Computational anatomy#Curve matching

At computational anatomy's heart is the comparison of shape by recognizing in one shape the other. This connects it to D'Arcy Wentworth Thompson's developments On Growth and Form which has led to scientific explanations of morphogenesis, the process by which patterns are formed in biology. Albrecht Durer's Four Books on Human Proportion were arguably the earliest works on computational anatomy.{{Cite book|url=https://archive.org/details/hierinnsindbegri00dure|title=Hierinn sind begriffen vier Bucher von menschlicher Proportion durch Albrechten Durer von Nurerberg [sic.] erfunden und beschuben zu nutz allen denen so zu diser kunst lieb tragen|last=Durer|first=Albrecht|year=1528|publisher=Hieronymus Andreae Formschneider}}{{Cite web|url=http://library.uthscsa.edu/2012/03/albrecht-durers-human-proportions/|title=Albrecht Dürer's Human Proportions " UT Health Science Center Library|last=Library|first=State of Texas, University of Texas Health Science Center at San Antonio|website=library.uthscsa.edu|date=27 March 2012 |access-date=2016-03-16}}{{Cite web|url=http://www.themorgan.org/collection/printed-books-and-bindings/73192|title=Albrecht Dürer|website=The Morgan Library & Museum|access-date=2016-03-16|date=2014-01-07}} The efforts of Noam Chomsky in his pioneering of computational linguistics inspired the original formulation of computational anatomy as a generative model of shape and form from exemplars acted upon via transformations.{{Cite journal|title=Computational Anatomy: An Emerging Discipline|journal=Q. Appl. Math.|volume=56|issue=4|pages=617–694|date=1998-12-01|first1=Ulf|last1=Grenander|first2=Michael I.|last2=Miller|doi=10.1090/qam/1668732|doi-access=free}}

Due to the availability of dense 3D measurements via technologies such as magnetic resonance imaging (MRI), computational anatomy has emerged as a subfield of medical imaging and bioengineering for extracting anatomical coordinate systems at the morphome scale in 3D. The spirit of this discipline shares strong overlap with areas such as computer vision and kinematics of rigid bodies, where objects are studied by analysing the groups responsible for the movement in question. Computational anatomy departs from computer vision with its focus on rigid motions, as the infinite-dimensional diffeomorphism group is central to the analysis of Biological shapes. It is a branch of the image analysis and pattern theory school at Brown University{{Cite web|title=Brown University – Pattern Theory Group: Home|url=http://www.dam.brown.edu/ptg/participants.shtml|website=www.dam.brown.edu|access-date=2015-12-27}} pioneered by Ulf Grenander. In Grenander's general metric pattern theory, making spaces of patterns into a metric space is one of the fundamental operations since being able to cluster and recognize anatomical configurations often requires a metric of close and far between shapes. The diffeomorphometry metric of computational anatomy measures how far two diffeomorphic changes of coordinates are from each other, which in turn induces a metric on the shapes and images indexed to them. The models of metric pattern theory,{{Cite book|title=General Pattern Theory: A Mathematical Study of Regular Structures|author=Grenander, Ulf|year=1993|publisher=Oxford University Press|isbn=9780198536710}}{{Cite book|title=Pattern Theory: From Representation to Inference|author=U. Grenander and M. I. Miller|publisher=Oxford University Press|date=2007-02-08|location=Oxford|isbn=9780199297061}} in particular group action on the orbit of shapes and forms is a central tool to the formal definitions in computational anatomy.

Computational anatomy is the study of shape and form at the morphome or gross anatomy millimeter, or morphology scale, focusing on the study of sub-manifolds of $\{\backslash mathbb\; R\}^3,$ points, curves surfaces and subvolumes of human anatomy.

An early modern computational neuro-anatomist was David Van Essen{{Cite journal|title=Two-dimensional maps of the cerebral cortex|journal=The Journal of Comparative Neurology|date=1980-05-15|pmid=7410593|pages=255–281|volume=191|issue=2|doi=10.1002/cne.901910208|first1=D. C.|last1=Van Essen|first2=J. H.|last2=Maunsell|s2cid=25729587}} performing some of the early physical unfoldings of the human brain based on printing of a human cortex and cutting. Jean Talairach's publication of Talairach coordinates is an important milestone at the morphome scale demonstrating the fundamental basis of local coordinate systems in studying neuroanatomy and therefore the clear link to charts of differential geometry. Concurrently, virtual mapping in computational anatomy across high resolution dense image coordinates was already happening in Ruzena Bajcy's{{Cite journal|title=Multiresolution Elastic Matching|journal=Comput. Vision Graph. Image Process.|date=1989-04-01|pages=1–21|volume=46|issue=1|doi=10.1016/S0734-189X(89)80014-3|first1=Ruzena|last1=Bajcsy|first2=Stane|last2=Kovačič}} and Fred Bookstein's{{Cite journal|title=Principal Warps: Thin-Plate Splines and the Decomposition of Deformations|journal= IEEE Transactions on Pattern Analysis and Machine Intelligence|date=1989-06-01|pages=567–585|volume=11|issue=6|doi=10.1109/34.24792|first=F. L.|last=Bookstein|s2cid=47302}} earliest developments based on computed axial tomography and magnetic resonance imagery.

The earliest introduction of the use of flows of diffeomorphisms for transformation of coordinate systems in image analysis and medical imaging was by Christensen, Joshi, Miller, and Rabbitt.{{Cite book|url=https://books.google.com/books?id=tVVGAQAAIAAJ|title=A deformable neuroanatomy textbook based on viscous fluid mechanics: Proceedings of the ... Conference on Information Sciences and Systems|last1=Christensen|first1=Gary|last2=Rabbitt|first2=Richard|last3=Miller|first3=Michael I.|date=1993-01-01|publisher=Department of Electrical Engineering, Johns Hopkins University|editor-last=Jerry Prince }}{{Cite journal|title=Deformable Templates Using Large Deformation Kinematics|journal= IEEE Transactions on Image Processing|date=1996-10-01|pages=1435–1447|volume=5|issue=10|doi=10.1109/83.536892|first1=G. E.|last1=Christensen|first2=R. D.|last2=Rabbitt|first3=M. I.|last3=Miller|pmid=18290061|bibcode=1996ITIP....5.1435C}}{{Cite book|url=https://books.google.com/books?id=8WdlWJepgWMC&q=info:fFDvh9Q-I8cJ:scholar.google.com&pg=PA115|title=Brain Warping: Chapter 7: Large Deformation Fluid Diffeomorphisms for Landmark and Image Matching|last1=Miller|first1=Michael|last2=Joshi|first2=Sarang|last3=Christensen|last4=Brain Warping Book Author: Toga|first4=Arthur|year=1997|isbn=9780080525549|pages=115|publisher=Elsevier }}

The first formalization of computational anatomy as an orbit of exemplar templates under diffeomorphism group action was in the original lecture given by Grenander and Miller with that title in May 1997 at the 50th Anniversary of the Division of Applied Mathematics at Brown University,{{Cite journal|editor=Walter Freiberger|title=Current and Future Challenges in the Applications of Mathematics|journal=Quarterly of Applied Mathematics }} and subsequent publication. This was the basis for the strong departure from much of the previous work on advanced methods for spatial normalization and image registration which were historically built on notions of addition and basis expansion. The structure preserving transformations central to the modern field of Computational Anatomy, homeomorphisms and diffeomorphisms carry smooth submanifolds smoothly. They are generated via Lagrangian and Eulerian flows which satisfy a law of composition of functions forming the group property, but are not additive.

The original model of computational anatomy was as the triple, $(\backslash mathcal\{G\},\; \backslash mathcal\{M\},\; \backslash mathcal\{P\})\; \backslash \; ,$

the group $g\; \backslash in\; \backslash mathcal\{G\}$, the orbit of shapes and forms $m\; \backslash in\; \backslash mathcal\{M\}$, and the probability laws $P$ which encode the variations of the objects in the orbit. The template or collection of templates are elements in the orbit $m\_\backslash mathrm\{temp\}\; \backslash in\; \backslash mathcal\; \{M\}$ of shapes.

The Lagrangian and Hamiltonian formulations of the equations of motion of computational anatomy took off post 1997 with several pivotal meetings including the 1997 Luminy meeting{{Cite web|title=Colloque Mathematiques et reconnaissance de formes|url=https://www.ceremade.dauphine.fr/~cohen/MSPCV/luminy.html|website=www.ceremade.dauphine.fr|access-date=2015-12-19}} organized by the Azencott{{Cite web|title=Robert Azencott, mathématicien polyglotte {{!}} La Recherche|url=http://www.larecherche.fr/actualite/aussi/robert-azencott-mathematicien-polyglotte-01-12-2000-70634|website=www.larecherche.fr|access-date=2016-02-20}} school at Ecole-Normale Cachan on the "Mathematics of Shape Recognition" and the 1998 Trimestre at Institute Henri Poincaré organized by David Mumford "Questions Mathématiques en Traitement du Signal et de l'Image" which catalyzed the Hopkins-Brown-ENS Cachan groups and subsequent developments and connections of computational anatomy to developments in global analysis.

The developments in computational anatomy included the establishment of the Sobolev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of variational problems in the space of diffeomorphisms,{{Cite web|url=http://www.dam.brown.edu/people/mariom/AM282-01/PAPERS/LANDMARK/an-approach-of-pattern.pdf|title=An Approach of Pattern Recognition through Infinite Dimensional Diffeomorphism|last=Trouve|first=Alain|archive-url=https://web.archive.org/web/20160304190038/http://www.dam.brown.edu/people/mariom/AM282-01/PAPERS/LANDMARK/an-approach-of-pattern.pdf|archive-date=2016-03-04|url-status=dead}}{{Cite journal|title=Variational Problems on Flows of Diffeomorphisms for Image Matching|journal=Q. Appl. Math.|date=1998-09-01|pages=587–600|volume=LVI|issue=3|first1=Paul|last1=Dupuis|first2=Ulf|last2=Grenander|doi = 10.1090/qam/1632326|doi-access=free}} the derivation of the Euler–Lagrange equations characterizing geodesics through the group and associated conservation laws,{{Cite journal|title=On the metrics and euler–lagrange equations of computational anatomy|journal=Annual Review of Biomedical Engineering|date=2002-01-01|pmid=12117763|pages=375–405|volume=4|doi=10.1146/annurev.bioeng.4.092101.125733|first1=Michael I.|last1=Miller|first2=Alain|last2=Trouve|first3=Laurent|last3=Younes|citeseerx=10.1.1.157.6533}}{{Cite journal|title=Geodesic Shooting for Computational Anatomy|journal=Journal of Mathematical Imaging and Vision|date=2006-01-28|pmc=2897162|pmid=20613972|pages=209–228|volume=24|issue=2|doi=10.1007/s10851-005-3624-0|first1=Michael I.|last1=Miller|first2=Alain|last2=Trouvé|first3=Laurent|last3=Younes|bibcode=2006JMIV...24..209M }}{{Cite journal|last1=Miller|first1=Michael I.|last2=Trouvé|first2=Alain|last3=Younes|first3=Laurent|date=2015-12-07|title=Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson|journal=Annual Review of Biomedical Engineering|volume=17|pages=447–509|doi=10.1146/annurev-bioeng-071114-040601|pmid=26643025}} the demonstration of the metric properties of the right invariant metric,{{Cite journal|title=Group Actions, Homeomorphisms, and Matching: A General Framework|journal=Int. J. Comput. Vis.|date=2001-01-01|pages=61–84|volume=41|issue=1–2|doi=10.1023/A:1011161132514|first1=M. I.|last1=Miller|first2=L.|last2=Younes|s2cid=15423783}} the demonstration that the Euler–Lagrange equations have a well-posed initial value problem with unique solutions for all time,{{Cite journal|title=Local Geometry of Deformable Templates|journal=SIAM Journal on Mathematical Analysis|date=2005-01-01|pages=17–59|volume=37|issue=1|doi=10.1137/S0036141002404838|first1=A.|last1=Trouvé|first2=L.|last2=Younes|citeseerx=10.1.1.158.302}} and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces.{{Cite journal|title=Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks|journal=SIAM J. Imaging Sci.|date=2012-03-01|pages=394–433|volume=5|issue=1|doi=10.1137/10081678X|first1=Mario|last1=Micheli|first2=Peter W.|last2=Michor|first3=David|last3=Mumford|arxiv=1009.2637|s2cid=2301243}} Following the Los Alamos meeting in 2002,{{Cite web|title=Home Page|url=http://cnls.lanl.gov/~rgarcia/Conferences/ImageAnalysis/|website=cnls.lanl.gov|access-date=2015-12-19}} Joshi's{{Cite journal|title=Landmark matching via large deformation diffeomorphisms|journal=IEEE Transactions on Image Processing|date=2000-01-01|pmid=18262973|pages=1357–1370|volume=9|issue=8|doi=10.1109/83.855431|first1=S. C.|last1=Joshi|first2=M. I.|last2=Miller|s2cid=6659707|bibcode=2000ITIP....9.1357J}} original large deformation singular Landmark solutions in computational anatomy were connected to peaked solitons or peakons{{Cite book|chapter=Peakons|title=Encyclopedia of Mathematical Physics|editor=J.-P. Francoise |editor2=G.L. Naber |editor3=S.T. Tsou|location=Oxford|publisher=Elsevier|volume=4|issue=2006|pages=12–20|arxiv=0908.4351|date=2009-08-29|first=Darryl D.|last=Holm|bibcode=2009arXiv0908.4351H }} as solutions for the Camassa–Holm equation. Subsequently, connections were made between computational anatomy's Euler–Lagrange equations for momentum densities for the right-invariant metric satisfying Sobolev smoothness to Vladimir Arnold's characterization of the Euler equation for incompressible flows as describing geodesics in the group of volume preserving diffeomorphisms.{{Cite journal|title=Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid|url=http://projecteuclid.org/euclid.bams/1183530815|journal=Bulletin of the American Mathematical Society|date=1969-09-01|pages=962–967|volume=75|issue=5|first1=David G.|last1=Ebin|first2=Jerrold E.|last2=Marsden|doi=10.1090/s0002-9904-1969-12315-3|doi-access=free}}{{Cite journal|title=On Euler's equation and 'EPDiff'|journal=Journal of Geometric Mechanics|volume=5|issue=3|pages=319–344|arxiv=1209.6576|date=2013|doi=10.3934/jgm.2013.5.319|first1=David|last1=Mumford|first2=Peter W.|last2=Michor|bibcode=2012arXiv1209.6576M }} The first algorithms, generally termed LDDMM for large deformation diffeomorphic mapping for computing connections between landmarks in volumes{{Cite book|title=Handbook of Mathematical Methods in Imaging|url=https://books.google.com/books?id=FzvNfkMjvPwC|publisher=Springer Science & Business Media|date=2010-11-23|isbn=9780387929194|first=Otmar|last=Scherzer}}{{Cite book|doi=10.1109/CVPR.2004.1315234|first1=J.|last1=Glaunes|first2=A.|last2=Trouve|first3=L.|last3=Younes|title=Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 |chapter=Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching |isbn=978-0-7695-2158-9|citeseerx=10.1.1.158.4209|volume=2|pages=712–718|year=2004}} and spherical manifolds,{{Cite journal|url=https://www.researchgate.net/publication/263559913|title=Landmark Matching via Large Deformation Diffeomorphisms on the Sphere: Special Issue on Mathematics and Image Analysis|journal=Journal of Mathematical Imaging and Vision|volume=20|pages=179–200|via=ResearchGate|doi=10.1023/B:JMIV.0000011326.88682.e5|access-date=2016-03-27|year=2004|last1=Glaunès|first1=Joan|last2=Vaillant|first2=Marc|last3=Miller|first3=Michael I|s2cid=21324161}} curves,{{Cite journal|title=Whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves, cortical surfaces, and images|journal=NeuroImage|date=2011-05-01|pmc=3119076|pmid=21281722|pages=162–173|volume=56|issue=1|doi=10.1016/j.neuroimage.2011.01.067|first1=Jia|last1=Du|first2=Laurent|last2=Younes|first3=Anqi|last3=Qiu}} currents and surfaces,{{Cite journal|title=Diffeomorphic Metric Surface Mapping in Superior Temporal Gyrus|journal=NeuroImage|date=2007-02-01|pmc=3140704|pmid=17185000|pages=1149–1159|volume=34|issue=3|doi=10.1016/j.neuroimage.2006.08.053|first1=Marc|last1=Vaillant|first2=Anqi|last2=Qiu|first3=Joan|last3=Glaunès|first4=Michael I.|last4=Miller}}{{Cite journal|title=Statistical models of sets of curves and surfaces based on currents|journal=Medical Image Analysis|date=2009-10-01|pmid=19679507|pages=793–808|volume=13|issue=5|doi=10.1016/j.media.2009.07.007|first1=Stanley|last1=Durrleman|first2=Xavier|last2=Pennec|first3=Alain|last3=Trouvé|first4=Nicholas|last4=Ayache|citeseerx=10.1.1.221.5224}} volumes,{{Cite journal|title=Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal=International Journal of Computer Vision|volume=61|issue=2|pages=139–157|author=M.F. Beg and M. I. Miller and A. Trouve and L. Younes|url=https://www.researchgate.net/publication/220660081|via=ResearchGate|access-date=2016-01-27|doi=10.1023/B:VISI.0000043755.93987.aa|year=2005 |s2cid=17772076}} tensors,{{Cite book |date=2006-07-05|pmc=2920614|pmid=20711423|pages=67|volume=2006|doi=10.1109/CVPRW.2006.65|first1=Yan|last1=Cao|first2=Michael I.|last2=Miller|first3=Susumu|last3=Mori|first4=Raimond L.|last4=Winslow|first5=Laurent|last5=Younes|title=2006 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW'06) |chapter=Diffeomorphic Matching of Diffusion Tensor Images |isbn=978-0-7695-2646-1}} varifolds,{{Cite journal|title=The varifold representation of non-oriented shapes for diffeomorphic registration|arxiv=1304.6108|journal=SIAM Journal on Imaging Sciences|pages=2547–2580|volume=6|issue=4|doi=10.1137/130918885|first1=Nicolas|last1=Charon|first2=Alain|last2=Trouvé|year=2013|bibcode=2013arXiv1304.6108C|s2cid=14335966}} and time-series{{Cite arXiv|title=Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View|eprint=1003.3895|date=2010-03-19|first1=Alain|last1=Trouvé|first2=François-Xavier|last2=Vialard|class=math.OC }}{{Cite journal|title=Principal geodesic analysis for the study of nonlinear statistics of shape|journal=IEEE Transactions on Medical Imaging|date=2004-08-01|pages=995–1005|volume=23|issue=8|doi=10.1109/TMI.2004.831793|pmid=15338733|first1=P.T.|last1=Fletcher|first2=C.|last2=Lu|first3=S.M.|last3=Pizer|first4=S.|last4=Joshi|citeseerx=10.1.1.76.539|s2cid=620015}} have followed.

