Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Spatial normalization
open-in-new
<h2 id="diffeomorphisms-as-compositional-transformations-of-coordinates">Diffeomorphisms as compositional transformations of coordinates</h2> <p>Alternatively, many advanced methods for spatial normalization are building on structure preserving transformations <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphisms</a> and <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> since they carry smooth submanifolds smoothly during transformation. Diffeomorphisms are generated in the modern field of <a href="/facts/Computational_anatomy/8lA9YzSB">Computational Anatomy</a> based on diffeomorphic flows, also called <a href="/facts/Diffeomorphic_Mapping_in_Computational_Anatomy/zWkUmrUI">diffeomorphic mapping</a>. However, such transformations via diffeomorphisms are not additive, although they form a <a href="/facts/Computational_anatomy/8lA9YzSB">group with function composition</a> and acting non-linearly on the images via <a href="/facts/Group_actions_in_computational_anatomy/hqHqXBxK">group action</a>. 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 <a href="/facts/Large_deformation_diffeomorphic_metric_mapping/zWkUmrUI">LDDMM</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> which provide flows of diffeomorphisms as the main computational tool for connecting coordinate systems corresponding to <a href="/facts/Computational_anatomy/8lA9YzSB">the geodesic flows of Computational Anatomy</a>. </p><p>There is a number of programs that implement both estimation and application of a warp-field. It is a part of the <a href="/facts/Statistical_parametric_mapping/NDJ7qx6M">SPM</a> and <a href="/facts/AIR_(program)/INe53dz2">AIR</a> programs as well as MRI Studio and MRI Cloud.org.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Voxel-based_morphometry/cbkfl9VV">Voxel-based morphometry</a></li> <li><a href="/facts/Computational_Anatomy/8lA9YzSB">Computational Anatomy</a></li> <li><a href="/facts/LDDMM/zWkUmrUI">LDDMM</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Toga, Arthur W. (November 17, 1998). Brain Warping. Academic Press. ISBN 9780080525549. <a href="9780080525549" target="_blank">9780080525549</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Landmark matching on brain surfaces via large deformation diffeomorphisms on the sphere — University of Utah". utah.pure.elsevier.com. Retrieved March 21, 2016. <a href="https://utah.pure.elsevier.com/en/publications/landmark-matching-on-brain-surfaces-via-large-deformation-diffeom-2" target="_blank">https://utah.pure.elsevier.com/en/publications/landmark-matching-on-brain-surfaces-via-large-deformation-diffeom-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Beg, M. Faisal; Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2005). "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". International Journal of Computer Vision. 61 (2): 139–157. doi:10.1023/B:VISI.0000043755.93987.aa. S2CID 17772076. Retrieved March 21, 2016. <a href="https://www.researchgate.net/publication/220660081" target="_blank">https://www.researchgate.net/publication/220660081</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Joshi, S. C.; Miller, M. I. (January 1, 2000). "Landmark matching via large deformation diffeomorphisms". IEEE Transactions on Image Processing. 9 (8): 1357–1370. Bibcode:2000ITIP....9.1357J. doi:10.1109/83.855431. ISSN 1057-7149. PMID 18262973. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"MRI Cloud". mricloud.org. Archived from the original on June 2, 2023. Retrieved July 6, 2023. <a href="https://mricloud.org/" target="_blank">https://mricloud.org/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"MRI Studio Wiki Documentation". www.mristudio.org. Archived from the original on April 1, 2016. Retrieved July 6, 2023. <a href="https://web.archive.org/web/20160401010232/https://www.mristudio.org/wiki/" target="_blank">https://web.archive.org/web/20160401010232/https://www.mristudio.org/wiki/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>