View source for Bayesian estimation of templates in computational anatomy

{{Further|Large 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 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|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 ==
{{Further|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>.
[[Group (mathematics)|Group]]s and [[Group action (mathematics)|group]] are familiar to the Engineering community with the universal popularization and standardization of [[linear algebra]] as a basic model
A popular  [[Group action (mathematics)|group action]] is on scalar images, <math>I(x),x \in {\mathbb R}^3</math>, with action on the right via the inverse.
:<math>
\phi \cdot I(x) = I \circ \phi^{-1} (x), x \in {\mathbb R}^3.
</math>
For sub-[[manifold]]s <math> X \subset {\mathbb R}^3 \in \mathcal{M} </math>, parametrized by a chart or [[immersion (mathematics)|immersion]] <math> m(u), u \in U </math>, the diffeomorphic action the flow of the position
:<math>
\phi \cdot m(u) \doteq \phi\circ m(u), u \in U.
</math>
Several [[group actions in computational anatomy]] have been defined.
== 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^*
</math> a generalized function or distribution, then <math> (\sigma\mid w)\doteq \int_{{\mathbb R}^3}  \sum_i w_i(x) \sigma_i(dx)
</math>. Since <math> A </math> is a differential operator, finiteness of the norm-square <math>  \int_X Av \cdot v dx < \infty
</math> includes derivatives from the differential operator implying smoothness of the vector fields.
To ensure smooth flows of diffeomorphisms with inverse,  the vector fields <math>  {\mathbb R}^3   </math> must be at least 1-time continuously differentiable in space<ref name=":22">P. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.</ref><ref name=":4">A. Trouvé. Action de groupe de dimension infinie et reconnaissance de formes. C R Acad Sci Paris Sér I Math, 321(8):1031– 1034, 1995.</ref> which are modelled as elements of the Hilbert space  <math>(V, \| \cdot \|_V )</math> using the [[Sobolev space|Sobolev]] embedding theorems so that each element <math>v_i \in H_0^3, i=1,2,3,</math> has 3-square-integrable derivatives. Thus <math>(V, \| \cdot \|_V )</math> embed smoothly in 1-time continuously differentiable functions.<ref name=":22" /><ref name=":4" />  The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:{{NumBlk|:|<math>
Diff_V \doteq \{\varphi=\phi_1: \dot \phi_t = v_t \circ \phi_t , \phi_0 = id, \int_0^1 \|v_t \|_V \, dt < \infty \} \ .
</math>|{{EquationRef|Diffeomorphism Group}}}}
== 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|journal = Statistical Methods in Medical Research|date = 1997-06-01|issn = 0962-2802|pmid = 9339500|pages = 267–299|volume = 6|issue = 3|doi = 10.1177/096228029700600305|language = en|first1 = Michael|last1 = Miller|first2 = Ayananshu|last2 = Banerjee|first3 = Gary|last3 = Christensen|first4 = Sarang|last4 = Joshi|first5 = Navin|last5 = Khaneja|first6 = Ulf|last6 = Grenander|first7 = Larissa|last7 = Matejic|s2cid = 35247542}}</ref><ref>{{Cite book|title = Pattern Theory: From Representation to Inference|author = U. Grenander and M. I. Miller |publisher = Oxford University Press|date = 2007-02-08|isbn = 978-0-19-929706-1|language = English}}</ref><ref>{{Cite book|title = Bayesian Multiple Atlas Deformable Templates|author = M. I. Miller and S. Mori and X. Tang and D. Tward and Y. Zhang | series = Brain Mapping: An Encyclopedic Reference|url = https://books.google.com/books?id=ysucBAAAQBAJ|publisher = Academic Press|date = 2015-02-14|isbn = 978-0-12-397316-0|language = en}}</ref> the source is the deformable template of images <math> I \in \mathcal {I} </math>, the channel outputs are the imaging sensors with observables <math> I^D \in {\mathcal I}^{\mathcal D} </math> . The variation in the anatomical configurations are modelled separately from the Medical imaging modalities [[Computed axial tomography|Computed Axial Tomography]] machine, [[Magnetic resonance imaging|MRI]] machine, [[Positron emission tomography|PET]] machine, and others. The [[Bayes' theorem|Bayes theory]] models the prior on the source of images <math> \pi_{\mathcal{I}} (\cdot) </math> on <math>I \in \mathcal{I} </math>, and the conditional density on the observable imagery <math> p(\cdot |I) \  \text{on} \ I^D \in {\mathcal I}^{\mathcal D} </math>, conditioned on <math> I \in \mathcal{I} </math>. For images with [[Computational anatomy#Groups and group actions|diffeomorphism group action]] <math>I \doteq \phi \cdot I_\mathrm{temp}, \phi \in Diff_V</math>,  then the prior on the group <math>\pi_{Diff_V} (\cdot)</math> induces the prior on images <math>\pi_{\mathcal{I}} (\cdot)</math>, written as densities the log-posterior takes the form
:<math>
\log p(\phi \cdot I\mid I^D) \simeq \log p(I^D \mid \phi \cdot I)+ \log \pi_{\operatorname{Diff}_V}(\phi) \ .