These contributions of computational anatomy to the global analysis associated to the infinite dimensional manifolds of subgroups of the diffeomorphism group is far from trivial. The original idea of doing differential geometry, curvature and geodesics on infinite dimensional manifolds goes back to Bernhard Riemann's Habilitation (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen{{Cite web|url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/|title=Ueber die Hypothesen, welche der Geometrie zu Grunde liegen|website=www.maths.tcd.ie|access-date=2016-03-16|url-status=dead|archive-url=https://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/|archive-date=2016-03-18}}{{Cite book|isbn=9783642351204|author=Bernhard Riemann|title=Über die Hypothesen, welche der Geometrie|date=4 May 2013|publisher=Springer}}); the key modern book laying the foundations of such ideas in global analysis are from Michor.{{Cite book|author=Peter W. Michor|title=Topics in Differential Geometry|date=2008-07-23|publisher=American Mathematical Society|isbn=9780821820032 }}

The applications within medical imaging of computational anatomy continued to flourish after two organized meetings at the Institute for Pure and Applied Mathematics conferences{{Cite journal|url=http://www.sciencedirect.com/science/journal/10538119/23/supp/S1|journal=NeuroImage|volume=23|issue=Supplement 1|pages=S1–S300|year=2004|title=Mathematics in Brain Imaging}}{{Cite journal|journal=NeuroImage|volume=45|issue=Supplement 1|pages=S1–S222|year=2009|url=http://www.sciencedirect.com/science/journal/10538119/45/1/supp/S1|title=Mathematics in Brain Imaging|doi=10.1016/j.neuroimage.2008.10.033|pmid=19027863|last1=Thompson|first1=Paul M.|last2=Miller|first2=Michael I.|last3=Poldrack|first3=Russell A.|last4=Nichols|first4=Thomas E.|last5=Taylor|first5=Jonathan E.|last6=Worsley|first6=Keith J.|last7=Ratnanather|first7=J. Tilak|s2cid=12143788|url-access=subscription}} at University of California, Los Angeles. Computational anatomy has been useful in creating accurate models of the atrophy of the human brain at the morphome scale, as well as Cardiac templates,{{Cite journal|last1=Fonseca|first1=Carissa G.|last2=Backhaus|first2=Michael|last3=Bluemke|first3=David A.|last4=Britten|first4=Randall D.|last5=Chung|first5=Jae Do|last6=Cowan|first6=Brett R.|last7=Dinov|first7=Ivo D.|last8=Finn|first8=J. Paul|last9=Hunter|first9=Peter J.|date=2011-08-15|title=The Cardiac Atlas Project—an imaging database for computational modeling and statistical atlases of the heart|journal=Bioinformatics|volume=27|issue=16|pages=2288–2295|doi=10.1093/bioinformatics/btr360|pmc=3150036|pmid=21737439}} as well as in modeling biological systems.{{Cite web|url=http://cellorganizer.org/Downloads/v1.8/RELEASE_NOTES.pdf|title=CellOrganizer 1.8 Release Notes}} Since the late 1990s, computational anatomy has become an important part of developing emerging technologies for the field of medical imaging. Digital atlases are a fundamental part of modern Medical-school education{{Cite book|author=Jamie Weir|edition=4th|title=Imaging Atlas of Human Anatomy|publisher=Mosby|date=2010-03-09|location=Edinburgh|isbn=9780723434573|display-authors=etal}}{{Cite web|title=The Whole Brain Atlas|url=http://www.med.harvard.edu/aanlib/home.html|website=www.med.harvard.edu|access-date=2016-01-26|url-status=dead|archive-url=https://web.archive.org/web/20160118132843/http://www.med.harvard.edu/aanlib/home.html|archive-date=2016-01-18}} and in neuroimaging research at the morphome scale.{{Cite journal|title=A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM)|journal=Philosophical Transactions of the Royal Society of London B|date=2001-08-29|pmc=1088516|pmid=11545704|pages=1293–1322|volume=356|issue=1412|doi=10.1098/rstb.2001.0915|first1=J|last1=Mazziotta|first2=A|last2=Toga|first3=A|last3=Evans|first4=P|last4=Fox|first5=J|last5=Lancaster|first6=K|last6=Zilles|first7=R|last7=Woods|first8=T|last8=Paus|first9=G|last9=Simpson}}{{Cite web|title=White Matter Atlas - Diffusion Tensor Imaging Atlas of the Brain's White Matter Tracts|url=http://www.dtiatlas.org/|website=www.dtiatlas.org|access-date=2016-01-26}} Atlas based methods and virtual textbooks{{Cite journal|title=Mathematical textbook of deformable neuroanatomies|journal=Proceedings of the National Academy of Sciences of the United States of America|date=1993-12-15|pmc=48101|pmid=8265653|pages=11944–11948|volume=90|issue=24|first1=M I|last1=Miller|first2=G E|last2=Christensen|first3=Y|last3=Amit|first4=U|last4=Grenander|doi=10.1073/pnas.90.24.11944|bibcode=1993PNAS...9011944M|doi-access=free}} which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer,{{Cite web|title=FreeSurfer|url=http://freesurfer.net/|website=freesurfer.net|access-date=2015-12-08}} FSL,{{Cite web|title=FSL - FslWiki|url=http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/|website=fsl.fmrib.ox.ac.uk|access-date=2015-12-08}} MRIStudio,{{Cite web|title=NITRC: MRI Studio: Tool/Resource Info|url=https://www.nitrc.org/projects/mri_studio/|website=www.nitrc.org|access-date=2015-12-08}} SPM.{{Cite web|title=SPM software - Statistical Parametric Mapping|url=http://www.fil.ion.ucl.ac.uk/spm/software/|website=www.fil.ion.ucl.ac.uk|access-date=2015-12-08}} Diffeomorphic registration, introduced in the 1990s, is now an important player with existing codes bases organized around ANTS,{{Cite web|title=stnava/ANTs|url=https://github.com/stnava/ANTs/blob/master/Scripts/antsIntroduction.sh|website=GitHub|access-date=2015-12-11}} DARTEL,{{Cite journal|title=A fast diffeomorphic image registration algorithm|journal=NeuroImage|date=2007-10-15|pmid=17761438|pages=95–113|volume=38|issue=1|doi=10.1016/j.neuroimage.2007.07.007|first=John|last=Ashburner|s2cid=545830}} DEMONS,{{Cite web|title=Software - Tom Vercauteren|url=https://sites.google.com/site/tomvercauteren/software|website=sites.google.com|access-date=2015-12-11}} LDDMM,{{Cite web|title=NITRC: LDDMM: Tool/Resource Info|url=https://www.nitrc.org/projects/lddmm-volume/|website=www.nitrc.org|access-date=2015-12-11}} StationaryLDDMM,{{Cite web|title=Publication:Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons|url=https://www.openaire.eu/search/publication?articleId=dedup_wf_001::ea7b28db1d4570e248acdffb6211d98d|website=www.openaire.eu|access-date=2015-12-11|archive-url=https://web.archive.org/web/20160216022906/https://www.openaire.eu/search/publication?articleId=dedup_wf_001::ea7b28db1d4570e248acdffb6211d98d|archive-date=2016-02-16|url-status=dead}} FastLDDMM,{{Cite book|last1=Zhang|first1=Miaomiao|last2=Fletcher|first2=P. Thomas|title=Information Processing in Medical Imaging |chapter=Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration |s2cid=10334673|date=2015-01-01 |volume=24|pages=249–259|issn=1011-2499|pmid=26221678|doi=10.1007/978-3-319-19992-4_19|series=Lecture Notes in Computer Science|isbn=978-3-319-19991-7}} are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry is an important technology built on many of these principles.

The model of human anatomy is a deformable template, an orbit of exemplars under group action. Deformable template models have been central to Grenander's metric pattern theory, accounting for typicality via templates, and accounting for variability via transformation of the template. An orbit under group action as the representation of the deformable template is a classic formulation from differential geometry. The space of shapes are denoted $m\; \backslash in\; \backslash mathcal\; \{M\}$, with the group $(\backslash mathcal\{G\},\; \backslash circ\; )$ with law of composition $\backslash circ$; the action of the group on shapes is denoted $g\; \backslash cdot\; m$, where the action of the group $g\; \backslash cdot\; m\; \backslash in\; \backslash mathcal\{M\},\; m\; \backslash in\; \backslash mathcal\; \{M\}$ is defined to satisfy

:$(g\; \backslash circ\; g^\backslash prime)\; \backslash cdot\; m=g\; \backslash cdot\; (g^\backslash prime\; \backslash cdot\; m)\backslash in\; \backslash mathcal\; \{\; M\}\; .$

The orbit $\backslash mathcal\{M\}$ of the template becomes the space of all shapes, $\backslash mathcal\{M\}\; \backslash doteq\; \backslash \{\; m=g\; \backslash cdot\; m\_\backslash mathrm\{temp\},\; g\; \backslash in\; \backslash mathcal\{G\}\; \backslash \}$, being homogenous under the action of the elements of $\backslash mathcal\{G\}$.

File:Medial-temporal-lobe-structures.png

The orbit model of computational anatomy is an abstract algebra – to be compared to linear algebra – since the groups act nonlinearly on the shapes. This is a generalization of the classical models of linear algebra, in which the set of finite dimensional $\{\backslash mathbb\; R\}^n$ vectors are replaced by the finite-dimensional anatomical submanifolds (points, curves, surfaces and volumes) and images of them, and the $n\; \backslash times\; n$ matrices of linear algebra are replaced by coordinate transformations based on linear and affine groups and the more general high-dimensional diffeomorphism groups.

The central objects are shapes or forms in computational anatomy, one set of examples being the 0,1,2,3-dimensional submanifolds of $\{\backslash mathbb\; R\}^3$, a second set of examples being images generated via medical imaging such as via magnetic resonance imaging (MRI) and functional magnetic resonance imaging. File:Subcortical-MRI-caudate-putamen.png The 0-dimensional manifolds are landmarks or fiducial points; 1-dimensional manifolds are curves such as sulcal and gyral curves in the brain; 2-dimensional manifolds correspond to boundaries of substructures in anatomy such as the subcortical structures of the midbrain or the gyral surface of the neocortex; subvolumes correspond to subregions of the human body, the heart, the thalamus, the kidney.

The landmarks $X\; \backslash doteq\; \backslash \{\; x\_1,\backslash dots,x\_n\; \backslash \}\; \backslash subset\; \{\backslash mathbb\; R\}^3\; \backslash in\; \backslash mathcal\{M\}$ are a collections of points with no other structure, delineating important fiducials within human shape and form (see associated landmarked image).

