Bayesian estimation of templates in computational anatomy

{{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 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 $\{\backslash mathbb\; R\}^3$ are the diffeomorphisms $\backslash mathcal\{G\}\; \backslash doteq\; Diff$ which are mappings with 3-components $\backslash phi(\backslash cdot)\; =\; (\backslash phi\_1(\backslash cdot),\backslash phi\_2\; (\backslash cdot),\backslash phi\_3\; (\backslash cdot))$, law of composition of functions $\backslash phi\; \backslash circ\; \backslash phi^\backslash prime\; (\backslash cdot)\backslash doteq\; \backslash phi\; (\backslash phi^\backslash prime(\backslash cdot))$, with inverse $\backslash phi\; \backslash circ\; \backslash phi^\{-1\}(\backslash cdot)\; =\backslash phi\; (\; \backslash phi^\{-1\}(\backslash cdot))=\; id$.

Groups and group are familiar to the Engineering community with the universal popularization and standardization of linear algebra as a basic model

A popular group action is on scalar images, $I(x),x\; \backslash in\; \{\backslash mathbb\; R\}^3$, with action on the right via the inverse.

:$$

\phi \cdot I(x) = I \circ \phi^{-1} (x), x \in {\mathbb R}^3.

For sub-manifolds $X\; \backslash subset\; \{\backslash mathbb\; R\}^3\; \backslash in\; \backslash mathcal\{M\}$, parametrized by a chart or immersion $m(u),\; u\; \backslash in\; U$, the diffeomorphic action the flow of the position

:$$

\phi \cdot m(u) \doteq \phi\circ m(u), u \in U.

Several group actions in computational anatomy have been defined.

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 $\backslash phi\_t,\; t\; \backslash in\; [0,1]$ which satisfy the Lagrangian and Eulerian specification of the flow fields satisfying the ordinary differential equation: File:Lagrangian flow.png

{{NumBlk|:|$$

\frac{d}{dt} \phi_t = v_t \circ \phi_t , \ \phi_0 = id \ ; |{{EquationRef|Lagrangian flow}}}}

with $v\; \backslash doteq\; (v\_1,v\_2,v\_3)$ the vector fields on $\{\backslash mathbb\; R\}^3$ termed the Eulerian velocity of the particles at position $\backslash phi$ of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert space with the vector fields having 1-continuous derivative . For $v\_t\; =\; \backslash dot\; \backslash phi\_t\; \backslash circ\; \backslash phi\_t^\{-1\},\; t\; \backslash in\; [0,1]$, with the inverse for the flow given by

{{NumBlk|:|$$

\frac{d}{dt} \phi_t^{-1} = -(D \phi_t^{-1}) v_t, \ \phi_0^{-1} = id \ , |{{EquationRef|Eulerianflow}}}}

and the $3\; \backslash times\; 3$ Jacobian matrix for flows in $\backslash mathbb\{R\}^3$ given as $\backslash \; D\backslash phi\; \backslash doteq\; \backslash left(\backslash frac\{\backslash partial\; \backslash phi\_i\}\{\backslash partial\; x\_j\}\backslash right).$

Flows were first introducedGE Christensen, RD Rabbitt, MI Miller, Deformable templates using large deformation kinematics, IEEE Trans Image Process. 1996;5(10):1435-47.GE Christensen, SC Joshi, MI Miller, Volumetric transformation of brain anatomy

IEEE Transactions on Medical Imaging,1997. for large deformations in image matching; $\backslash dot\; \backslash phi\_t(x)$ is the instantaneous velocity of particle $x$ at time $t$. 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 $(V,\; \backslash |\; \backslash cdot\; \backslash |\_V\; )$ as a reproducing kernel Hilbert space (RKHS), with the norm defined by a 1-1, differential operator$A:\; V\; \backslash rightarrow\; V^*$, Green's inverse $K\; =\; A^\{-1\}$. The norm according to $\backslash |\; v\backslash |\_V^2\; \backslash doteq\; \backslash int\_X\; Av\; \backslash cdot\; v\; dx\; ,\; v\; \backslash in\; V,$

where for $\backslash sigma(v)\; \backslash doteq\; Av\; \backslash in\; V^*$

a generalized function or distribution, then $(\backslash sigma\backslash mid\; w)\backslash doteq\; \backslash int\_\{\{\backslash mathbb\; R\}^3\}\; \backslash sum\_i\; w\_i(x)\; \backslash sigma\_i(dx)$

. Since $A$ is a differential operator, finiteness of the norm-square

includes derivatives from the differential operator implying smoothness of the vector fields.

