Bayesian estimation of templates in computational anatomy: Difference between revisions

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{{Further|LDDMMLarge |deformation diffeomorphic metric mapping|Bayesian model of computational anatomy}}

{{COI|date=December 2017}}

{{Main|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 Estimationestimation from Populationspopulations|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|Bayesian]] probability and statistics in the [[Computational anatomy#The random orbit model of computational anatomy|random orbit model of CA]] have emerged for submanifolds<ref>{{Cite journal|title = A Bayesian Generative Model for Surface Template Estimation|journal = International Journal of Biomedical Imaging|date = 2010-01-01|issn = 1687-4188|pmc = 2946602|pmid = 20885934|pages = 1–14|volume = 2010|doi = 10.1155/2010/974957|first1 = Jun|last1 = Ma|first2 = Michael I.|last2 = Miller|first3 = Laurent|last3 = Younes|doi-access = free}}</ref><ref>{{Cite journal|title = Atlas Generation for Subcortical and Ventricular Structures with its Applications in Shape Analysis|journal = IEEE Transactions on Image Processing|date = 2010-06-01|issn = 1057-7149|pmc = 2909363|pmid = 20129863|pages = 1539–1547|volume = 19|issue = 6|doi = 10.1109/TIP.2010.2042099|first1 = Anqi|last1 = Qiu|first2 = Timothy|last2 = Brown|first3 = Bruce|last3 = Fischl|first4 = Jun|last4 = Ma|first5 = Michael I.|last5 = Miller|bibcode = 2010ITIP...19.1539Q}}</ref> and dense image volumes.<ref>{{Cite journal|title = Bayesian Template Estimation in Computational Anatomy|journal = NeuroImage|date = 2008-08-01|issn = 1053-8119|pmc = 2602958|pmid = 18514544|pages = 252–261|volume = 42|issue = 1|doi = 10.1016/j.neuroimage.2008.03.056|first1 = Jun|last1 = Ma|first2 = Michael I.|last2 = Miller|first3 = Alain|last3 = Trouvé|first4 = Laurent|last4 = Younes}}</ref>

== The deformable template model of shapes and forms via diffeomorphic group actions ==

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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>.]]

: {{NumBlk|:|<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

: {{NumBlk|:|<math>

\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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== 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="Ardekani et al 2016">{{cite journal |last1=Ardekani |first1=Siamak |last2=Jain |first2=Saurabh |last3=Sanzi |first3=Alianna |last4=Corona-Villalobos |first4=Celia P. |last5=Abraham |first5=Theodore P. |last6=Abraham |first6=M. Roselle |last7=Zimmerman |first7=Stefan L. |last8=Wu |first8=Katherine C. |last9=Winslow |first9=Raimond L. |last10=Miller |first10=Michael I. |last11=Younes |first11=Laurent |title=Shape analysis of hypertrophic and hypertensive heart disease using MRI-based 3D surface models of left ventricular geometry |journal=Medical Image Analysis |date=April 2016 |volume=29 |pages=12–23 |doi=10.1016/j.media.2015.11.004 |pmid=26766206 |pmc=4850908 }}</ref>]]

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|last1=Ardekani|first1=Siamak|last2=Weiss|first2=Robert G.|last3=Lardo|first3=Albert C.|last4=George|first4=Richard T.|last5=Lima|first5=Joao A. C.|last6=Wu|first6=Katherine C.|last7=Miller|first7=Michael I.|last8=Winslow|first8=Raimond L.|last9=Younes|first9=Laurent|date=2009-06-01|title=Computational method for identifying and quantifying shape features of human left ventricular remodeling|journal=Annals of Biomedical Engineering|volume=37|issue=6|pages=1043–1054|doi=10.1007/s10439-009-9677-2|issn=1573-9686|pmc=2819012|pmid=19322659}}</ref><ref>{{Cite journal|last1=Steinert-Threlkeld|first1=Shane|last2=Ardekani|first2=Siamak|last3=Mejino|first3=Jose L. V.|last4=Detwiler|first4=Landon Todd|last5=Brinkley|first5=James F.|last6=Halle|first6=Michael|last7=Kikinis|first7=Ron|last8=Winslow|first8=Raimond L.|last9=Miller|first9=Michael I.|date=2012-06-01|title=Ontological labels for automated ___location of anatomical shape differences|journal=Journal of Biomedical Informatics|volume=45|issue=3|pages=522–527|doi=10.1016/j.jbi.2012.02.013|issn=1532-0480|pmc=3371096|pmid=22490168}}</ref><ref>{{Cite book |doi=10.1109/EMBC.2014.6944772|issn=1557-170X|pmc=4474039|pmid=25571140|isbn=978-1-4244-7929-0|chapter=Estimating dense cardiac 3D motion using sparse 2D tagged MRI cross-sections|title=2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society e|journal=Engineering in Medicine and Biology Society, 2008. Embs 2008. 30Th Annual International Conference of the IEEE|volume=2014|pages=5101–5104|year=2014|last1=Ardekani|first1=Siamak|last2=Gunter|first2=Geoffrey|last3=Jain|first3=Saurabh|last4=Weiss|first4=Robert G.|last5=Miller|first5=Michael I.|last6=Younes|first6=Laurent|title=2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society }}</ref> 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).<ref name="Ardekani et al 2016"/>

== 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 [[expectation-maximization]]expectation–maximization algorithm|expectation–maximization (EM) algorithm]].

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.