The sub-manifold shapes such as surfaces $X\; \backslash subset\; \{\backslash mathbb\; R\}^3\; \backslash in\; \backslash mathcal\{M\}$ are collections of points modeled as parametrized by a local chart or immersion $m:U\; \backslash subset\; \{\backslash mathbb\; R\}^\{1,2\}\; \backslash rightarrow\; \{\backslash mathbb\; R\}^3$, $m(u),\; u\; \backslash in\; U$ (see Figure showing shapes as mesh surfaces).

The images such as MR images or DTI images $I\; \backslash in\; \backslash mathcal\{M\}$, and are dense functions

$I(x),\; x\; \backslash in\; X\; \backslash subset\; \{\backslash mathbb\; R\}^\{1,2,3\}$ are scalars, vectors, and matrices (see Figure showing scalar image).

{{Further|Group actions in computational anatomy}}File:T1-weighted-MRI.pngGroups and group actions are familiar to the Engineering community with the universal popularization and standardization of linear algebra as a basic model for analyzing signals and systems in mechanical engineering, electrical engineering and applied mathematics. In linear algebra the matrix groups (matrices with inverses) are the central structure, with group action defined by the usual definition of $A$ as an $n\backslash times\; n$ matrix, acting on $x\; \backslash in\; \{\backslash mathbb\; R\}^n$ as $n\; \backslash times\; 1$ vectors; the orbit in linear-algebra is the set of $n$-vectors given by $y=A\; \backslash cdot\; x\; \backslash in\; \{\backslash mathbb\; R\}^n$, which is a group action of the matrices through the orbit of $\{\backslash mathbb\; R\}^n$.

The central group in computational anatomy defined on volumes in $\{\backslash mathbb\; R\}^3$ are the diffeomorphisms $\backslash mathcal\{G\}\; \backslash doteq\; \backslash operatorname\{Diff\}$ which are mappings with 3-components $\backslash varphi(\backslash cdot)=(\backslash varphi\_1(\backslash cdot),\backslash varphi\_2\; (\backslash cdot),\backslash varphi\_3\; (\backslash cdot))$, law of composition of functions $\backslash varphi\; \backslash circ\; \backslash varphi^\backslash prime\; (\backslash cdot)\backslash doteq\; \backslash varphi\; (\backslash varphi^\backslash prime(\backslash cdot))$, with inverse $\backslash varphi\; \backslash circ\; \backslash varphi^\{-1\}(\backslash cdot)=\backslash varphi\; (\; \backslash varphi^\{-1\}(\backslash cdot))=\{\backslash rm\; id\}$.

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

:$$

\varphi \cdot I(x)=I \circ \varphi^{-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

:$$

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

Several group actions in computational anatomy have been defined.{{citation needed|date=October 2016}}

For the study of rigid body kinematics, the low-dimensional matrix Lie groups have been the central focus. The matrix groups are low-dimensional mappings, which are diffeomorphisms that provide one-to-one correspondences between coordinate systems, with a smooth inverse. The matrix group of rotations and scales can be generated via a closed form finite-dimensional matrices which are solution of simple ordinary differential equations with solutions given by the matrix exponential.

For the study of deformable shape in computational anatomy, 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 $\backslash varphi\_t,\; t\; \backslash in\; [0,1]$ which satisfy the Lagrangian and Eulerian specification of the flow fields as first introduced in,{{Cite journal|last1=Christensen|first1=G. E.|last2=Rabbitt|first2=R. D.|last3=Miller|first3=M. I.|date=1996-10-01|title=Deformable Templates Using Large Deformation Kinematics|journal= IEEE Transactions on Image Processing|volume=5|issue=10|pages=1435–1447|doi=10.1109/83.536892|pmid=18290061|bibcode=1996ITIP....5.1435C}} satisfying the ordinary differential equation:File:Lagrangian flow.png

{{NumBlk|:|$$

\frac{d}{dt} \varphi_t=v_t \circ \varphi_t , \ \varphi_0={\rm 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 $\backslash varphi$ of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert space of high-dimension, with the Jacobian of the flow $\backslash \; D\backslash varphi\; \backslash doteq\; \backslash left(\backslash frac\{\backslash partial\; \backslash varphi\_i\}\{\backslash partial\; x\_j\}\; \backslash right)$ a high-dimensional field in a function space as well, rather than a low-dimensional matrix as in the matrix groups. Flows were first introduced{{cite journal | doi=10.1109/83.536892 | title=Deformable templates using large deformation kinematics | year=1996 | last1=Christensen | first1=G.E. | last2=Rabbitt | first2=R.D. | last3=Miller | first3=M.I. | journal=IEEE Transactions on Image Processing | volume=5 | issue=10 | pages=1435–1447 | pmid=18290061 | bibcode=1996ITIP....5.1435C }}{{cite journal | doi=10.1109/42.650882 | title=Volumetric transformation of brain anatomy | year=1997 | last1=Christensen | first1=G.E. | last2=Joshi | first2=S.C. | last3=Miller | first3=M.I. | journal=IEEE Transactions on Medical Imaging | volume=16 | issue=6 | pages=864–877 | pmid=9533586 | s2cid=14209020 }} for large deformations in image matching; $\backslash dot\; \backslash varphi\_t(x)$ is the instantaneous velocity of particle $x$ at time $t$ .

The inverse $\backslash varphi\_t^\{-1\},\; t\; \backslash in\; [0,1]$ required for the group is defined on the Eulerian vector-field with advective inverse flow

{{NumBlk|:|$$

\frac{d}{dt} \varphi_t^{-1}=- (D \varphi_t^{-1}) v_t , \ \varphi_0^{-1}={\rm id} \ .|{{EquationRef|inverse transport flow}}}}

The group of diffeomorphisms is very big. To ensure smooth flows of diffeomorphisms avoiding shock-like solutions for the inverse, the vector fields must be at least 1-time continuously differentiable in space.P. 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. For diffeomorphisms on $\{\backslash mathbb\; R\}^3$, vector fields are modelled as elements of the Hilbert space $(V,\; \backslash |\backslash cdot\; \backslash |\_V\; )$ using the Sobolev embedding theorems so that each element has strictly greater than 2 generalized square-integrable spatial derivatives (thus $v\_i\; \backslash in\; H\_0^3,\; i=1,2,3,$ is sufficient), yielding 1-time continuously differentiable functions.

The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:{{NumBlk|:|$$

\operatorname{Diff}_V \doteq \left\{\varphi=\varphi_1: \dot \varphi_t = v_t \circ \varphi_t , \varphi_0={\rm id}, \int_0^1 \|v_t \|_V \, dt < \infty \right\} \ ,|{{EquationRef|diffeomorphism group}}}}

where

$\backslash |v\backslash |\_V^2\; \backslash doteq\; \backslash int\_X\; Av\backslash cdot\; v\; \backslash ,\; dx,\; \backslash \; v\; \backslash in\; V\; \backslash \; ,$

with the linear operator $A$ mapping to the dual space $A:\; V\; \backslash mapsto\; V^*$, with the integral calculated by integration by parts when $Av\; \backslash in\; V^*$ is a generalized function in the dual space.

{{Hide|Sobolev smoothness and reproducing kernel Hilbert space with Green's kernel|

The modelling approach used in computational anatomy 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 of the Hilbert space is induced by the differential operator. For $\backslash sigma(v)\; \backslash doteq\; Av\; \backslash in\; V^*$

a generalized function or distribution, define the linear form as $(\backslash sigma|w)\backslash doteq\; \backslash int\_\{\{\backslash mathbb\; R\}^3\}\; \backslash sum\_i\; w\_i(x)\; \backslash sigma\_i(dx)$

. This determines the norm on $(V,\; \backslash |\backslash cdot\; \backslash |\_V\; )$ according to

:$\backslash langle\; v\; ,\; w\; \backslash rangle\_V\; \backslash doteq\; \backslash int\_X\; A\; v\; \backslash cdot\; w\; \backslash ,\; dx,\; \backslash \; \backslash |v\backslash |\_V^2\; \backslash doteq\; \backslash int\_X\; Av\; \backslash cdot\; v\; \backslash ,\; dx,\; \backslash \; v,w\; \backslash in\; V\; \backslash \; .$

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

includes derivatives from the differential operator implying smoothness of the vector fields.The Sobolev embedding theorem arguments were made in demonstrating that 1-continuous derivative is required for smooth flows.

For proper choice of $A$ then $(V,\backslash |\backslash cdot\; \backslash |\_V)$ is an RKHS with the operator $K=A^\{-1\}:\; V^*\; \backslash rightarrow\; V$ termed the Green's operator generated from the Green's function (scalar case) for the vector field case. The Green's kernels associated to the differential operator smooths since the kernel $k(\backslash cdot,\backslash cdot)$

is continuously differentiable in both variables implying

:$K\backslash sigma\; (x)\_i\; \backslash doteq\; \backslash sum\_j\; \backslash int\_\{\{\backslash mathbb\; R\}^3\}\; k\_\{ij\}(x,y)\; \backslash sigma\_j(dy)$

When $\backslash sigma\; \backslash doteq\; \backslash mu\; \backslash ,\; dx$, a vector density, $(\backslash sigma\backslash mid\; v)\; \backslash doteq\; \backslash int\; v\; \backslash cdot\; \backslash mu\; \backslash ,\; dx.$

}}

{{Further|Diffeomorphometry}}

The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation.{{Cite journal|title=Computable Elastic Distances Between Shapes|journal=SIAM Journal on Applied Mathematics|date=1998-04-01|pages=565–586|volume=58|issue=2|doi=10.1137/S0036139995287685|first=L.|last=Younes|citeseerx=10.1.1.45.503}}{{Cite journal|title=On Shape of Plane Elastic Curves|journal=International Journal of Computer Vision|date=2006-09-25|pages=307–324|volume=73|issue=3|doi=10.1007/s11263-006-9968-0|first1=Washington|last1=Mio|first2=Anuj|last2=Srivastava|first3=Shantanu|last3=Joshi|citeseerx=10.1.1.138.2219|s2cid=15202271}}{{Cite journal|title=A Metric on Shape Space with Explicit Geodesics|journal=Rend. Lincei Mat. Appl.|volume=9|issue=2008|pages=25–57|arxiv=0706.4299|year=2008|first1=Peter W.|last1=Michor|first2=David|last2=Mumford|first3=Jayant|last3=Shah|first4=Laurent|last4=Younes|bibcode=2007arXiv0706.4299M }}{{Cite journal|title=An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach|arxiv=math/0605009|journal=Applied and Computational Harmonic Analysis|pages=74–113|volume=23|issue=1|doi=10.1016/j.acha.2006.07.004|first1=Peter W.|last1=Michor|first2=David|last2=Mumford|year=2007|s2cid=732281}}{{Cite journal|title=Elastic geodesic paths in shape space of parameterized surfaces|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|date=2012-09-01|pmid=22144521|pages=1717–1730|volume=34|issue=9|doi=10.1109/TPAMI.2011.233|first1=Sebastian|last1=Kurtek|first2=Eric|last2=Klassen|first3=John C.|last3=Gore|first4=Zhaohua|last4=Ding|first5=Anuj|last5=Srivastava|s2cid=7178535}} The diffeomorphometry metric measures how close and far two shapes or images are from each other; the metric length is the shortest length of the flow which carries one coordinate system into the other.

Oftentimes, the familiar Euclidean metric is not directly applicable because the patterns of shapes and images do not form a vector space. In the Riemannian orbit model of computational anatomy, diffeomorphisms acting on the forms $\backslash varphi\; \backslash cdot\; m\; \backslash in\; \backslash mathcal\; \{M\},\; \backslash varphi\; \backslash in\; \backslash operatorname\{Diff\}\_V,\; m\; \backslash in\; \backslash mathcal\{M\}$ do not act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric is another. The method we use to induce the Riemannian metric is used to induce the metric on the orbit of shapes by defining it in terms of the metric length between diffeomorphic coordinate system transformations of the flows. Measuring the lengths of the geodesic flow between coordinates systems in the orbit of shapes is called diffeomorphometry.

Define the distance on the group of diffeomorphisms

{{NumBlk|:|$$

d_{\operatorname{Diff}_V}(\psi, \varphi)=\inf_{v_t} \left(\int_0^1 \int_X Av_t \cdot v_t \, dx \ dt: \varphi_0=\psi, \varphi_1=\varphi, \dot \varphi_t=v_t \circ \varphi_t \right)^{1/2} \ ;

|{{EquationRef|metric-diffeomorphisms}}}}

this is the right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all $\backslash varphi\; \backslash in\; \backslash operatorname\{Diff\}\_V$,

: $d\_\{\backslash operatorname\{Diff\}\_V\}(\backslash psi,\; \backslash varphi)\; =\; d\_\{\backslash operatorname\{Diff\}\_V\}(\backslash psi\; \backslash circ\; \backslash varphi,\; \backslash varphi\; \backslash circ\; \backslash varphi)$.

The distance on shapes and forms,$d\_\{\backslash mathcal\{M\}\}:\backslash mathcal\{M\}\; \backslash times\; \backslash mathcal\{M\}\backslash rightarrow\; \backslash R^+$,

{{NumBlk|:| $$

d_{\mathcal{M}}(m,n)=\inf \left\{ d_{\operatorname{Diff}_V}({\rm id},\varphi) : \varphi \in \operatorname{Diff}_V,\, \varphi \cdot m=n \right\}

|{{EquationRef|metric-shapes-forms}}}}

the images are denoted with the orbit as $I\; \backslash in\; \backslash mathcal\{I\}$ and metric $,\; d\_\{\backslash mathcal\{I\}\}$.

In classical mechanics the evolution of physical systems is described by solutions to the Euler–Lagrange equations associated to the Least-action principle of Hamilton. This is a standard way, for example of obtaining Newton's laws of motion of free particles. More generally, the Euler–Lagrange equations can be derived for systems of generalized coordinates. The Euler–Lagrange equation in computational anatomy describes the geodesic shortest path flows between coordinate systems of the diffeomorphism metric. In computational anatomy the generalized coordinates are the flow of the diffeomorphism and its Lagrangian velocity $\backslash varphi,\; \backslash dot\{\backslash varphi\}$, the two related via the Eulerian velocity $v\; \backslash doteq\; \backslash dot\{\backslash varphi\}\; \backslash circ\; \backslash varphi^\{-1\}$.

