Bayesian estimation of templates in computational anatomy: Difference between revisions
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{{Further
{{COI|date=December 2017}}
▲{{Further information|LDDMM | Bayesian model of computational anatomy}}
{{Main article|Computational anatomy}}▼
[[Statistical shape analysis]] and [[Computational anatomy#Statistical shape theory in computational anatomy|statistical shape theory]] in [[computational anatomy]] (CA) is performed relative to templates, therefore it is a local theory of statistics on shape. [[Computational anatomy#Template
== The deformable template model of shapes and forms via diffeomorphic group actions ==
{{Further
The central group acting CA defined on volumes in <math>{\mathbb R}^3</math> are the [[diffeomorphisms]] <math>\mathcal{G} \doteq Diff</math> which are mappings with 3-components <math>\phi(\cdot) = (\phi_1(\cdot),\phi_2 (\cdot),\phi_3 (\cdot))</math>, law of composition of functions <math> \phi \circ \phi^\prime (\cdot)\doteq \phi (\phi^\prime(\cdot)) </math>, with inverse <math> \phi \circ \phi^{-1}(\cdot) =\phi ( \phi^{-1}(\cdot))= id</math>.
[[Group (mathematics)|Group]]s and [[
A popular [[
:<math>
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</math>
For sub-[[manifold]]
:<math>
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== Geodesic positioning via the Riemannian exponential ==
For the study of deformable shape in CA, a more general diffeomorphism group has been the group of choice, which is the infinite dimensional analogue. The high-dimensional diffeomorphism groups used in computational anatomy are generated via smooth flows <math> \phi_t, t \in [0,1] </math> which satisfy the Lagrangian and Eulerian specification of the flow fields satisfying the ordinary differential equation: [[File:Lagrangian flow.png|thumb|Showing the Lagrangian flow of coordinates <math>x \in X</math> with associated vector fields <math>v_t, t \in [0,1]</math> satisfying ordinary differential equation <math>\dot \phi_t = v_t(\phi_t), \phi_0=id</math>.]]
\frac{d}{dt} \phi_t = v_t \circ \phi_t , \ \phi_0 = id \ ; </math>|{{EquationRef|Lagrangian flow}}}}
with <math> v \doteq (v_1,v_2,v_3) </math> the vector fields on <math> {\mathbb R}^3 </math> termed the [[Lagrangian and Eulerian specification of the flow field|Eulerian]] velocity of the particles at position <math>\phi</math> of the flow. The vector fields are functions in a function space, modelled as a smooth [[Hilbert space|Hilbert]] space with the vector fields having 1-continuous derivative . For <math>v_t = \dot \phi_t \circ \phi_t^{-1}, t \in [0,1]</math>, with the inverse for the flow given by
\frac{d}{dt} \phi_t^{-1} = -(D \phi_t^{-1}) v_t, \ \phi_0^{-1} = id \ , </math>|{{EquationRef|Eulerianflow}}}}
and the <math>3 \times 3</math> Jacobian matrix for flows in <math>\mathbb{R}^3</math> given as <math> \ D\phi \doteq \left(\frac{\partial \phi_i}{\partial x_j}\right). </math>
Flows were first introduced<ref>GE Christensen, RD Rabbitt, MI Miller, Deformable templates using large deformation kinematics, IEEE Trans Image Process. 1996;5(10):1435-47.</ref><ref>GE Christensen, SC Joshi, MI Miller, Volumetric transformation of brain anatomy
IEEE Transactions on Medical Imaging,1997.</ref> for large deformations in image matching; <math>\dot \phi_t(x)</math> is the instantaneous velocity of particle <math>x</math> at time <math>t</math>. with the vector fields termed the Eulerian velocity of the particles at position of the flow. The modelling approach used in CA enforces a continuous differentiability condition on the vector fields by modelling the space of vector fields <math>(V, \| \cdot \|_V )</math> as a [[reproducing kernel Hilbert space]] (RKHS), with the norm defined by a 1-1, differential operator<math> A: V \rightarrow V^* </math>, Green's inverse <math>K = A^{-1}</math>. The norm according to <math> \| v\|_V^2 \doteq \int_X Av \cdot v dx , v \in V,
</math> where for <math> \sigma(v) \doteq Av \in V^*
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== The Bayes model of computational anatomy ==
The central statistical model of [[computational anatomy]] in the context of [[medical imaging]] is the source-channel model of [[Shannon theory]];<ref>{{Cite journal|title = Statistical methods in computational anatomy
:<math>
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</math>
[[Maximum a posteriori estimation]] (MAP) estimation is central to modern [[statistical theory]]. Parameters of interest <math> \theta \in \Theta </math> take many forms including (i) disease type such as [[neurodegenerative]] or [[neurodevelopmental]] diseases, (ii) structure type such as cortical or
:<math>
\hat \theta \doteq \arg \max_{\theta \in \Theta} \log p(\theta \mid I^D). </math>
[[File:Xiaoying Tang ADNI template.png|thumb|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.<ref name="Tang 599–611">{{Cite journal|
]]This requires computation of the conditional probabilities <math>p(\theta\mid I^D) = \frac{p(I^D,\theta)}{p(I^D)}</math>. The multiple atlas orbit model randomizes over the denumerable set of atlases <math>\{ I_a, a \in \mathcal{A} \}</math>. The model on images in the orbit take the form of a multi-modal mixture distribution
:<math>p(I^D, \theta) = \textstyle \sum_{a \in \mathcal{A}} p(I^D,\theta\mid I_a) \pi_{\mathcal A}(a) \ .</math>
== Surface templates for computational neuroanatomy and subcortical structures ==
The study of sub-cortical
Shown in the accompanying Figure is an example of subcortical structure templates generated from T1-weighted [[Magnetic resonance imaging|magnetic resonance imagery]] by Tang et al.<ref name="Tang 599–611"/><ref name="Tang 2093–2117"/><ref name="Tang 645–660"/> for the study of Alzheimer's disease in the ADNI population of subjects.
== Surface estimation in cardiac computational anatomy ==
[[File:Siamak atlas.tif|alt=Showing population heart atlases with superimposed hypertrophy.|thumb|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.<ref name="
Numerous studies have now been done on cardiac hypertrophy and the role of the structural integraties in the functional mechanics of the heart. Siamak Ardekani has been working on populations of Cardiac anatomies reconstructing atlas coordinate systems from populations.<ref>{{Cite journal|
== MAP Estimation of volume templates from populations and the EM algorithm ==
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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 [[
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.
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