Editing Morphometrics (section)

===Landmark-based geometric morphometrics===
{{Further|Geometric data analysis|Statistical shape analysis}}
[[File:Onymacris unguicularis with landmarks for morphometric analysis - ZooKeys-353-047-g005.jpg|thumb|''[[Onymacris unguicularis]]'' beetle with landmarks for morphometric analysis]]

In landmark-based geometric morphometrics, the spatial information missing from traditional morphometrics is contained in the data, because the data are coordinates of [[Landmark point|landmarks]]: discrete anatomical loci that are arguably ''homologous'' in all individuals in the analysis (i.e. they can be regarded as the "same" point in each specimens in the study). For example, where two specific [[Suture (anatomy)|sutures]] intersect is a landmark, as are intersections between veins on an insect wing or leaf, or [[foramina]], small holes through which veins and blood vessels pass. Landmark-based studies have traditionally analyzed 2D data, but with the increasing availability of 3D imaging techniques, 3D analyses are becoming more feasible even for small structures such as teeth.<ref>{{cite journal|last=Singleton|first=M.|author2=Rosenberger, A. L. |author3=Robinson, C. |author4=O'Neill, R. |title=Allometric and metameric shape variation in ''Pan'' mandibular molars: A digital morphometric analysis|journal=Anatomical Record|year=2011|volume=294|issue=2|pages=322–334|doi=10.1002/ar.21315|pmid=21235007|s2cid=17561423|doi-access=free}}</ref> Finding enough landmarks to provide a comprehensive description of shape can be difficult when working with fossils or easily damaged specimens. That is because all landmarks must be present in all specimens, although coordinates of missing landmarks can be estimated. The data for each individual consists of a ''configuration'' of landmarks.

There are three recognized categories of landmarks.<ref name="Bookstein1991">{{cite book|last=Bookstein|first=F. L.|title=Morphometric Tools for Landmark Data: Geometry and Biology|url=https://archive.org/details/morphometrictool0000book|url-access=registration|year=1991|publisher=Cambridge University Press|location=Cambridge}}</ref> ''Type 1 landmarks'' are defined locally, i.e. in terms of structures close to that point; for example, an intersection between three sutures, or intersections between veins on an insect wing are locally defined and surrounded by tissue on all sides. ''Type 3 landmarks'', in contrast, are defined in terms of points far away from the landmark, and are often defined in terms of a point "furthest away" from another point. ''Type 2 landmarks'' are intermediate; this category includes points such as  the tip structure, or local minima and maxima of curvature.  They are defined in terms of local features, but they are not surrounded on all sides. In addition to landmarks, there are ''semilandmarks'', points whose position along a curve is arbitrary but which provide information about curvature in two<ref>{{cite journal|last=Zelditch|first=M. |author2=Wood, A. R. |author3=Bonnet, R. M. |author4=Swiderski, D. L. |title=Modularity of the rodent mandible: Integrating muscles, bones and teeth|journal=Evolution & Development|year=2008|volume=10|issue=6|pages=756–768|doi=10.1111/j.1525-142X.2008.00290.x|pmid=19021747 |hdl=2027.42/73767 |s2cid=112076 |url=https://deepblue.lib.umich.edu/bitstream/2027.42/73767/1/j.1525-142X.2008.00290.x.pdf|hdl-access=free}}</ref> or three dimensions.<ref>{{cite journal|last=Mitteroecker|first=P|author2=Bookstein, F.L.
|title=The evolutionary role of modularity and integration in the hominoid cranium
|journal=Evolution|year=2008
|volume=62
|issue=4
|pages=943–958|doi=10.1111/j.1558-5646.2008.00321.x|pmid=18194472|s2cid=23716467|doi-access=free}}</ref>

====Procrustes-based geometric morphometrics====
[[File:Procrustes superimposition.png|thumb|Procrustes superimposition]]
Shape analysis begins by removing the information that is not about shape. By definition, shape is not altered by translation, scaling or rotation.<ref>{{cite journal|last=Kendall|first=D.G.|title=The diffusion of shape|journal=Advances in Applied Probability|year=1977|volume=9|pages=428–430|doi=10.2307/1426091|issue=3|jstor=1426091|s2cid=197438611 }}</ref>  Thus, to compare shapes, the non-shape information is removed from the coordinates of landmarks.  There is more than one way to do these three operations. One method is to fix the coordinates of two points to (0,0) and (0,1), which are the two ends of a baseline. In one step, the shapes are translated to the same position (the same two coordinates are fixed to those values), the shapes are scaled (to unit baseline length) and the shapes are rotated.<ref name=Bookstein1991 />  An alternative, and preferred method, is [[Procrustes superimposition]].  This method translates the centroid of the shapes to (0,0); the ''x'' coordinate of the centroid is the average of the ''x'' coordinates of the landmarks, and the ''y'' coordinate of the centroid is the average of the ''y''-coordinates. Shapes are scaled to unit centroid size, which is the square root of the summed squared distances of each landmark to the centroid.  The configuration is rotated to minimize the deviation between it and a reference, typically the mean shape. In the case of semi-landmarks, variation in position along the curve is also removed. Because shape space is curved, analyses are done by projecting shapes onto a space tangent to shape space.  Within the tangent space, conventional multivariate statistical methods such as multivariate analysis of variance and multivariate regression, can be used to test statistical hypotheses about shape.

Procrustes-based analyses have some limitations.  One is that the Procrustes superimposition uses a least-squares criterion to find the optimal rotation; consequently, variation that is localized to a single landmark will be smeared out across many.  This is called the 'Pinocchio effect'. Another is that the superimposition may itself impose a pattern of covariation on the landmarks.<ref>{{cite journal|last=Rohlf|first=F. J.|author2=Slice, D.|title=Extensions of the Procrustes method for the optimal superimposition of landmarks|journal=Systematic Zoology|year=1990|volume=39|pages=40–59|doi=10.2307/2992207|issue=1|jstor=2992207|citeseerx=10.1.1.547.626}}</ref><ref>{{cite journal|last=Walker|first=J.|title=The ability of geometric morphometric methods to estimate a known covariance matrix|journal=Systematic Biology|year=2000|volume=49|pages=686–696|doi=10.1080/106351500750049770|pmid=12116434|issue=4|doi-access=free}}</ref> Additionally, any information that cannot be captured by landmarks and semilandmarks cannot be analyzed, including classical measurements like "greatest skull breadth".  Moreover, there are criticisms of Procrustes-based methods that motivate an alternative approach to analyzing landmark data.

====Euclidean distance matrix analysis====

====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>