Hamilton's principle for generating the Euler–Lagrange equation requires the action integral on the Lagrangian given by

{{NumBlk|::|$J(\backslash varphi)\; \backslash doteq\; \backslash int\_0^1\; L(\backslash varphi\_t,\backslash dot\; \backslash varphi\_t)\; \backslash ,\; dt\; \backslash \; ;$|{{EquationRef|Hamiltonian-integrated-Lagrangian}}}}

the Lagrangian is given by the kinetic energy:{{NumBlk|:|$$

L(\varphi_t, \dot \varphi_t ) \doteq \frac{1}{2} \int_X A (\dot \varphi_t \circ \varphi_t^{-1}) \cdot (\dot \varphi_t \circ \varphi_t^{-1} ) dx

=\frac{1}{2} \int_X Av_t \cdot v_t \ dx \ .

|{{EquationRef|Lagrangian-kinetic-energy}}}}

In computational anatomy, $Av$ was first called the Eulerian or diffeomorphic shape momentum{{Cite journal|last1=MILLER|first1=MICHAEL I.|last2=TROUVÉ|first2=ALAIN|last3=YOUNES|first3=LAURENT|date=2006-01-31|title=Geodesic Shooting for Computational Anatomy|journal=Journal of Mathematical Imaging and Vision|volume=24|issue=2|pages=209–228|doi=10.1007/s10851-005-3624-0|pmc=2897162|pmid=20613972|bibcode=2006JMIV...24..209M }} since when integrated against Eulerian velocity $v$ gives energy density, and since there is a conservation of diffeomorphic shape momentum which holds. The operator $A$ is the generalized moment of inertia or inertial operator.

{{Further|Riemannian metric and Lie-bracket in computational anatomy}}

Classical calculation of the Euler–Lagrange equation from Hamilton's principle requires the perturbation of the Lagrangian on the vector field in the kinetic energy with respect to first order perturbation of the flow. This requires adjustment by the Lie bracket of vector field, given by operator $ad\_v:\; w\; \backslash in\; V\; \backslash mapsto\; V$ which involves the Jacobian given by

:$ad\_v[w]\backslash doteq\; [v,w]\backslash doteq\; (Dv)w-(Dw)v\; \backslash in\; V$.

Defining the adjoint $ad\_v^*:\; V^*\; \backslash rightarrow\; V^*,$ then the first order variation gives the Eulerian shape momentum $Av\; \backslash in\; V^*$ satisfying the generalized equation:

{{NumBlk|:|$\backslash frac\{d\}\{dt\}\; Av\_t\; +\; ad\_\{v\_t\}^*\; (Av\_t)=0\; \backslash \; ,\; \backslash \; t\; \backslash in\; [0,1]\; \backslash \; ;$

|{{EquationRef|EL-general}}}}

meaning for all smooth $w\; \backslash in\; V\; ,$

:$\backslash int\_X\; \backslash left(\; \backslash frac\{d\}\{dt\}\; Av\_t\; +\; ad\_\{v\_t\}^*\; (Av\_t)\; \backslash right)\; \backslash cdot\; w\; \backslash ,\; dx=\backslash int\_X\; \backslash frac\{d\}\{dt\}\; Av\_t\; \backslash cdot\; w\; \backslash ,\; dx\; +\; \backslash int\_X\; Av\_t\; \backslash cdot\; ((Dv\_t)w-(Dw)v\_t)\; dx=0\; .$

Computational anatomy is the study of the motions of submanifolds, points, curves, surfaces and volumes.

Momentum associated to points, curves and surfaces are all singular, implying the momentum is concentrated on subsets of $\{\backslash mathbb\; R\}^3$ which are dimension $\backslash leq\; 2$ in Lebesgue measure. In such cases, the energy is still well defined $(Av\_t\backslash mid\; v\_t)$ since although $Av\_t$ is a generalized function, the vector fields are smooth and the Eulerian momentum is understood via its action on smooth functions. The perfect illustration of this is even when it is a superposition of delta-diracs, the velocity of the coordinates in the entire volume move smoothly. The Euler–Lagrange equation ({{EquationNote|EL-general}}) on diffeomorphisms for generalized functions $Av\; \backslash in\; V^*$ was derived in.M.I. Miller, A. Trouve, L. Younes, Geodesic Shooting in Computational Anatomy, IJCV, 2006. In Riemannian Metric and Lie-Bracket Interpretation of the Euler–Lagrange Equation on Geodesics derivations are provided in terms of the adjoint operator and the Lie bracket for the group of diffeomorphisms. It has come to be called EPDiff equation for diffeomorphisms connecting to the Euler-Poincare method having been studied in the context of the inertial operator $A=identity$ for incompressible, divergence free, fluids.{{cite journal

| last1=Holm | first1=Darryl D.

| last2=Marsden | first2=Jerrold E.

| last3=Ratiu | first3=Tudor S. | authorlink3=Tudor Ratiu

| year=1998

| title=The Euler-Poincaré equations and semidirect products with applications to continuum theories

| journal=Advances in Mathematics

| volume=137

| pages=1–81

| doi=10.1006/aima.1998.1721 | doi-access=free

| arxiv=chao-dyn/9801015

| s2cid=163598}}

For the momentum density case $(Av\_t\; \backslash mid\; w)=\backslash int\_X\; \backslash mu\_t\; \backslash cdot\; w\; \backslash ,\; dx$, then Euler–Lagrange equation has a classical solution:{{NumBlk|:|

: $$

\frac{d}{dt} \mu_t + (Dv_t)^T \mu_t +(D\mu_t)v_t + ( \nabla \cdot v) \mu_t=0 \ , t \in [0,1].

|{{EquationRef|EL-Classic}}}}The Euler–Lagrange equation on diffeomorphisms, classically defined for momentum densities first appeared in{{cite journal|last1=Miller|first1=M.I.|last2=Trouve|first2=A.|last3=Younes|first3=L|year=2002|title=On the Metrics and Euler–Lagrange equations of Computational Anatomy|journal=Annu. Rev. Biomed. Eng.|volume=4|pages=375–405|doi=10.1146/annurev.bioeng.4.092101.125733|pmid=12117763|citeseerx=10.1.1.157.6533 }} for medical image analysis.

{{Further|Riemannian metric and Lie-bracket in computational anatomy}}

In medical imaging and computational anatomy, positioning and coordinatizing shapes are fundamental operations; the system for positioning anatomical coordinates and shapes built on the metric and the Euler–Lagrange equation a geodesic positioning system as first explicated in Miller Trouve and Younes.

Solving the geodesic from the initial condition $v\_0$ is termed the Riemannian-exponential, a mapping $\backslash operatorname\{Exp\}\_\{\backslash rm\; id\}(\backslash cdot):\; V\; \backslash to\; \backslash operatorname\{Diff\}\_V$

at identity to the group.

The Riemannian exponential satisfies $$

\operatorname{Exp}_{\rm id} (v_0)=\varphi_1 for initial condition $\backslash dot\; \backslash varphi\_0=v\_0$, vector field dynamics $\backslash dot\; \backslash varphi\_t=v\_t\; \backslash circ\; \backslash varphi\_t,\; t\; \backslash in\; [0,1]$

,

• for classical equation diffeomorphic shape momentum $\backslash int\_X\; Av\_t\; \backslash cdot\; w\; \backslash ,\; dx$, $Av\; \backslash in\; V$, then

:$$

\ \ \ \frac{d}{dt} Av_t + (Dv_t)^T Av_t +(DAv_t)v_t + ( \nabla \cdot v) Av_t=0 \ ;

• for generalized equation, then $$

Av \in V^* , $w\; \backslash in\; V$,

:$\backslash \; \backslash \; \backslash \; \backslash int\_X\; \backslash frac\{d\}\{dt\}\; Av\_t\; \backslash cdot\; w\; \backslash ,\; dx\; +\; \backslash int\_X\; Av\_t\; \backslash cdot\; ((Dv\_t)w-(Dw)v\_t)\; \backslash ,\; dx=0\; .$

Computing the flow $v\_0$ onto coordinates Riemannian logarithm,{{Cite journal|title=Diffeomorphometry and geodesic positioning systems for human anatomy|journal=Technology|date=2014-03-01|pmc=4041578|pmid=24904924|pages=36–43|volume=2|issue=1|doi=10.1142/S2339547814500010|first1=Michael I.|last1=Miller|first2=Laurent|last2=Younes|first3=Alain|last3=Trouvé}} mapping $Log\_\{\backslash rm\; id\}(\backslash cdot):\; \backslash operatorname\{Diff\}\_V\; \backslash to\; V$ at identity from $\backslash varphi$ to vector field $v\_0\; \backslash in\; V$;

: $$

\log_{\rm id}( \varphi)=v_0 \ \text{initial condition of EL geodesic} \dot \varphi_0=v_0, \varphi_0=id, \varphi_1=\varphi \ .

Extended to the entire group they become

: $$

\varphi=\operatorname{Exp}_\varphi(v_0\circ \varphi) \doteq \operatorname{Exp}_{\rm id} (v_0) \circ \varphi

; $$

\log_\varphi(\varphi)\doteq \log_{\rm id}( \varphi \circ\varphi^{-1}) \circ \varphi

.

These are inverses of each other for unique solutions of Logarithm; the first is called geodesic positioning, the latter geodesic coordinates (see exponential map, Riemannian geometry for the finite dimensional version). The geodesic metric is a local flattening of the Riemannian coordinate system (see figure).

File:FigBrain.png

In computational anatomy the diffeomorphisms are used to push the coordinate systems, and the vector fields are used

as the control within the

anatomical orbit or morphological space. The model is that of a dynamical system, the flow of coordinates $t\; \backslash mapsto\; \backslash varphi\_t\; \backslash in\; \backslash operatorname\{Diff\}\_V$ and the control the vector field $t\; \backslash mapsto\; v\_t\; \backslash in\; V$ related via $\backslash dot\; \backslash varphi\_t=v\_t\; \backslash cdot\; \backslash varphi\_t,\backslash varphi\_0=\{\backslash rm\; id\}.$ The Hamiltonian view

{{Cite journal|title=Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'arcy Thompson|journal=Annual Review of Biomedical Engineering|date=2015-01-01|volume=17|issue=1|doi=10.1146/annurev-bioeng-071114-040601|first1=Michael I.|last1=Miller|first2=Alain|last2=Trouvé|first3=Laurent|last3=Younes|pages=447–509|pmid=26643025}}

Glaunès J, Trouvé A, Younes L. 2006. Modeling planar shape variation via Hamiltonian flows of curves.

In Statistics and Analysis of Shapes, ed. H Krim, A Yezzi Jr, pp. 335–61. Model. Simul. Sci. Eng. Technol.

Boston: Birkhauser

{{cite arXiv|eprint=1401.0661|last1=Micheli|first1=Mario|title=Shape deformation analysis from the optimal control viewpoint|last2=Michor|first2=Peter W.|last3=Mumford|first3=David|last4=Younes|first4=Laurent|class=math.OC|year=2014}}

{{cite journal | doi = 10.1142/S2339547814500010 | pmid=24904924 | pmc=4041578 | volume=2 | title=Diffeomorphometry and geodesic positioning systems for human anatomy | year=2014 | journal=Technology (Singap World Sci) | pages=36–43 | last1 = Miller | first1 = MI | last2 = Younes | first2 = L | last3 = Trouvé | first3 = A| issue=1 }}

{{Cite journal|title=An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach|journal=Applied and Computational Harmonic Analysis|date=2007-07-01|pages=74–113|volume=23|series=Special Issue on Mathematical Imaging|issue=1|doi=10.1016/j.acha.2006.07.004|first1=Peter W.|last1=Michor|first2=David|last2=Mumford|arxiv=math/0605009|s2cid=732281}} reparameterizes the momentum distribution $Av\; \backslash in\; V^*$ in terms of the conjugate momentum or canonical momentum, introduced as a Lagrange multiplier $p:\; \backslash dot\; \backslash varphi\; \backslash mapsto\; (p\backslash mid\backslash dot\; \backslash varphi)$ constraining the Lagrangian velocity $\backslash dot\; \backslash varphi\_t=v\_t\; \backslash circ\; \backslash varphi\_t$.accordingly:

:$$

H(\varphi_t,p_t,v_t)=\int_X p_t \cdot (v_t \circ \varphi_t) \, dx-\frac{1}{2}\int_X Av_t \cdot v_t \, dx .

This function is the extended Hamiltonian. The Pontryagin maximum principle gives the optimizing vector field which determines the geodesic flow satisfying $\backslash dot\; \backslash varphi\_t=v\_t\; \backslash circ\; \backslash varphi\_t\; ,\; \backslash varphi\_0=\{\backslash rm\; id\},$ as well as the reduced Hamiltonian

:$H(\backslash varphi\_t,p\_t)\; \backslash doteq\; \backslash max\_v\; H(\; \backslash varphi\_t,\; p\_t,v)\; \backslash \; .$

The Lagrange multiplier in its action as a linear form has its own inner product of the canonical momentum acting on the velocity of the flow which is dependent on the shape, e.g. for landmarks a sum, for surfaces a surface integral, and. for volumes it is a volume integral with respect to $dx$ on $\{\backslash mathbb\; R\}^3$. In all cases the Greens kernels carry weights which are the canonical momentum evolving according to an ordinary differential equation which corresponds to EL but is the geodesic reparameterization in canonical momentum. The optimizing vector field is given by

:$v\_t\; \backslash doteq\; \backslash operatorname\{argmax\}\_v\; H(\backslash varphi\_t,p\_t,v)$

with dynamics of canonical momentum reparameterizing the vector field along the geodesic

{{NumBlk|:|$\backslash begin\{cases\}\; \backslash displaystyle$

\dot \varphi_t=\frac{\partial H( \varphi_t, p_t)}{\partial p} \\[6pt]

\displaystyle \dot p_t=-\frac{\partial H(\varphi_t,p_t)}{\partial \varphi}

\end{cases}

|{{EquationRef|Hamiltonian-dynamics}}}}

Whereas the vector fields are extended across the entire background space of $\{\backslash mathbb\; R\}^3$, the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function $Av\_t\; \backslash in\; V^*$ concentrated to the submanifolds. For landmarks{{cite journal | last1 = Joshi | first1 = S. | last2 = Miller | first2 = M.I. | year = 2000 | title = Landmark matching via large deformation diffeomorphisms | journal = IEEE Transactions on Image Processing| volume = 9 | issue = 8| pages = 1357–70 | doi = 10.1109/83.855431 | pmid = 18262973 | bibcode = 2000ITIP....9.1357J }}{{cite book | doi=10.1007/3-540-44745-8_34 | chapter=Geodesic Interpolating Splines | title=Energy Minimization Methods in Computer Vision and Pattern Recognition | series=Lecture Notes in Computer Science | year=2001 | last1=Camion | first1=Vincent | last2=Younes | first2=Laurent | volume=2134 | pages=513–527 | isbn=978-3-540-42523-6 }}{{cite journal | doi=10.1023/B:JMIV.0000011323.32914.f3 | title=Affine Invariant Flows in the Beltrami Framework | year=2004 | last1=Sochen | first1=Nir | journal=Journal of Mathematical Imaging and Vision | volume=20 | pages=133–146 | s2cid=11969555 }} the geodesics have Eulerian shape momentum which are a superposition of delta distributions travelling with the finite numbers of particles; the diffeomorphic flow of coordinates have velocities in the range of weighted Green's Kernels. For surfaces, the momentum is a surface integral of delta distributions travelling with the surface.