To ensure smooth flows of diffeomorphisms with inverse, the vector fields $\{\backslash mathbb\; R\}^3$ must be at least 1-time continuously differentiable in spaceP. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.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. which are modelled as elements of the Hilbert space $(V,\; \backslash |\; \backslash cdot\; \backslash |\_V\; )$ using the Sobolev embedding theorems so that each element $v\_i\; \backslash in\; H\_0^3,\; i=1,2,3,$ has 3-square-integrable derivatives. Thus $(V,\; \backslash |\; \backslash cdot\; \backslash |\_V\; )$ embed smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:{{NumBlk|:|$$

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 \} \ .

|{{EquationRef|Diffeomorphism Group}}}}

The central statistical model of computational anatomy in the context of medical imaging is the source-channel model of Shannon theory;{{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}}{{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}}{{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}} the source is the deformable template of images $I\; \backslash in\; \backslash mathcal\; \{I\}$, the channel outputs are the imaging sensors with observables $I^D\; \backslash in\; \{\backslash mathcal\; I\}^\{\backslash mathcal\; D\}$ . The variation in the anatomical configurations are modelled separately from the Medical imaging modalities Computed Axial Tomography machine, MRI machine, PET machine, and others. The Bayes theory models the prior on the source of images $\backslash pi\_\{\backslash mathcal\{I\}\}\; (\backslash cdot)$ on $I\; \backslash in\; \backslash mathcal\{I\}$, and the conditional density on the observable imagery $p(\backslash cdot\; |I)\; \backslash \; \backslash text\{on\}\; \backslash \; I^D\; \backslash in\; \{\backslash mathcal\; I\}^\{\backslash mathcal\; D\}$, conditioned on $I\; \backslash in\; \backslash mathcal\{I\}$. For images with diffeomorphism group action $I\; \backslash doteq\; \backslash phi\; \backslash cdot\; I\_\backslash mathrm\{temp\},\; \backslash phi\; \backslash in\; Diff\_V$, then the prior on the group $\backslash pi\_\{Diff\_V\}\; (\backslash cdot)$ induces the prior on images $\backslash pi\_\{\backslash mathcal\{I\}\}\; (\backslash cdot)$, written as densities the log-posterior takes the form

:$$

\log p(\phi \cdot I\mid I^D) \simeq \log p(I^D \mid \phi \cdot I)+ \log \pi_{\operatorname{Diff}_V}(\phi) \ .

Maximum a posteriori estimation (MAP) estimation is central to modern statistical theory. Parameters of interest $\backslash theta\; \backslash in\; \backslash Theta$ 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 $I^D$, MAP estimation maximizes the posterior:

:$$

\hat \theta \doteq \arg \max_{\theta \in \Theta} \log p(\theta \mid I^D).

[[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.{{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}}{{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}}{{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}}

]]This requires computation of the conditional probabilities $p(\backslash theta\backslash mid\; I^D)\; =\; \backslash frac\{p(I^D,\backslash theta)\}\{p(I^D)\}$. The multiple atlas orbit model randomizes over the denumerable set of atlases $\backslash \{\; I\_a,\; a\; \backslash in\; \backslash mathcal\{A\}\; \backslash \}$. The model on images in the orbit take the form of a multi-modal mixture distribution

:$p(I^D,\; \backslash theta)\; =\; \backslash textstyle\; \backslash sum\_\{a\; \backslash in\; \backslash mathcal\{A\}\}\; p(I^D,\backslash theta\backslash mid\; I\_a)\; \backslash pi\_\{\backslash mathcal\; A\}(a)\; \backslash \; .$

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,{{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}}{{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}}{{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}}{{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}} Alzheimer's disease,{{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}}{{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}}{{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}} and Depression,{{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}}{{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}} many neuroanatomical shape statistical studies have now been completed using templates built from all of the subcortical structures for depression,{{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}} Alzheimer's,{{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}}{{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}}{{Cite journal|last1=Miller|first1=Michael I.|last2=Younes|first2=Laurent|last3=Ratnanather|first3=J. 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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}} Bipolar disorder, ADHD,{{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}} autism,{{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}} and Huntington's Disease.{{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}}{{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}} Templates were generated using Bayesian template estimation data back to Ma, Younes and Miller.{{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}}

Shown in the accompanying Figure is an example of subcortical structure templates generated from T1-weighted magnetic resonance imagery by Tang et al. for the study of Alzheimer's disease in the ADNI population of subjects.

File:Siamak atlas.tif

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.{{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}}{{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}}{{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 }} 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).