</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 subcortical structures in problems associated to segmentation of images, and (iii) template reconstruction from populations.  Given the observed image <math> I^D  </math>, MAP estimation maximizes the posterior:
:<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|last1=Tang|first1=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=Journal 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|last1=Tang|first1=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|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|last1=Tang|first1=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|pmc=5479937}}</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>
== Surface templates for computational neuroanatomy and subcortical structures ==
The study of sub-cortical neuroanatomy has been the focus of many studies. Since the original publications by Csernansky and colleagues of hippocampal change in Schizophrenia,<ref>{{Cite journal|last1=Csernansky|first1=John G.|last2=Joshi|first2=Sarang|last3=Wang|first3=Lei|last4=Haller|first4=John W.|last5=Gado|first5=Mokhtar|last6=Miller|first6=J. Philip|last7=Grenander|first7=Ulf|last8=Miller|first8=Michael I.|date=1998-09-15|title=Hippocampal morphometry in schizophrenia by high dimensional brain mapping|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=95|issue=19|pages=11406–11411|issn=0027-8424|pmc=21655|pmid=9736749|doi=10.1073/pnas.95.19.11406|bibcode=1998PNAS...9511406C|doi-access=free}}</ref><ref>{{Cite journal|last1=Csernansky|first1=John G.|last2=Wang|first2=Lei|last3=Jones|first3=Donald|last4=Rastogi-Cruz|first4=Devna|last5=Posener|first5=Joel A.|last6=Heydebrand|first6=Gitry|last7=Miller|first7=J. Philip|last8=Miller|first8=Michael I.|s2cid=14924093|date=2002-12-01|title=Hippocampal deformities in schizophrenia characterized by high dimensional brain mapping|journal=The American Journal of Psychiatry|volume=159|issue=12|pages=2000–2006|doi=10.1176/appi.ajp.159.12.2000|issn=0002-953X|pmid=12450948}}</ref><ref>{{Cite journal|last1=Wang|first1=L.|last2=Joshi|first2=S. C.|last3=Miller|first3=M. I.|last4=Csernansky|first4=J. G.|s2cid=16573767|date=2001-09-01|title=Statistical analysis of hippocampal asymmetry in schizophrenia|journal=NeuroImage|volume=14|issue=3|pages=531–545|doi=10.1006/nimg.2001.0830|issn=1053-8119|pmid=11506528}}</ref><ref>{{Cite journal|last1=Csernansky|first1=John G.|last2=Schindler|first2=Mathew K.|last3=Splinter|first3=N. Reagan|last4=Wang|first4=Lei|last5=Gado|first5=Mohktar|last6=Selemon|first6=Lynn D.|last7=Rastogi-Cruz|first7=Devna|last8=Posener|first8=Joel A.|last9=Thompson|first9=Paul A.|date=2004-05-01|title=Abnormalities of thalamic volume and shape in schizophrenia|journal=The American Journal of Psychiatry|volume=161|issue=5|pages=896–902|doi=10.1176/appi.ajp.161.5.896|issn=0002-953X|pmid=15121656}}</ref> Alzheimer's disease,<ref>{{Cite journal|last1=Csernansky|first1=J. G.|last2=Wang|first2=L.|last3=Swank|first3=J.|last4=Miller|first4=J. P.|last5=Gado|first5=M.|last6=McKeel|first6=D.|last7=Miller|first7=M. I.|last8=Morris|first8=J. C.|date=2005-04-15|title=Preclinical detection of Alzheimer's disease: hippocampal shape and volume predict dementia onset in the elderly|journal=NeuroImage|volume=25|issue=3|pages=783–792|doi=10.1016/j.neuroimage.2004.12.036|issn=1053-8119|pmid=15808979|s2cid=207164390}}</ref><ref>{{Cite journal|last1=Wang|first1=Lei|last2=Miller|first2=J. Philp|last3=Gado|first3=Mokhtar H.|last4=McKeel|first4=Daniel W.