The geodesics connecting coordinate systems satisfying {{EquationNote|EL-general}} have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path $t\; \backslash in\; [0,1]$, $H(\backslash varphi,p)=\backslash max\_v\; H(\backslash varphi,p,v)$

, equalling the {{EquationNote|Lagrangian-kinetic-energy}} and is stationary along {{EquationNote|EL-general}}. Defining the geodesic velocity at the identity $v\_0=\backslash arg\; \backslash max\_v\; H(\backslash varphi\_0,p\_0,v)$

, then along the geodesic

{{NumBlk|::|$H(\backslash varphi\_t,p\_t)=H(\backslash varphi\_0,p\_0)=\backslash frac\{1\}\{2\}\; \backslash int\_X\; p\_0\; \backslash cdot\; v\_0\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\; \backslash int\_X\; Av\_0\; \backslash cdot\; v\_0\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\; \backslash int\_X\; Av\_t\; \backslash cdot\; v\_t\; \backslash ,\; dx$

|{{EquationRef|Hamiltonian-geodesics}}}}The stationarity of the Hamiltonian demonstrates the interpretation of the Lagrange multiplier as momentum; integrated against velocity $\backslash dot\; \backslash varphi$ gives energy density. The canonical momentum has many names. In optimal control, the flows $\backslash varphi$ is interpreted as the state, and $p$ is interpreted as conjugate state, or conjugate momentum.{{Cite journal|title=Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson|journal=Annual Review of Biomedical Engineering|date=2015-01-01|pmid=26643025|pages=447–509|volume=17|issue=1|doi=10.1146/annurev-bioeng-071114-040601|first1=Michael I.|last1=Miller|first2=Alain|last2=Trouvé|first3=Laurent|last3=Younes}} The geodesi of EL implies specification of the vector fields $v\_0$ or Eulerian momentum $Av\_0$ at $t=0$, or specification of canonical momentum $p\_0$ determines the flow.

In computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitives. The geodesic flows between the submanifolds determine the distance, and form the basic measuring and transporting tools of diffeomorphometry. At $t=0$ the geodesic has vector field $v\_0=K\; p\_0$ determined by the conjugate momentum and the Green's kernel of the inertial operator defining the Eulerian momentum $K=A^\{-1\}$. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

:$d\_\{\backslash operatorname\{Diff\}\_V\}(\{\backslash rm\; id\},\backslash varphi)=\backslash |\backslash log\_\{\backslash rm\; id\}(\backslash varphi)\backslash |\_V=\backslash |v\_0\; \backslash |\_V=\backslash sqrt\{2H(\{\backslash rm\; id\},p\_0)\}$

Given the least-action there is a natural definition of momentum associated to generalized coordinates; the quantity acting against velocity gives energy. The field has studied two forms, the momentum associated to the Eulerian vector field termed Eulerian diffeomorphic shape momentum, and the momentum associated to the initial coordinates or canonical coordinates termed canonical diffeomorphic shape momentum. Each has a conservation law. The conservation of momentum goes hand in hand with the {{EquationNote|EL-general}}. In computational anatomy, $Av$ is the Eulerian momentum since when integrated against Eulerian velocity $v$ gives energy density; operator $A$ the generalized moment of inertia or inertial operator which acting on the Eulerian velocity gives momentum which is conserved along the geodesic: {{NumBlk|:|

\begin{align}

\text{Eulerian } &

\frac{d}{dt} \int_X Av_t \cdot ((D \varphi_t) w) \circ \varphi_t^{-1}) \, dx = 0 \ , \ t \in [0,1] .

\\[8pt]

\text{Canonical } &

\frac{d}{dt} \int_X p_t \cdot ( (D\varphi_t) w) \, dx=0 \ , \ t \in [0,1] \ \text{ for all} \ w \in V \ .

\end{align}

|{{EquationRef|Euler-conservation-constant-energy}}}}

Conservation of Eulerian shape momentum was shown in{{Cite journal|title=Geodesic Shooting for Computational Anatomy|journal=Journal of Mathematical Imaging and Vision|date=2006-01-31|pmc=2897162|pmid=20613972|pages=209–228|volume=24|issue=2|doi=10.1007/s10851-005-3624-0|first1=MICHAEL I.|last1=MILLER|first2=ALAIN|last2=TROUVÉ|first3=LAURENT|last3=YOUNES|bibcode=2006JMIV...24..209M }} and follows from {{EquationNote|EL-general}}; conservation of canonical momentum was shown in

{{hidden

|Proof of conservation

|The proof follow from defining $w\_t=((D\; \backslash varphi\_t)\; w)\; \backslash circ\; \backslash varphi\_t^\{-1\}$, $\backslash frac\{d\}\{dt\}\; w\_t=(Dv\_t)\; w\_t\; -\; (D\; w\_t)\; v\_t$ implying

$\backslash frac\{d\}\{dt\}\; (Av\_t\backslash mid((D\; \backslash varphi\_t)\; w)\; \backslash circ\; \backslash varphi\_t^\{-1\})=(\backslash frac\{d\}\{dt\}Av\_t\backslash mid((D\; \backslash varphi\_t)\; w)\; \backslash circ\; \backslash varphi\_t^\{-1\})+(Av\_t\backslash mid\backslash frac\{d\}\{dt\}\; ((D\; \backslash varphi\_t)\; w)\; \backslash circ\; \backslash varphi\_t^\{-1\})=(\backslash frac\{d\}\{dt\}\; Av\_t\backslash mid\; w\_t\; )+(Av\_t\backslash mid(D\; v\_t)\; w\_t-(Dw\_t)v\_t)=0.$

The proof on canonical momentum is shown from $\backslash dot\; p\_t=-\; (Dv\_t)\_\{|\_\{\backslash varphi\_t\}\}^T\; p\_t$:

: $\backslash frac\{d\}\{dt\}\; (p\_t\backslash mid(D\backslash varphi\_t)\; w)=(\backslash dot\; p\_t\; \backslash mid\; (D\backslash varphi\_t)\; w\; )\; +(p\_t\backslash mid\backslash frac\{d\}\{dt\}\; (D\backslash varphi\_t)\; w)$

=(\dot p_t\mid(D \varphi_t) w)+(p_t\mid(Dv_t)_{|_{\varphi_t} }(D \varphi_t) w)=0.

}}

{{Further|Large deformation diffeomorphic metric mapping}}

Construction of diffeomorphic correspondences between shapes calculates the initial vector field coordinates $v\_0\; \backslash in\; V$ and associated weights on the Greens kernels $p\_0$. These initial coordinates are determined by matching of shapes, called large-deformation diffeomorphic metric mapping (LDDMM). LDDMM has been solved for landmarks with and without correspondence{{Cite book|title=Geodesic Interpolating Splines|url=http://dl.acm.org/citation.cfm?id=646596.756898|journal=Proceedings of the Third International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition|date=2001-01-01|isbn=978-3-540-42523-6|pages=513–527|series=EMMCVPR '01|first1=Vincent|last1=Camion|first2=Laurent|last2=Younes|doi = 10.1007/3-540-44745-8_34}}{{Cite journal|title=Statistics on diffeomorphisms via tangent space representations|journal=NeuroImage|date=2004-01-01|pmid=15501085|pages=S161–169|volume=23|doi=10.1016/j.neuroimage.2004.07.023|first1=M.|last1=Vaillant|first2=M. I.|last2=Miller|first3=L.|last3=Younes|first4=A.|last4=Trouvé|issue=Suppl 1 |citeseerx=10.1.1.132.6802|s2cid=8255538}}{{Cite book |date=2007-01-01|pmid=17633716|pages=396–407|volume=20|first1=Stephen|last1=Marsland|first2=Robert|last2=McLachlan|title=Information Processing in Medical Imaging |chapter=A Hamiltonian Particle Method for Diffeomorphic Image Registration |doi=10.1007/978-3-540-73273-0_33|series=Lecture Notes in Computer Science|isbn=978-3-540-73272-3}}{{Cite book|volume=2|pages=712–718|url=https://www.researchgate.net/publication/4082354 |access-date=2015-11-25|doi=10.1109/CVPR.2004.1315234|year=2004|last1=Glaunes|first1=J|last2=Trouve|first2=A|last3=Younes|first3=L|title=Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 |chapter=Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching |isbn=978-0-7695-2158-9|citeseerx=10.1.1.158.4209}} and for dense image matchings.{{Cite journal|title=Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal=International Journal of Computer Vision|date=2005-02-01|pages=139–157|volume=61|issue=2|doi=10.1023/B:VISI.0000043755.93987.aa|first1=M. Faisal|last1=Beg|first2=Michael I.|last2=Miller|first3=Alain|last3=Trouvé|first4=Laurent|last4=Younes|s2cid=17772076}} curves,{{Cite journal|title=Large Deformation Diffeomorphic Metric Curve Mapping|journal=International Journal of Computer Vision|date=2008-12-01|pmc=2858418|pmid=20419045|pages=317–336|volume=80|issue=3|doi=10.1007/s11263-008-0141-9|first1=Joan|last1=Glaunès|first2=Anqi|last2=Qiu|first3=Michael I.|last3=Miller|first4=Laurent|last4=Younes}} surfaces,{{Cite book|date=2005-01-01|pages=381–392|first1=Marc|last1=Vaillant|first2=Joan|last2=Glaunès|title=Information Processing in Medical Imaging |chapter=Surface Matching via Currents |journal=Information Processing in Medical Imaging: Proceedings of the ... Conference |volume=19|citeseerx=10.1.1.88.4666|pmid=17354711|doi=10.1007/11505730_32|isbn=978-3-540-26545-0|series=Lecture Notes in Computer Science}} dense vector{{Cite book|date=2005-10-01|pages=1379–1386 Vol. 2|volume=2|doi=10.1109/ICCV.2005.132|first1=Yan|last1=Cao|first2=M.I.|last2=Miller|first3=R.L.|last3=Winslow|first4=L.|last4=Younes|title=Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1 |chapter=Large deformation diffeomorphic metric mapping of fiber orientations |isbn=978-0-7695-2334-7|citeseerx=10.1.1.158.1582|s2cid=13019795}} and tensor{{Cite journal|title=Large deformation diffeomorphic metric mapping of vector fields|journal=IEEE Transactions on Medical Imaging|date=2005-09-01|pmid=16156359|pages=1216–1230|volume=24|issue=9|pmc=2848689|doi=10.1109/TMI.2005.853923|first1=Yan|last1=Cao|first2=M.I.|last2=Miller|first3=R.L.|last3=Winslow|first4=L.|last4=Younes|citeseerx=10.1.1.157.8377|s2cid=7046743}} imagery, and varifolds removing orientation.{{Cite journal|title=The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration|journal=SIAM Journal on Imaging Sciences|date=2013-01-01|pages=2547–2580|volume=6|issue=4|doi=10.1137/130918885|first1=N.|last1=Charon|first2=A.|last2=Trouvé|arxiv=1304.6108|bibcode=2013arXiv1304.6108C|s2cid=14335966}} LDDMM calculates geodesic flows of the {{EquationNote|EL-general}} onto target coordinates, adding to the action integral $\backslash frac\{1\}\{2\}\; \backslash int\_0^1\; \backslash int\_X\; Av\_t\backslash cdot\; v\_t\; \backslash ,\; dx$

\, dt an endpoint matching condition $E:\; \backslash varphi\_1\; \backslash rightarrow\; R^+$ measuring the correspondence of elements in the orbit under coordinate system transformation. Existence of solutions were examined for image matching. The solution of the variational problem satisfies the {{EquationNote|EL-general}} for $t\; \backslash in\; [0,1)$ with boundary condition.

: $$

\begin{align}

& \min \left\{ C(\varphi) : v=\dot \varphi \circ \varphi^{-1}, \varphi_0={\rm id} \right\} \\[5pt]

\doteq {} & \frac{1}{2} \int_0^1 \int_X Av_t \cdot v_t \, dx \, dt +E( \varphi_1)

\end{align}

: $$

\begin{cases}

\text{Euler conservation } & \displaystyle \frac{d}{dt} Av_t + ad_{v_t}^* (Av_t)=0 , \ t \in [0,1) \ ,

\\

\text{Boundary condition } & \displaystyle \varphi_0={\rm id}, Av_1 = \left. - \frac{\partial E(\varphi)}{\partial \varphi} \right|_{\varphi=\varphi_1} \ .

\end{cases}

Conservation from {{EquationNote|EL-general}} extends the B.C. at $t=1$ to the rest of the path $$

t \in [0,1) . The inexact matching problem with the endpoint matching term $E(\backslash varphi\_1)$ has several alternative forms. One of the key ideas of the stationarity of the Hamiltonian along the geodesic solution is the integrated running cost reduces to initial cost at t = 0, geodesics of the {{EquationNote|EL-general}} are determined by their initial condition $v\_0$.

The running cost is reduced to the initial cost determined by $v\_0=Kp\_0$ of {{EquationNote|Kernel-Surf.-Land.-Geodesics}}.

:$$

\begin{align}

& \min_{v_0} C(v_0) \doteq \frac{1}{2} \int_X Av_0 \cdot v_0 \, dx +E(\mathrm{Exp}_\mathrm{id}(v_0) \cdot I_0) \\[6pt]

& \min_{p_0} C(p_0) = \frac{1}{2}\int_X p_0 \cdot Kp_0 \, dx+ E(\mathrm{Exp}_\text{id}(Kp_0) \cdot I_0)

\end{align}

The matching problem explicitly indexed to initial condition $v\_0$ is called shooting, which can also be reparamerized via the conjugate momentum $p\_0$.

