Bayesian estimation of templates in computational anatomy: Difference between revisions

Revision as of 22:22, 13 January 2018

Revision as of 03:34, 14 October 2018

Line 1:

{{Further information|LDDMM | Bayesian model of computational anatomy}}

{{Multiple issues|

{{COI|date=December 2017}}

Line 4 ⟶ 5:

}}

{{Main article|Computational anatomy}}

{{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 Estimation from Populations|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|url = https://dx.doi.org/10.1155/2010/974957|journal = Journal of Biomedical Imaging|date = 2010-01-01|issn = 1687-4188|pmc = 2946602|pmid = 20885934|pages = 16:1–16:14|volume = 2010|doi = 10.1155/2010/974957|first = Jun|last = Ma|first2 = Michael I.|last2 = Miller|first3 = Laurent|last3 = Younes}}</ref><ref>{{Cite journal|title = Atlas Generation for Subcortical and Ventricular Structures with its Applications in Shape Analysis|journal = IEEE transactionsTransactions on image processing : a publication of the IEEE SignalImage Processing Society|date = 2010-06-01|issn = 1057-7149|pmc = 2909363|pmid = 20129863|pages = 1539–1547|volume = 19|issue = 6|doi = 10.1109/TIP.2010.2042099|first = Anqi|last = Qiu|first2 = Timothy|last2 = Brown|first3 = Bruce|last3 = Fischl|first4 = Jun|last4 = Ma|first5 = Michael I.|last5 = Miller}}</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|first = Jun|last = 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 ==

{{Further information|Group actions in computational anatomy}}[[Linear algebra]] is one of the central tools to modern engineering. Central to linear algebra is the notion of an orbit of vectors, with the matrices forming [[Matrix group|groups]] (matrices with inverses and identity) which act on the vectors. In linear algebra the equations describing the orbit elements the vectors are linear in the vectors being acted upon by the matrices. 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 <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>.