|last5=Rothermich|first5=Marcus|last6=Miller|first6=Michael I.|last7=Morris|first7=John C.|last8=Csernansky|first8=John G.|date=2006-03-01|title=Abnormalities of hippocampal surface structure in very mild dementia of the Alzheimer type|journal=NeuroImage|volume=30|issue=1|pages=52–60|doi=10.1016/j.neuroimage.2005.09.017|issn=1053-8119|pmc=2853193|pmid=16243546}}</ref><ref>{{Cite journal|last1=Wang|first1=Lei|last2=Swank|first2=Jeffrey S.|last3=Glick|first3=Irena E.|last4=Gado|first4=Mokhtar H.|last5=Miller|first5=Michael I.|last6=Morris|first6=John C.|last7=Csernansky|first7=John G.|date=2003-10-01|title=Changes in hippocampal volume and shape across time distinguish dementia of the Alzheimer type from healthy aging|journal=NeuroImage|volume=20|issue=2|pages=667–682|doi=10.1016/S1053-8119(03)00361-6|issn=1053-8119|pmid=14568443|s2cid=21246081}}</ref> and Depression,<ref>{{Cite journal|last1=Posener|first1=Joel A.|last2=Wang|first2=Lei|last3=Price|first3=Joseph L.|last4=Gado|first4=Mokhtar H.|last5=Province|first5=Michael A.|last6=Miller|first6=Michael I.|last7=Babb|first7=Casey M.|last8=Csernansky|first8=John G.|s2cid=12131077|date=2003-01-01|title=High-dimensional mapping of the hippocampus in depression|journal=The American Journal of Psychiatry|volume=160|issue=1|pages=83–89|doi=10.1176/appi.ajp.160.1.83|issn=0002-953X|pmid=12505805}}</ref><ref>{{Cite journal|last1=Munn|first1=Melissa A.|last2=Alexopoulos|first2=Jim|last3=Nishino|first3=Tomoyuki|last4=Babb|first4=Casey M.|last5=Flake|first5=Lisa A.|last6=Singer|first6=Tisha|last7=Ratnanather|first7=J. Tilak|last8=Huang|first8=Hongyan|last9=Todd|first9=Richard D.|date=2007-09-01|title=Amygdala Volume Analysis in Female Twins with Major Depression|journal=Biological Psychiatry|volume=62|issue=5|pages=415–422|doi=10.1016/j.biopsych.2006.11.031|issn=0006-3223|pmc=2904677|pmid=17511971}}</ref> many neuroanatomical shape statistical studies have now been completed using templates built from all of the subcortical structures for depression,<ref>{{Cite web|url=https://www.researchgate.net/publication/271515359|title=Amygdala and Hippocampal in ADHD: Volumetric and Morphometric Analysis and Relation to Mood Symptoms.|website=ResearchGate|access-date=2016-03-22}}</ref> Alzheimer's,<ref name="Tang 599–611"/><ref name="Tang 2093–2117"/><ref>{{Cite journal|last1=Qiu|first1=Anqi|last2=Fennema-Notestine|first2=Christine|last3=Dale|first3=Anders M.|last4=Miller|first4=Michael I.|date=2009-04-15|title=Regional shape abnormalities in mild cognitive impairment and Alzheimer's disease|journal=NeuroImage|volume=45|issue=3|pages=656–661|issn=1053-8119|pmc=2847795|pmid=19280688|doi=10.1016/j.neuroimage.2009.01.013}}</ref><ref>{{Cite journal|last1=Qiu|first1=Anqi|last2=Younes|first2=Laurent|last3=Miller|first3=Michael I.|last4=Csernansky|first4=John G.|date=2008-03-01|title=Parallel Transport in Diffeomorphisms Distinguishes the Time-Dependent Pattern of Hippocampal Surface Deformation due to Healthy Aging and the Dementia of the Alzheimer's Type|journal=NeuroImage|volume=40|issue=1|pages=68–76|doi=10.1016/j.neuroimage.2007.11.041|issn=1053-8119|pmc=3517912|pmid=18249009}}</ref><ref>{{Cite journal|last1=Miller|first1=Michael I.|last2=Younes|first2=Laurent|last3=Ratnanather|first3=J. Tilak|last4=Brown|first4=Timothy|last5=Reigel|first5=Tommy|last6=Trinh|first6=Huong|last7=Tang|first7=Xiaoying|last8=Barker|first8=Peter|last9=Mori|first9=Susumu|date=2012-10-01|title=Amygdala Atrophy in MCI/Alzheimer's Disease in the BIOCARD cohort based on Diffeomorphic Morphometry|journal=Medical Image Computing and Computer-Assisted Intervention|volume=2012|pages=155–166|pmc=4063307|pmid=24955432}}</ref><ref>{{Cite journal|last1=Miller|first1=Michael I.