{{Further|LDDMM|Large deformation diffeomorphic metric mapping#The variational problem of dense image matching}}

Dense image matching has a long history now with the earliest efforts{{Cite journal|title=A computerized system for the elastic matching of deformed radiographic images to idealized atlas images|journal=Journal of Computer Assisted Tomography|date=1983-08-01|pmid=6602820|pages=618–625|volume=7|issue=4|first1=R.|last1=Bajcsy|first2=R.|last2=Lieberson|first3=M.|last3=Reivich|doi=10.1097/00004728-198308000-00008}}{{Cite journal|title=Structural Image Restoration through Deformable Templates|journal=Journal of the American Statistical Association|date=1991-06-01|pages=376–387|volume=86|issue=414|doi=10.1080/01621459.1991.10475053|first1=Yali|last1=Amit|first2=Ulf|last2=Grenander|first3=Mauro|last3=Piccioni}} exploiting a small deformation framework. Large deformations began in the early 1990s, with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense image matching established in. Beg solved via one of the earliest LDDMM algorithms based on solving the variational matching with endpoint defined by the dense imagery with respect to the vector fields, taking variations with respect to the vector fields. Another solution for dense image matching reparameterizes the optimization problem in terms of the state $q\_t\; \backslash doteq\; I\; \backslash circ\; \backslash varphi\_t^\{-1\},\; q\_0=I$ giving the solution in terms of the infinitesimal action defined by the advection equation.{{Cite journal|title=Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation|journal=Int. J. Comput. Vis.|date=2012-04-01|pages=229–241|volume=97|issue=2|doi=10.1007/s11263-011-0481-8|first1=François-Xavier|last1=Vialard|first2=Laurent|last2=Risser|first3=Daniel|last3=Rueckert|first4=Colin J.|last4=Cotter|s2cid=18251140}}

{{Further|Large deformation diffeomorphic metric mapping#Beg's LDDMM algorithm for image matching}}

For Beg's LDDMM, denote the Image $I(x),\; x\; \backslash in\; X$ with group action $\backslash varphi\; \backslash cdot\; I\; \backslash doteq\; I\; \backslash circ\; \backslash varphi^\{-1\}$.

Viewing this as an optimal control problem, the state of the system is the diffeomorphic flow of coordinates $\backslash varphi\_t,\; t\; \backslash in\; [0,1]$, with the dynamics relating the control $v\_t,\; t\; \backslash in\; [0,1]$ to the state given by $\backslash dot\; \backslash varphi=v\; \backslash circ\; \backslash varphi$. The endpoint matching condition $E(\backslash varphi\_1)\; \backslash doteq\; \backslash |I\backslash circ\; \backslash varphi\_1^\{-1\}\; -I^\backslash prime\backslash |^2$ gives the variational problem

{{NumBlk|:|

\begin{align}

& \ \ \ \ \ \min_{v: \dot \varphi=v \circ \varphi} C(v) \doteq \frac{1}{2} \int_0^1 \int_X Av_t \cdot v_t \, dx \, dt +\frac{1}{2} \int_{{\mathbb R}^3}| I\circ \varphi_1^{-1}(x)-I^\prime (x)|^2 \, dx

\end{align}

|{{EquationRef|Dense-image-matching}}}}

:$$

\begin{cases}

\text{Endpoint condition:} & Av_1=\mu_1 \, dx, \mu_1=(I\circ \varphi_1^{-1}-I^\prime ) \nabla (I \circ \varphi_1^{-1}) \ ,\\[5pt]

\text{Conservation:} & Av_t=\mu_t \, dx, \ \mu_t=(D \varphi_t^{-1})^T \mu_0 \circ \varphi_t^{-1}|D \varphi_t^{-1}|\ . \\[5pt]

& \mu_0=(I - I^\prime \circ \varphi_1) \nabla I|D \varphi_1|\ . \\

\end{cases}

Beg's iterative LDDMM algorithm has fixed points which satisfy the necessary optimizer conditions. The iterative algorithm is given in Beg's LDDMM algorithm for dense image matching.

{{Further|Large Deformation Diffeomorphic Metric Mapping#Hamiltonian LDDMM for Dense Image Matching}}

Denote the Image $I(x),\; x\; \backslash in\; X$, with state $q\_t\; \backslash doteq\; I\; \backslash circ\; \backslash varphi\_t^\{-1\}$ and the dynamics related state and control given by the advective term $\backslash dot\; q\_t=-\; \backslash nabla\; q\_t\; \backslash cdot\; v\_t$. The endpoint $$

E(q_1) \doteq \|q_1-I^\prime\|^2

gives the variational problem

{{NumBlk|::|

\min_{q: \dot q=v \circ q} C(v) \doteq \frac{1}{2} \int_0^1 \int_X Av_t \cdot v_t \, dx \, dt +\frac{1}{2} \int_{{\mathbb R}^3}|q_1(x)-I^\prime(x)|^2 \, dx

|{{EquationRef|Dense-image-matching}}}}

Viallard's iterative Hamiltonian LDDMM has fixed points which satisfy the necessary optimizer conditions.

{{Further|Group actions in computational anatomy|LDDMM}}

File:Dti-MRI-brain-section.png

Dense LDDMM tensor matching{{Cite book|date=2006-06-01|pmc=2920614|pmid=20711423|pages=67|doi=10.1109/CVPRW.2006.65|first1=Yan|last1=Cao|first2=M.I.|last2=Miller|first3=Susumu|last3=Mori|first4=R.L.|last4=Winslow|first5=L.|last5=Younes|title=2006 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW'06) |chapter=Diffeomorphic Matching of Diffusion Tensor Images |volume=2006|isbn=978-0-7695-2646-1}} takes the images as 3x1 vectors and 3x3 tensors solving the variational problem matching between coordinate system based on the principle eigenvectors of the diffusion tensor MRI image (DTI) denoted $M(x),\; x\; \backslash in\; \{\backslash mathbb\; R\}^3$ consisting of the $3\; \backslash times\; 3$-tensor at every voxel. Several of the group actions defined based on the Frobenius matrix norm between square matrices $\backslash |A\; \backslash |\_F^2\; \backslash doteq\; \backslash operatorname\{trace\}\; A^T\; A$. Shown in the accompanying figure is a DTI image illustrated via its color map depicting the eigenvector orientations of the DTI matrix at each voxel with color determined by the orientation of the directions.

Denote the $3\; \backslash times\; 3$ tensor image $M(x),\; x\; \backslash in\; \{\backslash mathbb\; R\}^3$ with eigen-elements $\backslash \{\backslash lambda\_i(x),\; e\_i(x),\; i=1,2,3\backslash \}$, $\backslash lambda\_1\; \backslash geq\; \backslash lambda\_2\; \backslash geq\; \backslash lambda\_3$.

Coordinate system transformation based on DTI imaging has exploited two actions

one based on the principle eigen-vector or entire matrix.

LDDMM matching based on the principal eigenvector of the diffusion tensor matrix

takes the image $I(x),\; x\; \backslash in\; \{\backslash mathbb\; R\}^3$ as a unit vector field defined by the first eigenvector. The group action becomes

:$$

\varphi \cdot I=

\begin{cases}

\frac{D_{\varphi^{-1}} \varphi I \circ \varphi^{-1} \|I \circ \varphi^{-1}\|}{\|D_{\varphi^{-1}} \varphi I \circ \varphi^{-1}\|} & I\circ \varphi \neq 0; \\

0 & \text{otherwise.}

\end{cases}

LDDMM matching based on the entire tensor matrix

has group action becomes $\backslash varphi\; \backslash cdot\; M=(\backslash lambda\_1\; \backslash hat\{e\}\_1\; \backslash hat\{e\}\_1^\{T\}+\backslash lambda\_2\; \backslash hat\{e\}\_2\; \backslash hat\{e\}\_2^\{T\}\; +\backslash lambda\_3\; \backslash hat\{e\}\_3\; \backslash hat\{e\}\_3^\{T\})\; \backslash circ\; \backslash varphi^\{-1\}\; ,$ transformed eigenvectors

:$\backslash begin\{align\}$

\hat{e}_1 &=\frac{D \varphi e_1}{\|D \varphi e_1\|} \ , \ \ \

\hat{e}_2=\frac{D \varphi e_2-\langle \hat{e}_1,D \varphi e_2\rangle \hat{e}_1}{\sqrt{\|D \varphi e_2\|^2-\langle \hat{e}_1,D \varphi e_2\rangle^2}}\ , \ \ \

\hat{e}_3=\hat{e}_1 \times \hat{e}_2

\end{align}.

The variational problem matching onto the principal eigenvector or the matrix is described

LDDMM Tensor Image Matching.

{{Further|Group actions in computational anatomy|LDDMM#LDDMM ODF}}

High angular resolution diffusion imaging (HARDI) addresses the well-known limitation of DTI, that is, DTI can only reveal one dominant fiber orientation at each location. HARDI measures diffusion along $n$ uniformly distributed directions on the sphere and can characterize more complex fiber geometries. HARDI can be used to reconstruct an orientation distribution function (ODF) that characterizes the angular profile of the diffusion probability density function of water molecules. The ODF is a function defined on a unit sphere, $\{\backslash mathbb\{S\}\}^2$.

Dense LDDMM ODF matching {{cite journal|last1=Du|first1=J|last2=Goh|first2=A|last3=Qiu|first3=A|title=Diffeomorphic metric mapping of high angular resolution diffusion imaging based on Riemannian structure of orientation distribution functions|journal= IEEE Transactions on Medical Imaging|date=2012|volume=31|issue=5|pages=1021–1033|doi=10.1109/TMI.2011.2178253|pmid=22156979|s2cid=11533837}} takes the HARDI data as ODF at each voxel and solves the LDDMM variational problem in the space of ODF. In the field of information geometry,{{cite book|last1=Amari|first1=S|title=Differential-Geometrical Methods in Statistics|date=1985|publisher=Springer}} the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. For the purpose of LDDMM ODF mapping, the square-root representation is chosen because it is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form. In the following, denote square-root ODF ($\{\backslash sqrt\{\backslash text\{ODF\}\}\}$) as $\backslash psi(\{\backslash bf\; s\})$, where $\backslash psi(\{\backslash bf\; s\})$ is non-negative to ensure uniqueness and $\backslash int\_\{\{\backslash bf\; s\}\backslash in\; \{\backslash mathbb\{S\}\}^2\}\; \backslash psi^2(\{\backslash bf\; s\})\; \backslash ,\; d\{\backslash bf\; s\}=1$. The variational problem for matching assumes that two ODF volumes can be generated from one to another via flows of diffeomorphisms $\backslash varphi\_t$, which are solutions of ordinary differential equations $\backslash dot\; \backslash varphi\_t=v\_t\; (\backslash varphi\_t),\; t\; \backslash in\; [0,1],$ starting from the identity map $\backslash varphi\_0=\{\backslash rm\; id\}$. Denote the action of the diffeomorphism on template as $\backslash varphi\_1\; \backslash cdot\; \backslash psi\_\{\backslash mathrm\{temp\}\}(\{\backslash bf\; s\},x)$, $\{\backslash bf\; s\}\; \backslash in\; \{\backslash mathbb\{S\}\}^2$, $x\; \backslash in\; X$ are respectively the coordinates of the unit sphere, $\{\{\backslash mathbb\{S\}\}^2\}$ and the image domain, with the target indexed similarly, $\backslash psi\_\{\backslash mathrm\{targ\}\}(\{\backslash bf\; s\},\; x)$,$\{\backslash bf\; s\}\; \backslash in\; \{\backslash mathbb\{S\}\}^2$,$x\; \backslash in\; X$.

The group action of the diffeomorphism on the template is given according to

:$\backslash varphi\_1\; \backslash cdot\; \backslash psi\; (x)\backslash doteq\; (D\backslash varphi\_1)\; \backslash psi\; \backslash circ\; \backslash varphi\_1^\{-1\}(x),\; x\; \backslash in\; X$,

where $(D\backslash varphi\_1)$ is the Jacobian of the affine-transformed ODF and is defined as

: $$

(D \varphi_1) \psi\circ \varphi_1^{-1}(x)=\sqrt{\frac{\det{\bigl(D_{\varphi_1^{-1}}\varphi_1 \bigr)^{-1}} }{\left\|{\bigl(D_{\varphi_1^{-1}}\varphi_1 \bigr)^{-1} } {\bf s} \right\|^3} } \quad

\psi \left( \frac{(D_{\varphi_1^{-1}}\varphi_1 \bigr)^{-1} {\bf s}}{\|(D_{\varphi_1^{-1}}\varphi_1 \bigr)^{-1} {\bf s}\|}, \varphi_1^{-1}(x) \right) .

This group action of diffeomorphisms on ODF reorients the ODF and reflects changes in both the magnitude of $\backslash psi$ and the sampling directions of $\backslash bf\; s$ due to affine transformation. It guarantees that the volume fraction of fibers oriented toward a small patch must remain the same after the patch is transformed.

The LDDMM variational problem is defined as

:$$

C(v)=\inf_{v: \dot \varphi_t=

v_t \circ \varphi_t,\varphi_0= {\rm id}} \int_0^1 \int_X Av_t \cdot v_t \, dx \, dt +\lambda \int_{x\in\Omega} \|\log_{(D\varphi_1) \psi_{\mathrm{temp}} \circ \varphi_1^{-1}(x)}(\psi_{\mathrm{targ}}(x))\|^2_{(D\varphi_1) \psi_{\mathrm{temp}}\circ \varphi_1^{-1}(x)} \, dx.

where the logarithm of $\backslash psi\_1,\; \backslash psi\_2\; \backslash in\; \backslash Psi$ is defined as

:$$

\|\log_{\psi_1}(\psi_2)\|_{\psi_1}

=\cos^{-1} \langle \psi_1, \psi_2 \rangle=

\cos^{-1} \left(\int_{{\bf s}\in {\mathbb{S}}^2} \psi_1({\bf s}) \psi_2({\bf s})d{\bf s}\right),

where $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is the normal dot product between points in the sphere under the $\backslash mathrm\{L\}^2$ metric.

This LDDMM-ODF mapping algorithm has been widely used to study brain white matter degeneration in aging, Alzheimer's disease, and vascular dementia.{{cite journal|last1=Thong|first1=JY|last2=Du|first2=J|last3=Ratnarajah|first3=N|last4=Dong|first4=Y|last5=Soon|first5=HW|last6=Saini|first6=M|last7=Tan|first7=MZ|last8=Ta|first8=AT|last9=Chen|first9=C|last10=Qiu|first10=A|s2cid=15230668|title=Abnormalities of cortical thickness, subcortical shapes, and white matter integrity in subcortical vascular cognitive impairment|journal=Hum. Brain Mapp.|date=2014|volume=35|issue=5|pages=2320–2332|doi=10.1002/hbm.22330|pmid=23861356|pmc=6869364}} The brain white matter atlas generated based on ODF is constructed via Bayesian estimation.{{Cite book|last1=DU|first1=J|last2=Goh|first2=A|last3=Qiu|first3=A|title=Geometric Science of Information|chapter=Bayesian Atlas Estimation from High Angular Resolution Diffusion Imaging (HARDI)|series=Lecture Notes in Computer Science|s2cid=8571740|date=2013|volume=8085|pages=149–157|doi=10.1007/978-3-642-40020-9_15|isbn=978-3-642-40019-3}} Regression analysis on ODF is developed in the ODF manifold space in.{{cite journal|last1=Du|first1=J|last2=Goh|first2=A|last3=Kushnarev|first3=S|last4=Qiu|first4=A|title=Geodesic regression on orientation distribution functions with its application to an aging study|journal=NeuroImage|date=2014|volume=87|pages=416–426|doi=10.1016/j.neuroimage.2013.06.081|pmid=23851325|s2cid=26942635}}

{{Further|Metamorphosis}}

File:Metamorphosis-Tumor-Genesis.png Michael Jackson video. Notice the insertion of tumor gray level intensity which does not exist in template.]]

