Bayesian estimation of templates in computational anatomy: Difference between revisions
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{{Further|
{{COI|date=December 2017}}
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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>
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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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