Editing Morphometrics (section)

====Diffeomorphometry====
[[Diffeomorphometry]]<ref>{{Cite journal|last1=Miller|first1=Michael I.|last2=Younes|first2=Laurent|last3=Trouvé|first3=Alain|date=2013-11-18|title=Diffeomorphometry and geodesic positioning systems for human anatomy|journal=Technology|volume=2|issue=1|pages=36–43|doi=10.1142/S2339547814500010|issn=2339-5478|pmc=4041578|pmid=24904924}}</ref> is the focus on comparison of shapes and forms with a metric structure based on diffeomorphisms, and is central to the field of [[computational anatomy]].<ref>{{Cite journal|last1=Grenander|first1=Ulf|last2=Miller|first2=Michael I.|date=1998-12-01|title=Computational Anatomy: An Emerging Discipline|journal=Q. Appl. Math.|volume=LVI|issue=4|pages=617–694|doi=10.1090/qam/1668732|issn=0033-569X|doi-access=free}}</ref> Diffeomorphic registration,<ref>{{Cite journal|last1=Christensen|first1=G. E.|last2=Rabbitt|first2=R. D.|last3=Miller|first3=M. I.|date=1996-01-01|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|issn=1057-7149|pmid=18290061|bibcode=1996ITIP....5.1435C}}</ref> introduced  in the 90s, is now an important player with existing code bases organized around ANTS,<ref>{{Cite web|title = stnava/ANTs|url = https://github.com/stnava/ANTs/blob/master/Scripts/antsIntroduction.sh|website = GitHub|access-date = 2015-12-11}}</ref> DARTEL,<ref>{{Cite journal|title = A fast diffeomorphic image registration algorithm|journal = NeuroImage|date = 2007-10-15|issn = 1053-8119|pmid = 17761438|pages = 95–113|volume = 38|issue = 1|doi = 10.1016/j.neuroimage.2007.07.007|first = John|last = Ashburner|s2cid = 545830}}</ref> DEMONS,<ref>{{Cite web|title = Software - Tom Vercauteren|url = https://sites.google.com/site/tomvercauteren/software|website = sites.google.com|access-date = 2015-12-11}}</ref> [[Large deformation diffeomorphic metric mapping|LDDMM]],<ref>{{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}}</ref> StationaryLDDMM<ref>{{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}}</ref> are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. [[Voxel-based morphometry]] (VBM) is an important technology built on many of these principles. Methods based on diffeomorphic flows are used in For example, deformations could be diffeomorphisms of the ambient space, resulting in the LDDMM ([[Large deformation diffeomorphic metric mapping|Large Deformation Diffeomorphic Metric Mapping]]) framework for shape comparison.<ref name="LDDMM">{{cite journal|author1=F. Beg |author2=M. Miller |author3=A. Trouvé |author4=L. Younes |title=Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal=International Journal of Computer Vision|volume=61|issue=2|date=February 2005|doi=10.1023/b:visi.0000043755.93987.aa|pages=139–157|s2cid=17772076 }}</ref> On such deformations is the right invariant metric of [[Computational anatomy|Computational Anatomy]] which generalizes the metric of non-compressible Eulerian flows but to include the Sobolev norm ensuring smoothness of the flows,<ref>{{Cite journal|last1=Miller|first1=M. I.|last2=Younes|first2=L.|date=2001-01-01|title=Group Actions, Homeomorphisms, And Matching: A General Framework|journal=International Journal of Computer Vision|volume=41|pages=61–84|doi=10.1023/A:1011161132514|citeseerx=10.1.1.37.4816|s2cid=15423783}}</ref>  metrics have now been defined associated to Hamiltonian controls of diffeomorphic flows.<ref>{{Cite journal|last1=Miller|first1=Michael I.|last2=Trouvé|first2=Alain|last3=Younes|first3=Laurent|date=2015-01-01|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|issn=1545-4274|pmid=26643025}}</ref>