The principle mode of variation represented by the orbit model is change of coordinates. For setting in which pairs of images are not related by diffeomorphisms but have photometric variation or image variation not represented by the template, active appearance modelling has been introduced, originally by Edwards-Cootes-Taylor{{Cite book|title=Active appearance models|publisher=Springer Berlin Heidelberg|date=1998-06-02|isbn=9783540646136|pages=484–498|series=Lecture Notes in Computer Science|first1=T. F.|last1=Cootes|first2=G. J.|last2=Edwards|first3=C. J.|last3=Taylor|editor-first=Hans|editor-last=Burkhardt|editor-first2=Bernd|editor-last2=Neumann}} and in 3D medical imaging in.{{Cite journal|title=Morphological appearance manifolds for group-wise morphometric analysis|journal=Medical Image Analysis|date=2011-12-01|pmc=4392008|pmid=21873104|pages=814–829|volume=15|issue=6|doi=10.1016/j.media.2011.06.003|first1=Nai-Xiang|last1=Lian|first2=Christos|last2=Davatzikos}} In the context of computational anatomy in which metrics on the anatomical orbit has been studied, metamorphosis for modelling structures such as tumors and photometric changes which are not resident in the template was introduced in for magnetic resonance image models, with many subsequent developments extending the metamorphosis framework.{{Cite book|title=Metamorphoses Through Lie Group Action|date=2005-01-01|first1=Alain|last1=Trouvé|first2=Laurent|last2=Younes|citeseerx=10.1.1.157.8752}}{{Cite arXiv|title=The Euler-Poincare theory of Metamorphosis|eprint=0806.0870|date=2008-06-04|first1=Darryl D.|last1=Holm|first2=Alain|last2=Trouve|first3=Laurent|last3=Younes|class=cs.CV }}{{Cite arXiv|title=Metamorphosis of Images in Reproducing Kernel Hilbert Spaces|eprint=1409.6573|date=2014-09-23|first1=Casey L.|last1=Richardson|first2=Laurent|last2=Younes|class=math.OC}}

For image matching the image metamorphosis framework enlarges the action so that $t\; \backslash mapsto\; (\backslash varphi\_t,I\_t)$ with action $\backslash varphi\_t\; \backslash cdot\; I\_t\; \backslash doteq\; I\_t\; \backslash circ\; \backslash varphi\_t^\{-1\}$. In this setting metamorphosis combines both the diffeomorphic coordinate system transformation of computational anatomy as well as the early morphing technologies which only faded or modified the photometric or image intensity alone.

Then the matching problem takes a form with equality boundary conditions:

:$$

\min_{(v,I)}

\frac{1}{2} \int_0^1 \left( \int_X A v_t \cdot v_t \, dx +

\|\dot I_t \circ \varphi_t^{-1} \|^2 / \sigma^2 \right) \, dt \text{ subject to} \ \varphi_0={\rm id}, I_0=\text{fixed}, I_1=\text{fixed}

Transforming coordinate systems based on Landmark point or fiducial marker features dates back to Bookstein's early work on small deformation spline methods{{Cite journal|title=Principal Warps: Thin-Plate Splines and the Decomposition of Deformations|url=http://www.computer.org/csdl/trans/tp/1989/06/i0567.pdf|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|date=1989-01-01|pages=567–585|volume=11|issue=6|doi=10.1109/34.24792|first=F. L.|last=Bookstein}} for interpolating correspondences defined by fiducial points to the two-dimensional or three-dimensional background space in which the fiducials are defined. Large deformation landmark methods came on in the late 1990s.{{Cite book|doi=10.1007/3-540-44745-8_34|publisher=Springer Berlin Heidelberg|date=2001-09-03|isbn=9783540425236|pages=513–527|series=Lecture Notes in Computer Science|first1=Vincent|last1=Camion|first2=Laurent|last2=Younes|title=Energy Minimization Methods in Computer Vision and Pattern Recognition |chapter=Geodesic Interpolating Splines |volume=2134 |editor-first=Mário|editor-last=Figueiredo|editor-first2=Josiane|editor-last2=Zerubia|editor-first3=Anil K.|editor-last3=Jain|citeseerx=10.1.1.57.7394}} The above Figure depicts a series of landmarks associated three brain structures, the amygdala, entorhinal cortex, and hippocampus.

Matching geometrical objects like unlabelled point distributions, curves or surfaces is another common problem in computational anatomy. Even in the discrete setting where these are commonly given as vertices with meshes, there are no predetermined correspondences between points as opposed to the situation of landmarks described above. From the theoretical point of view, while any submanifold $X$ in $\{\backslash mathbb\; R\}^3$, $d=1,2,3$ can be parameterized in local charts $m:\; u\; \backslash in\; U\; \backslash subset\; \{\backslash mathbb\; R\}^\{0,1,2,3\}\; \backslash rightarrow\; \{\backslash mathbb\; R\}^3$, all reparametrizations of these charts give geometrically the same manifold. Therefore, early on in computational anatomy, investigators have identified the necessity of parametrization invariant representations. One indispensable requirement is that the end-point matching term between two submanifolds is itself independent of their parametrizations. This can be achieved via concepts and methods borrowed from Geometric measure theory, in particular currents{{Cite book |date=2005-01-01|pmid=17354711|pages=381–392|volume=19|first1=Marc|last1=Vaillant|first2=Joan|last2=Glaunès|title=Information Processing in Medical Imaging |chapter=Surface Matching via Currents |journal=Information Processing in Medical Imaging: Proceedings of the ... Conference |s2cid=5103312|doi=10.1007/11505730_32|series=Lecture Notes in Computer Science|isbn=978-3-540-26545-0}} and varifolds which have been used extensively for curve and surface matching.

{{Further|Large deformation diffeomorphic metric mapping#LDDMM registered landmark matching}}

File:ComputationalAnatomy LandmarkGeodesic Single withLabel.gif

[[File:Joshi Landmark Matching.png|thumb|Figure showing landmark matching with correspondence.

Left and right panels depict two different kernel with solutions.

]]

Denoted the landmarked shape $X\backslash doteq\; \backslash \{\; x\_1,\backslash dots,\; x\_n\; \backslash \}\; \backslash subset\; \{\backslash mathbb\; R\}^3$ with endpoint $$

E(\varphi_1) \doteq \textstyle \sum_i \displaystyle \|\varphi_1(x_i)-x_i^\prime \|^2 , the variational problem becomes

{{NumBlk|:|

:$$

\min_{\varphi: v=\dot \varphi \circ \varphi^{-1}} C(\varphi) \doteq \frac{1}{2} \int (Av_t\mid v_t) \, dt +\frac{1}{2} \sum_i \|\varphi_1(x_i) - x_i^\prime \|^2

|{{EquationRef|Landmark-Matching}}}}The geodesic Eulerian momentum is a generalized function $$

\displaystyle Av_t \in V^*\textstyle, t \in [0,1]

, supported on the landmarked set in the variational problem. The endpoint condition with conservation implies the initial momentum at the identity of the group:

:$$

\begin{cases}

\text{Endpoint condition: } & Av_1=\sum_{i=1}^n p_1(i)\delta_{\varphi_1(x_i)}, p_1(i)=(x_i^\prime-\varphi_1(x_i)) \ ,

\\[5pt]

\text{Conservation: } & Av_t=\sum_{i=1}^n p_t(i) \delta_{\varphi_t(x_i)} ,

\ p_t(i)=(D\varphi_{t1})_{|\varphi_t(x_i)}^T p_1(i) \ , \ \varphi_{t1} \doteq \varphi_1 \circ \varphi_t^{-1} \ ,

\\[5pt]

& Av_0=\sum_i \delta_{x_i} (\cdot) p_0(i) \ \text{ with } p_0(i)=(D\varphi_1)^T_{|x_i}(x_i^\prime-\varphi_1(x_i))

\end{cases}

The iterative algorithm

for large deformation diffeomorphic metric mapping for landmarks is given.

Glaunes and co-workers first introduced diffeomorphic matching of pointsets in the general setting of matching distributions.{{Cite book|last1=Glaunes|first1=J.|last2=Trouve|first2=A.|last3=Younes|first3=L.|title=Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 |chapter=Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching |date=2004-06-01 |volume=2|pages=II–712–II–718 Vol.2|doi=10.1109/CVPR.2004.1315234|isbn=978-0-7695-2158-9|citeseerx=10.1.1.158.4209}} As opposed to landmarks, this includes in particular the situation of weighted point clouds with no predefined correspondences and possibly different cardinalities. The template and target discrete point clouds are represented as two weighted sums of Diracs $\backslash mu\_\{m\}=\backslash sum\_\{i=1\}^\{n\}\; \backslash rho\_i\; \backslash delta\_\{x\_i\}$ and $\backslash mu\_\{m^\backslash prime\}=\backslash sum\_\{i=1\}^\{n^\backslash prime\}\; \backslash rho^\{\backslash prime\}\_\{i\}\; \backslash delta\_\{x^\{\backslash prime\}\_\{i\}\}$ living in the space of signed measures of $\backslash mathbb\{R\}^3$. The space is equipped with a Hilbert metric obtained from a real positive kernel $k(x,y)$ on $\backslash mathbb\{R\}^3$, giving the following norm:

: $$

\|\mu_{m} \|_\mathrm{mea}^2=\sum_{i,j=1}^{n} \rho_i \rho_j k(x_i,x_j)

The matching problem between a template and target point cloud may be then formulated using this kernel metric for the endpoint matching term:

: $$

\min_{\varphi: v=\dot \varphi \circ \varphi^{-1}} C(\varphi) \doteq \frac{1}{2} \int (Av_t\mid v_t) \, dt +\frac{1}{2} \|\mu_{\varphi_1 \cdot m} - \mu_{m^\prime} \|_\mathrm{mea}^2

where $\backslash mu\_\{\backslash varphi\_1\; \backslash cdot\; m\}=\backslash sum\_\{i=1\}^\{n\}\; \backslash rho\_i\; \backslash delta\_\{\backslash varphi\_1(x\_i)\}$ is the distribution transported by the deformation.

In the one dimensional case, a curve in 3D can be represented by an embedding $m:\; u\; \backslash in\; [0,1]\; \backslash rightarrow\; \{\backslash mathbb\; R\}^3$, and the group action of Diff becomes $\backslash varphi\; \backslash cdot\; m=\backslash varphi\; \backslash circ\; m$. However, the correspondence between curves and embeddings is not one to one as the any reparametrization $m\; \backslash circ\; \backslash gamma$, for $\backslash gamma$ a diffeomorphism of the interval [0,1], represents geometrically the same curve. In order to preserve this invariance in the end-point matching term, several extensions of the previous 0-dimensional measure matching approach can be considered.

• Curve matching with currents

In the situation of oriented curves, currents give an efficient setting to construct invariant matching terms. In such representation, curves are interpreted as elements of a functional space dual to the space vector fields, and compared through kernel norms on these spaces. Matching of two curves $m$ and $m^\backslash prime$ writes eventually as the variational problem

: $$

\min_{\varphi: v=\dot \varphi \circ \varphi^{-1}} C(\varphi) \doteq \frac{1}{2} \int

(Av_t\mid v_t) \, dt +\frac{1}{2} \|\mathcal{C}_{\varphi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2

with the endpoint term $E(\backslash varphi\_1)=\backslash |\backslash mathcal\{C\}\_\{\backslash varphi\_1\; \backslash cdot\; m\}\; -\; \backslash mathcal\{C\}\_\{m^\backslash prime\}\; \backslash |\_\backslash mathrm\{cur\}^2/2$ is obtained from the norm

: $$

\|\mathcal{C}_{m} \|_\mathrm{cur}^2=\int_0^1 \int_0^1 K_C(m(u),m(v)) \partial m(u) \cdot \partial m(v) \, du \, dv

the derivative $\backslash partial\; m(u)$ being the tangent vector to the curve and $K\_\{\backslash mathcal\{C\}\}$ a given matrix kernel of $\{\backslash mathbb\; R\}^3$. Such expressions are invariant to any positive reparametrizations of $m$ and $m\text{'}$, and thus still depend on the orientation of the two curves.

• Curve matching with varifolds

Varifold is an alternative to currents when orientation becomes an issue as for instance in situations involving multiple bundles of curves for which no "consistent" orientation may be defined. Varifolds directly extend 0-dimensional measures by adding an extra tangent space direction to the position of points, leading to represent curves as measures on the product of $\{\backslash mathbb\; R\}^3$ and the Grassmannian of all straight lines in $\{\backslash mathbb\; R\}^3$. The matching problem between two curves then consists in replacing the endpoint matching term by $E(\backslash varphi\_1)=\backslash |\backslash mathcal\{V\}\_\{\backslash varphi\_1\; \backslash cdot\; m\}\; -\; \backslash mathcal\{V\}\_\{m^\backslash prime\}\; \backslash |\_\backslash text\{cur\}^2/2$ with varifold norms of the form:

: $$

\|\mathcal{V}_m \|_{var}^2=\int_{0}^1 \int_{0}^1 k_{\mathbb{R}^3}(m(u),m(v)) k_{\mathbf{Gr}}\left([\partial m(u)],[\partial m(v)] \right) \partial m(u) {

where $[\backslash partial\; m(u)]$ is the non-oriented line directed by tangent vector $\backslash partial\; m(u)$ and $k\_\{\backslash mathbb\{R\}^3\},\; k\_\{\backslash mathbf\{Gr\}\}$ two scalar kernels respectively on $\backslash mathbb\{R\}^3$ and the Grassmannian. Due to the inherent non-oriented nature of the Grassmannian representation, such expressions are invariant to positive and negative reparametrizations.

Surface matching share many similarities with the case of curves. Surfaces in $\{\backslash mathbb\; R\}^3$ are parametrized in local charts by embeddings $m:\; u\; \backslash in\; U\; \backslash subset\; \{\backslash mathbb\; R\}^\{2\}\; \backslash rightarrow\; \{\backslash mathbb\; R\}^3$, with all reparametrizations $m\backslash circ\; \backslash gamma$ with $\backslash gamma$ a diffeomorphism of U being equivalent geometrically. Currents and varifolds can be also used to formalize surface matching.

