 via the Riemannian geodesic log and exponential for computational anatomy the initial vector field in the tangent space at the identity so that formula_109, with formula_110 the mapping of the hyper-template.
The MAP estimation problem becomes
The EM algorithm takes as complete data the vector-field coordinates parameterizing the mapping, formula_108 and compute iteratively the conditional-expectation