|last2=Ratnanather|first2=J. Tilak|last3=Tward|first3=Daniel J.|last4=Brown|first4=Timothy|last5=Lee|first5=David S.|last6=Ketcha|first6=Michael|last7=Mori|first7=Kanami|last8=Wang|first8=Mei-Cheng|author8-link= Mei-Cheng Wang |last9=Mori|first9=Susumu|date=2015-01-01|title=Network neurodegeneration in Alzheimer's disease via MRI based shape diffeomorphometry and high-field atlasing|journal= Frontiers in Bioengineering and Biotechnology|page=54|doi=10.3389/fbioe.2015.00054|pmc=4515983|pmid=26284236|volume=3|doi-access=free}}</ref> Bipolar disorder, ADHD,<ref>{{Cite journal|last1=Qiu|first1=Anqi|last2=Crocetti|first2=Deana|last3=Adler|first3=Marcy|last4=Mahone|first4=E. Mark|last5=Denckla|first5=Martha B.|last6=Miller|first6=Michael I.|last7=Mostofsky|first7=Stewart H.|date=2009-01-01|title=Basal Ganglia Volume and Shape in Children With Attention Deficit Hyperactivity Disorder|journal=The American Journal of Psychiatry|volume=166|issue=1|pages=74–82|doi=10.1176/appi.ajp.2008.08030426|issn=0002-953X|pmc=2890266|pmid=19015232}}</ref> autism,<ref>{{Cite journal|url=http://www.jaacap.com/article/S0890-8567(10)00248-0/abstract|title=Basal Ganglia Shapes Predict Social, Communication, and Motor Dysfunctions in Boys With Autism Spectrum Disorder - Journal of the American Academy of Child & Adolescent Psychiatry|volume=49|issue=6|pages=539–51, 551.e1–4|journal=Journal of the American Academy of Child and Adolescent Psychiatry|access-date=2016-03-22|pmid=20494264|year=2010|last1=Qiu|first1=A.|last2=Adler|first2=M.|last3=Crocetti|first3=D.|last4=Miller|first4=M. I.|last5=Mostofsky|first5=S. H.|doi=10.1016/j.jaac.2010.02.012}}</ref> and Huntington's Disease.<ref>{{Cite journal|last1=Younes|first1=Laurent|last2=Ratnanather|first2=J. Tilak|last3=Brown|first3=Timothy|last4=Aylward|first4=Elizabeth|last5=Nopoulos|first5=Peg|last6=Johnson|first6=Hans|last7=Magnotta|first7=Vincent A.|last8=Paulsen|first8=Jane S.|last9=Margolis|first9=Russell L.|date=2014-03-01|title=Regionally selective atrophy of subcortical structures in prodromal HD as revealed by statistical shape analysis|journal=Human Brain Mapping|volume=35|issue=3|pages=792–809|doi=10.1002/hbm.22214|issn=1097-0193|pmc=3715588|pmid=23281100}}</ref><ref>{{Cite AV media|url=http://f1000research.com/posters/1096125|title=Anatomical connectivity in prodromal Huntington Disease|website=F1000Research|date=2014-07-18 |author=Frieda van den Noort |author2=Frieda van den Noort |author3=Andreia Faria |author4=Tilak Ratnanather |author5=Christopher Ross |author6=Susumu Mori |author7=Laurent Younes |author8=Michael Miller|type=Poster}}</ref> Templates were generated using Bayesian template estimation data back to Ma, Younes and Miller.<ref>{{Cite journal|last1=Ma|first1=Jun|last2=Miller|first2=Michael I.|last3=Younes|first3=Laurent|date=2010-01-01|title=A Bayesian Generative Model for Surface Template Estimation|journal=International Journal of Biomedical Imaging|volume=2010|doi=10.1155/2010/974957|issn=1687-4188|pmc=2946602|pmid=20885934|pages=1–14|doi-access=free}}</ref>
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="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|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 ==
Generating templates empirically from populations is a fundamental operation ubiquitous to the discipline.
