Editing Image registration (section)

=== Transformations of coordinates via the law of function composition rather than addition ===
Alternatively, many advanced methods for spatial normalization are building on structure preserving transformations [[homeomorphism]]s and [[diffeomorphism]]s since they carry smooth submanifolds smoothly during transformation. Diffeomorphisms are generated in the modern field of [[Computational anatomy|Computational Anatomy]] based on flows since diffeomorphisms are not additive although they form a group, but a group under the law of function composition. For this reason, flows which generalize the ideas of additive groups allow for generating large deformations that preserve topology, providing 1-1 and onto transformations. Computational methods for generating such transformation are often called [[Large deformation diffeomorphic metric mapping|LDDMM]]<ref>{{Cite book|url=https://books.google.com/books?id=8WdlWJepgWMC|title=Brain Warping|last=Toga|first=Arthur W.|date=1998-11-17|publisher=Academic Press|isbn=9780080525549|language=en}}</ref><ref>{{Cite web|url=https://utah.pure.elsevier.com/en/publications/landmark-matching-on-brain-surfaces-via-large-deformation-diffeom-2|title=Landmark matching on brain surfaces via large deformation diffeomorphisms on the sphere — University of Utah|website=utah.pure.elsevier.com|access-date=2016-03-21|archive-url=https://web.archive.org/web/20180629235930/https://utah.pure.elsevier.com/en/publications/landmark-matching-on-brain-surfaces-via-large-deformation-diffeom-2|archive-date=2018-06-29|url-status=dead}}</ref><ref>{{Cite journal|url=https://www.researchgate.net/publication/220660081|title=Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal=International Journal of Computer Vision|volume=61|issue=2|pages=139–157|doi=10.1023/B:VISI.0000043755.93987.aa|access-date=2016-03-21|year=2005|last1=Beg|first1=M. Faisal|last2=Miller|first2=Michael I.|last3=Trouvé|first3=Alain|last4=Younes|first4=Laurent|s2cid=17772076}}</ref><ref>{{Cite journal|last1=Joshi|first1=S. C.|last2=Miller|first2=M. I.|date=2000-01-01|title=Landmark matching via large deformation diffeomorphisms|journal=IEEE Transactions on Image Processing|volume=9|issue=8|pages=1357–1370|doi=10.1109/83.855431|issn=1057-7149|pmid=18262973|bibcode=2000ITIP....9.1357J}}</ref> which provide flows of diffeomorphisms as the main computational tool for connecting coordinate systems corresponding to [[Computational anatomy#The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit|the geodesic flows of Computational Anatomy]].

There are a number of programs which generate diffeomorphic transformations of coordinates via [[Diffeomorphic Mapping in Computational Anatomy|diffeomorphic mapping]] including MRI Studio<ref>{{Cite web|url=https://www.mristudio.org|title=MRI Studio}}</ref> and MRI Cloud.org<ref>{{Cite web|url=https://mricloud.org/|title=MRICloud Brain Mapping}}</ref>