Bayesian estimation of templates in computational anatomy: Difference between revisions

Revision as of 01:16, 16 October 2018

Revision as of 00:13, 11 November 2018

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\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|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Younes|first4=Laurent|last5=Miller|first5=Michael I.|date=2015-01-01|title=Baseline Shape Diffeomorphometry Patterns of Subcortical and Ventricular Structures in Predicting Conversion of Mild Cognitive Impairment to Alzheimer’s Disease|journal=JJournalJournal of Alzheimer's Disease|volume=44|issue=2|pages=599–611|doi=10.3233/JAD-141605|issn=1387-2877|pmc=4474004|pmid=25318546}}</ref><ref name="Tang 2093–2117">{{Cite journal|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Younes|first4=Laurent|last5=Miller|first5=Michael I.|last6=for the Alzheimer's Disease Neuroimaging Initiative|date=2015-06-01|title=The diffeomorphometry of regional shape change rates and its relevance to cognitive deterioration in mild cognitive impairment and Alzheimer's disease|url=http://onlinelibrary.wiley.com/doi/10.1002/hbm.22758/abstract|journal=Human Brain Mapping|language=en|volume=36|issue=6|pages=2093–2117|doi=10.1002/hbm.22758|issn=1097-0193|pmc=4474005|pmid=25644981}}</ref><ref name="Tang 645–660">{{Cite journal|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Miller|first4=Michael I.|last5=Alzheimer's Disease Neuroimaging Initiative|date=2015-01-01|title=APOE Affects the Volume and Shape of the Amygdala and the Hippocampus in Mild Cognitive Impairment and Alzheimer's Disease: Age Matters|journal=Journal of Alzheimer's Disease|volume=47|issue=3|pages=645–660|doi=10.3233/JAD-150262|issn=1875-8908|pmid=26401700}}</ref>

]]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>