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Geometric morphometrics
GMM is a subdiscipline of statistics that is concerned with the analysis of shape and shape changes of geometrical objects. Geometric morphometrics, has been conceived by many researches as a "revolution" of the field of morphometrics in the 1980s. The "modern synthesis" of geometric morphometrics has fused two separate morphometric themes: Deformation grids and the application of standard multivariate statistics.
 Thin plate spline of a midsaggital profil with landmarks |
The foundation for any comparison is to objectify the characteristics of morphology. Morphometry (Greek: "morphe" meaning "shape", and "métron" meaning "measurement") - taking measurements such as length, surface area, volume, angles, or curvature to characterize the appearance of organisms - has a long history that goes back at least to the 17th century in Europe. In Virtual Anthropology, with its emphasis on digital data and configurations of landmarks, Geometric Morphometrics (GMM) is a core technology.
Landmarks are loci that have names ('bridge of the nose', 'tip of the chin') as well as Cartesian coordinates. The names are intended to imply true homology (biological correspondence) from form to form. That is, landmark points not only have their own locations but also have 'corresponding' locations in every other form of the study and in the average of all the forms of the data set.
The application of standard multivariate statistics to a variety of geometric quantities measured on specimens one by one is a traditional morphometric approach. This approach, while statistically sophisticated, omitted much of the geometric information actually available. Late in the last century - as a part of the morphometric revolution - a mathematical statistical theory of shape was developed by several authors more or less coincidentally. Ubiquitous in this theory is the classic distinction between size and shape. Shape is the information about a geometric object that is invariant to overall location, size, and orientation. The choice of a suitable size measure is a crucial step in that course, since size itself is a vague concept when objects differ in shape. The size measure should reflect the observer's intuitive notion of scale, while being consistent in a statistical sense. In geometric morphometrics, centroid size is used as the main size measure. This measure derived from the spread of the landmarks around their centre of gravity, is approximately uncorrelated with shape when landmark position vary only by "pure" digitising noise.
Thin plate splines |
Procrustes superimposition |
Semilandmarks |
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An interpolation algorithm, the so called "thin
plate spline" (TPS), was borrowed from material physics and was
introduced to morphometrics by Fred Bookstein. It generates a
deformation grid between two point configurations ...
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Shape variables are usually constructed by a
Procrustes superimposition. The raw coordinates are superimposed by
translating the configurations to a common centroid, scaling to unit
centroid size, and rotating until the sum of the squared distances
...
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A recent extension of the Procrustes
superimposition is the sliding landmark algorithm. Traditional
landmarks need to be identifiable in all two or three dimensions of
physical space in order to precisely locate them on every
specimen.
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