Bayesian estimation of templates in computational anatomy

Template estimation in computational anatomy from populations of observations is a fundamental operation ubiquitous to the discipline.

Several methods for template estimation based on Bayesian probability and statistics in the random orbit model of CA have emerged for submanifolds[1][2] and dense image volumes.

[3] Linear algebra is one of the central tools to modern engineering.

In computational anatomy the space of all shapes and forms is modeled as an orbit similar to the vectors in linear-algebra, however the groups do not act linear as the matrices do, and the shapes and forms are not additive.

In computational anatomy addition is essentially replaced by the law of composition.

The central group acting CA defined on volumes in

The high-dimensional diffeomorphism groups used in computational anatomy are generated via smooth flows

with the vector fields termed the Eulerian velocity of the particles at position of the flow.

as a reproducing kernel Hilbert space (RKHS), with the norm defined by a 1-1, differential operator

includes derivatives from the differential operator implying smoothness of the vector fields.

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

[6][7] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:The central statistical model of computational anatomy in the context of medical imaging is the source-channel model of Shannon theory;[8][9][10] the source is the deformable template of images

The Bayes theory models the prior on the source of images

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

Since the original publications by Csernansky and colleagues of hippocampal change in Schizophrenia,[14][15][16][17] Alzheimer's disease,[18][19][20] and Depression,[21][22] many neuroanatomical shape statistical studies have now been completed using templates built from all of the subcortical structures for depression,[23] Alzheimer's,[11][12][24][25][26][27] Bipolar disorder, ADHD,[28] autism,[29] and Huntington's Disease.

[32] Shown in the accompanying Figure is an example of subcortical structure templates generated from T1-weighted magnetic resonance imagery by Tang et al.[11][12][13] for the study of Alzheimer's disease in the ADNI population of subjects.

Numerous studies have now been done on cardiac hypertrophy and the role of the structural integraties in the functional mechanics of the heart.

[34][35][36] The figure on the right shows the computational cardiac anatomy method being used to identify regional differences in radial thickness at end-systolic cardiac phase between patients with hypertrophic cardiomyopathy (left) and hypertensive heart disease (right).

Color map that is placed on a common surface template (gray mesh) represents region ( basilar septal and the anterior epicardial wall) that has on average significantly larger radial thickness in patients with hypertrophic cardiomyopathy vs. hypertensive heart disease (reference below).

[33] 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.

the problem is to estimate the template in the orbit of dense images

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

Showing the Lagrangian flow of coordinates with associated vector fields satisfying ordinary differential equation .
Shown are shape templates of amygdala, hippocampus, and ventricle generated from 754 ADNI samples` Top panel denotes the localized surface area group differences between normal aging and Alzheimer disease (positive represents atrophy in Alzheimer whereas negative suggests expansion). Bottom panel denotes the group differences in the annualized rates of change in the localized surface areas (positive represents faster atrophy rates (or slower expansion rates) in Alzheimer whereas negative suggests faster expansion rates (or slower atrophy rates) in Alzheimer); taken from Tang et al. [ 11 ] [ 12 ] [ 13 ]
Showing population heart atlases with superimposed hypertrophy.
Showing population atlases identifying regional differences in radial thickness at end-systolic cardiac phase between patients with hypertrophic cardiomyopathy (left) and hypertensive heart disease (right). Gray mesh shows the common surface template to the population, with the color map representing basilar septal and anterior epicardial wall with larger radial thickness in patients with hypertrophic cardiomyopathy vs. hypertensive heart disease. [ 33 ]