Several methods based on Bayesian statistics have emerged for submanifolds and dense image volumes.
For the dense image volume case, given the observable <math> I^{D_1}, I^{D_2}, \dots </math> the problem is to estimate the template in the orbit of dense images <math> I \in \mathcal{I} </math>.  Ma's procedure 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>, with the parameters to be estimated the log-coordinates <math>\theta \doteq v_0</math> determining the geodesic mapping of the hyper-template <math>\mathrm{Exp}_\mathrm{id}(v_0) \cdot I_0 = I \in \mathcal{I}</math>.
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 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.
The MAP estimation problem becomes
:<math>
\max_{v_0} p(I^D, \theta = v_0) = \int p(I^D, \theta= v_0\mid v_1,v_2, \dots ) \pi(v_1,v_2, \dots ) \, dv
</math>
The EM algorithm takes as complete data the vector-field coordinates parameterizing the mapping, <math>v_i,i=1,\dots</math> and compute iteratively the conditional-expectation
:<math>
\begin{matrix}
Q(\theta=v_0; \theta^\text{old}=v_0^\text{old}) &
= - E ( \log p(I^D, \theta=v_0 \mid v_1,v_2,\dots)|I^D, \theta^\text{old}) \\
& = - \| ( \bar I^\text{old} - I_0 \circ \mathrm{Exp}_\mathrm{id}(v_0)^{-1} ) \sqrt{\beta^\text{old}} \|^2 - \|v_0\|_V^2
\end{matrix}
</math>
* Compute new template maximizing Q-function, setting <math>\theta^\text{new}\doteq v_0^\text{new} = \arg \max_{\theta = v_0} Q(\theta ; \theta^\text{old}=v_0^\text{old}) = - \| ( \bar I^\text{old} - I_0 \circ \mathrm{Exp}_\mathrm{id}(v_0)^{-1} ) \sqrt{\beta^\text{old}}\|^2
- \| v_0\|_V^2</math>
* Compute the mode-approximation for the expectation updating the expected-values for the mode values:
:: <math>v_i^\text{new} = \arg \max_{v: \dot \phi = v \circ \phi } - \int_0^1 \| v_t \|_V^2 \, dt - \| I^{D_i} - I_0 \circ \mathrm{Exp}_\mathrm{id}(v_0^\text{old})^{-1} \circ \mathrm{Exp}_\mathrm{id}(v)^{-1}  \|^2. i=1,2,\dots</math>
:: <math>\beta^\text{new} (x) = \sum_{i=1}^n | D \mathrm{Exp}_\mathrm{id}(v_i^\text{new})(x) |, \text{ with } \bar I^\text{new} (x) = \frac{ \sum_{i=1}^n I^{D_i} \circ \mathrm{Exp}_\mathrm{id}(v_i^\text{new}) | D \mathrm{Exp}_\mathrm{id}(v_i^\text{new})(x) |  }{ \beta^\text{old}(x)}</math>
== References ==
{{Reflist|30em}}
[[Category:Bayesian estimation]]
[[Category:Philosophy of science]]