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 $I^\{D\_1\},\; I^\{D\_2\},\; \backslash dots$ the problem is to estimate the template in the orbit of dense images $I\; \backslash in\; \backslash mathcal\{I\}$. Ma's procedure takes an initial hypertemplate $I\_0\; \backslash in\; \backslash mathcal\{I\}$ as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism $I\; \backslash doteq\; \backslash phi\_0\; \backslash cdot\; I\_0$, with the parameters to be estimated the log-coordinates $\backslash theta\; \backslash doteq\; v\_0$ determining the geodesic mapping of the hyper-template $\backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_0)\; \backslash cdot\; I\_0\; =\; I\; \backslash in\; \backslash mathcal\{I\}$.

In the Bayesian random orbit model of computational anatomy the observed MRI images $I^\{D\_i\}$ are modelled as a conditionally Gaussian random field with mean field $\backslash phi\_i\; \backslash cdot\; I$, with $\backslash phi\_i$ a random unknown transformation of the template. The MAP estimation problem is to estimate the unknown template $I\; \backslash in\; \backslash mathcal\{I\}$ given the observed MRI images.

Ma's procedure for dense imagery takes an initial hypertemplate $I\_0\; \backslash in\; \backslash mathcal\{I\}$ as the starting point, and models the template in the orbit under the unknown to be estimated diffeomorphism $I\; \backslash doteq\; \backslash phi\_0\; \backslash cdot\; I\_0$. The observables are modelled as conditional random fields, $I^\{D\_i\}$ a {{EquationNote|conditional-Gaussian}} random field with mean field $\backslash phi\_i\; \backslash cdot\; I\; \backslash doteq\; \backslash phi\_i\; \backslash cdot\; \backslash phi\_0\; \backslash cdot\; I\_0$. The unknown variable to be estimated explicitly by MAP is the mapping of the hyper-template $\backslash phi\_0$, 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 (EM) algorithm.

The orbit-model is exploited by associating the unknown to be estimated flows to their log-coordinates $v\_i,i=1,\backslash dots$ via the Riemannian geodesic log and exponential for computational anatomy the initial vector field in the tangent space at the identity so that $\backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_\{i\})\; \backslash doteq\; \backslash phi\_i$, with $\backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_\{0\})$ the mapping of the hyper-template.

The MAP estimation problem becomes

:$$

\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

The EM algorithm takes as complete data the vector-field coordinates parameterizing the mapping, $v\_i,i=1,\backslash dots$ and compute iteratively the conditional-expectation

:$$

\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}

• Compute new template maximizing Q-function, setting $\backslash theta^\backslash text\{new\}\backslash doteq\; v\_0^\backslash text\{new\}\; =\; \backslash arg\; \backslash max\_\{\backslash theta\; =\; v\_0\}\; Q(\backslash theta\; ;\; \backslash theta^\backslash text\{old\}=v\_0^\backslash text\{old\})\; =\; -\; \backslash |\; (\; \backslash bar\; I^\backslash text\{old\}\; -\; I\_0\; \backslash circ\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_0)^\{-1\}\; )\; \backslash sqrt\{\backslash beta^\backslash text\{old\}\}\backslash |^2$

- \| v_0\|_V^2

• Compute the mode-approximation for the expectation updating the expected-values for the mode values:

:: $v\_i^\backslash text\{new\}\; =\; \backslash arg\; \backslash max\_\{v:\; \backslash dot\; \backslash phi\; =\; v\; \backslash circ\; \backslash phi\; \}\; -\; \backslash int\_0^1\; \backslash |\; v\_t\; \backslash |\_V^2\; \backslash ,\; dt\; -\; \backslash |\; I^\{D\_i\}\; -\; I\_0\; \backslash circ\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_0^\backslash text\{old\})^\{-1\}\; \backslash circ\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v)^\{-1\}\; \backslash |^2.\; i=1,2,\backslash dots$

:: $\backslash beta^\backslash text\{new\}\; (x)\; =\; \backslash sum\_\{i=1\}^n\; |\; D\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_i^\backslash text\{new\})(x)\; |,\; \backslash text\{\; with\; \}\; \backslash bar\; I^\backslash text\{new\}\; (x)\; =\; \backslash frac\{\; \backslash sum\_\{i=1\}^n\; I^\{D\_i\}\; \backslash circ\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_i^\backslash text\{new\})\; |\; D\; \backslash mathrm\{Exp\}\_\backslash mathrm\{id\}(v\_i^\backslash text\{new\})(x)\; |\; \}\{\; \backslash beta^\backslash text\{old\}(x)\}$

{{Reflist|30em}}

Category:Bayesian estimation

Category:Philosophy of science