• Surface matching with currents

Oriented surfaces can be represented as 2-currents which are dual to differential 2-forms. In $\{\backslash mathbb\; R\}^3$, one can further identify 2-forms with vector fields through the standard wedge product of 3D vectors. In that setting, surface matching writes again:

: $$

\min_{\varphi: v=\dot \varphi \circ \varphi^{-1}} C(\varphi) \doteq \frac{1}{2} \int

(Av_t\mid v_t) \, dt +\frac{1}{2} \|\mathcal{C}_{\varphi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2

with the endpoint term $E(\backslash varphi\_1)=\backslash |\backslash mathcal\{C\}\_\{\backslash varphi\_1\; \backslash cdot\; m\}\; -\; \backslash mathcal\{C\}\_\{m^\backslash prime\}\; \backslash |\_\backslash mathrm\{cur\}^2/2$ given through the norm

: $$

\|\mathcal{C}_{m} \|_\mathrm{cur}^2=\iint_{U\times U} K_C(m(u),m(v)) \vec{n}(u) \cdot \vec{n}(v) \, du \, dv

with $\backslash vec\{n\}=\backslash partial\_\{u\_1\}\; m\; \backslash wedge\; \backslash partial\_\{u\_2\}\; m$ the normal vector to the surface parametrized by $m$.

This surface mapping algorithm has been validated for brain cortical surfaces against CARET and FreeSurfer.{{cite journal|last1=Zhong|first1=J|last2=Phua|first2=DY|last3=Qiu|first3=A|title=Quantitative evaluation of LDDMM, FreeSurfer, and CARET for cortical surface mapping|journal=NeuroImage|date=2010|volume=52|issue=1|pages=131–141|doi=10.1016/j.neuroimage.2010.03.085|pmid=20381626|s2cid=6767322}} LDDMM mapping for multiscale surfaces is discussed in.{{cite journal|last1=Tan|first1=M|last2=Qiu|first2=A|title=Large Deformation Multiresolution Diffeomorphic Metric Mapping for Multiresolution Cortical Surfaces: A Coarse-to-Fine Approach|journal= IEEE Transactions on Image Processing|date=2016|volume=25|issue=9|pages=4061–4074|doi=10.1109/TIP.2016.2574982|pmid=27254865|bibcode=2016ITIP...25.4061T|s2cid=16307639}}

• Surface matching with varifolds

For non-orientable or non-oriented surfaces, the varifold framework is often more adequate. Identifying the parametric surface $m$ with a varifold $\backslash mathcal\{V\}\_m$ in the space of measures on the product of $\{\backslash mathbb\; R\}^3$ and the Grassmannian, one simply replaces the previous current metric $\backslash |\backslash mathcal\{C\}\_\{m\}\; \backslash |\_\backslash mathrm\{cur\}^2$ by:

: $$

\|\mathcal{V}_{m} \|_\mathrm{var}^2=\iint_{U\times U} k_{\mathbb{R}^3}(m(u),m(v)) k_{\mathbf{Gr}}\left([\vec{n}(u)],[\vec{n}(v)] \right) {

where $[\backslash vec\{n\}(u)]$ is the (non-oriented) line directed by the normal vector to the surface.

There are many settings in which there are a series of measurements, a time-series to which the underlying

coordinate systems will be matched and flowed onto. This occurs for example

in the dynamic growth and atrophy models and motion tracking such as have been explored in{{Cite journal|title=Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms|journal=NeuroImage|date=2004-01-01|pmid=15501089|pages=S19–33|volume=23|doi=10.1016/j.neuroimage.2004.07.021|first=Michael I.|last=Miller|issue=Suppl 1 |citeseerx=10.1.1.121.4222|s2cid=13365411}}{{Cite journal|title=Geodesic regression for image time-series|journal=Medical Image Computing and Computer-Assisted Intervention |date=2011-01-01|pmc=4339064|pmid=21995085|pages=655–662|volume=14|issue=Pt 2|first1=Marc|last1=Niethammer|first2=Yang|last2=Huang|first3=François-Xavier|last3=Vialard|doi=10.1007/978-3-642-23629-7_80}}{{Cite arXiv|title=Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View|eprint=1003.3895|last1=Trouvé|first1=Alain|last2=Vialard|first2=François-Xavier|class=math.OC|year=2010 }}{{Cite journal|title=Principal geodesic analysis for the study of nonlinear statistics of shape|journal=IEEE Transactions on Medical Imaging|date=2004-08-01|pages=995–1005|volume=23|issue=8|doi=10.1109/TMI.2004.831793|pmid=15338733|first1=P.T.|last1=Fletcher|first2=C.|last2=Lu|first3=S.M.|last3=Pizer|first4=S.|last4=Joshi|citeseerx=10.1.1.76.539|s2cid=620015}} An observed time sequence is given and the goal is to infer the time flow of geometric change of coordinates carrying the exemplars or templars through the period of observations.

The generic time-series matching problem considers the series of times is . The flow optimizes at the series of costs $E(t\_k),\; k=1,\; \backslash ldots,\; K$ giving optimization problems of the form

: $\backslash min\_\{\backslash varphi:v=\backslash dot\; \backslash varphi\; \backslash circ\; \backslash varphi^\{-1\},\; \backslash varphi\_0=id\}\; C(\backslash varphi)\; \backslash doteq\; \backslash frac\{1\}\{2\}\; \backslash int\_0^1\; (Av\_t\backslash mid\; v\_t)\; \backslash ,\; dt\; +\backslash sum\_\{k=1\}^K\; E(\; \backslash varphi\_\{t\_k\})$

.

There have been at least three solutions offered thus far, piecewise geodesic, principal geodesic and splines.{{Cite journal|title=Shape splines and stochastic shape evolutions: A second order point of view|journal=Quarterly of Applied Mathematics|date=2012-01-01|pages=219–251|volume=70|issue=2|doi=10.1090/S0033-569X-2012-01250-4|first1=Alain|last1=Trouvé|first2=François-Xavier|last2=Vialard|arxiv=1003.3895|s2cid=96421820}}

File:Showing orbit as a surface.jpg

The random orbit model of computational anatomy first appeared in{{Cite journal|title=Statistical methods in computational anatomy|journal=Statistical Methods in Medical Research|date=1997-06-01|pmid=9339500|pages=267–299|volume=6|issue=3|doi=10.1177/096228029700600305|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=9780199297061}}{{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=9780123973160}} modelling the change in coordinates associated to the randomness of the group acting on the templates, which induces the randomness on the source of images in the anatomical orbit of shapes and forms and resulting observations through the medical imaging devices. Such a random orbit model in which randomness on the group induces randomness on the images was examined for the Special Euclidean Group for object recognition in.{{Cite book|doi=10.1007/978-1-4612-4120-1_18|publisher=Birkhäuser Boston|date=1997-01-01|isbn=978-1-4612-8662-2|pages=327–350|series=Systems & Control: Foundations & Applications|first1=S.|last1=Srivastava|first2=M. I.|last2=Miller|first3=U.|last3=Grenander|title=Systems and Control in the Twenty-First Century |chapter=Ergodic Algorithms on Special Euclidean Groups for ATR |editor-first=Christopher I.|editor-last=Byrnes|editor-first2=Biswa N.|editor-last2=Datta|editor-first3=Clyde F.|editor-last3=Martin|editor-first4=David S.|editor-last4=Gilliam|citeseerx=10.1.1.44.4751}}

Depicted in the figure is a depiction of the random orbits around each exemplar, $m\_0\; \backslash in\; \backslash mathcal\{M\}$, generated by randomizing the flow by generating the initial tangent space vector field at the identity $v\_0\; \backslash in\; V$, and then generating random object $n\; \backslash doteq\; \backslash operatorname\{Exp\}\_\{\backslash rm\; id\}(v\_0)\; \backslash cdot\; m\_0\; \backslash in\; \backslash mathcal\{M\}$.

The random orbit model induces the prior on shapes and images $$

I \in \mathcal{I}

conditioned on a particular atlas $$

I_a \in \mathcal{I}

. For this the generative model generates the mean field $$

I

as a random change in coordinates of the template according to $$

I \doteq \varphi \cdot I_a

, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows. The prior on random transformations $$

\pi_\text{Diff} (d\varphi)

on $\backslash operatorname\{Diff\}\_V$ is induced by the flow $$

\operatorname{Exp}_{\rm id}(v)

, with $$

v \in V

constructed as a Gaussian random field prior $$

\pi_V(dv)

. The density on the random observables at the output of the sensor $I^D\; \backslash in\; \backslash mathcal\{I\}^D$ are given by

File:Synthesized cortical structures from common template.png

: $$

p(I^D\mid I_a)=\int_V p(I^D\mid \operatorname{Exp}_{\rm id}(v) \cdot I_a ) \pi_V (dv) \ .

Shown in the Figure on the right the cartoon orbit are a random spray of the subcortical manifolds generated by randomizing the vector fields $v\_0$ supported over the submanifolds.

{{main|The Bayesian model of computational anatomy}}

File:Source-channel-model-Shannon.png

The central statistical model of computational anatomy in the context of medical imaging has been the source-channel model of Shannon theory; the source is the deformable template of images $I\; \backslash in\; \backslash mathcal\; \{I\}$, the channel outputs are the imaging sensors with observables $I^D\; \backslash in\; \{\backslash mathcal\; I\}^\{\backslash mathcal\; D\}$ (see Figure).

See The Bayesian model of computational anatomy for discussions (i) MAP estimation with multiple atlases, (ii)

MAP segmentation with multiple atlases, MAP estimation of templates from populations.

{{Further|Statistical shape analysis|Shape theory (mathematics)}}

Shape in computational anatomy is a local theory, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape in computational anatomy is the empirical study of diffeomorphic correspondences between populations and common template coordinate systems. This is a strong departure from Procrustes Analyses and shape theories pioneered by David G. Kendall{{Cite journal|title=A Survey of the Statistical Theory of Shape|jstor=2245331|journal=Statistical Science|date=1989-01-01|pages=87–99|volume=4|issue=2|first=David G.|last=Kendall|doi=10.1214/ss/1177012582|doi-access=free}} in that the central group of Kendall's theories are the finite-dimensional Lie groups, whereas the theories of shape in computational anatomy{{Cite book|doi=10.1007/978-88-7642-387-1_4|publisher=Scuola Normale Superiore|date=2012-01-01|isbn=9788876423888|pages=43–53|series=Colloquia|first=David|last=Mumford|s2cid=116135355|editor-first=Umberto|editor-last=Zannier|title=Colloquium de Giorgi 2009|chapter=The geometry and curvature of shape spaces}}{{Cite book|edition=1st|title=Shapes and Diffeomorphisms|author=Laurent Younes|publisher=Springer|date=2010-05-25|isbn=9783642120541 }}{{Cite journal|title=Spaces and Manifolds of Shapes in Computer Vision: An Overview|journal=Image Vision Comput.|date=2012-06-01|pages=389–397|volume=30|issue=6–7|doi=10.1016/j.imavis.2011.09.009|first=Laurent|last=Younes}} have focused on the diffeomorphism group, which to first order via the Jacobian can be thought of as a field–thus infinite dimensional–of low-dimensional Lie groups of scale and rotations.

File:Subcortical structures embedded in two-dimensional momentum space.png

The random orbit model provides the natural setting to understand empirical shape and shape statistics within computational anatomy since the non-linearity of the induced probability law on anatomical shapes and forms $m\; \backslash in\; \backslash mathcal\{M\}$ is induced via the reduction to the vector fields $v\_0\; \backslash in\; V$ at the tangent space at the identity of the diffeomorphism group. The successive flow of the Euler equation induces the random space of shapes and forms $\backslash operatorname\{Exp\}\_\{\backslash rm\; id\}(v\_0)\; \backslash cdot\; m\; \backslash in\; \backslash mathcal\{M\}$.

Performing empirical statistics on this tangent space at the identity is the natural way for inducing probability laws on the statistics of shape. Since both the vector fields and the Eulerian momentum $Av\_0$ are in a Hilbert space the natural model is one of a Gaussian random field, so that given test function $w\; \backslash in\; V$, then the inner-products with the test functions are Gaussian distributed with mean and covariance.

This is depicted in the accompanying figure where sub-cortical brain structures are depicted in a two-dimensional coordinate system based on inner products of their initial vector fields that generate them from the template is shown in a 2-dimensional span of the Hilbert space.

File:Template-generation.png

{{Further|The Bayesian model of computational anatomy|Bayesian estimation of templates in computational anatomy}}

The study of shape and statistics in populations are local theories, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape is then the study of diffeomorphic correspondences relative to the template. A core operation is the generation of templates from populations, estimating a shape that is matched to the population. There are several important methods for generating templates including methods based on Frechet averaging,{{Cite journal|title=Unbiased diffeomorphic atlas construction for computational anatomy|journal=NeuroImage|date=2004-01-01|pages=151–160|volume=23|first1=S.|last1=Joshi|first2=Brad|last2=Davis|first3=B. Matthieu|last3=Jomier|first4=Guido Gerig|last4=B|doi=10.1016/j.neuroimage.2004.07.068|pmid=15501084|citeseerx=10.1.1.104.3808|s2cid=2271742}} and statistical approaches based on the expectation-maximization algorithm and the Bayes Random orbit models of computational anatomy.{{Cite journal|title=Bayesian Template Estimation in Computational Anatomy|journal=NeuroImage|date=2008-08-01|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}} Shown in the accompanying figure is a subcortical template reconstruction from the population of MRI subjects.{{cite journal|last1=Qiu|first1=Anqi|last2=Miller|first2=Michael I.|title=Multi-structure network shape analysis via normal surface momentum maps|journal=NeuroImage|date=2008|volume=42|issue=4|pages=1430–1438|doi=10.1016/j.neuroimage.2008.04.257|pmid=18675553|citeseerx=10.1.1.463.7231|s2cid=10434173}}

Software suites containing a variety of diffeomorphic mapping algorithms include the following:

• ANTS
• DARTEL Voxel-based morphometry
• DEFORMETRICA{{Cite web|url=http://www.deformetrica.org/|title=Deformetrica|access-date=12 January 2017}}
• DEMONS
• LDDMM Large deformation diffeomorphic metric mapping
• LDDMM based on frame-based kernel {{cite web|last1=Tan|first1=Mingzhen|last2=Qiu|first2=Anqi|title=LDDMM with frame-based kernel|url=http://www.bioeng.nus.edu.sg/cfa/brainmapping.html|website=Computational Functional Anatomy}}
• StationaryLDDMM

• MRICloud{{Cite web|url=http://www.mricloud.org/|title=MriCloud|access-date=26 October 2016}}

• Bayesian estimation of templates in computational anatomy
• Computational neuroanatomy
• Geometric data analysis
• Large deformation diffeomorphic metric mapping
• Procrustes analysis
• Riemannian metric and Lie-bracket in computational anatomy
• Shape analysis (disambiguation)
• Statistical shape analysis

{{Reflist}}

Category:Geometry

Category:Fluid mechanics

Category:Bayesian estimation

Category:Neuroscience

Category:Neural engineering

Category:Biomedical engineering

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