Three general approaches to form are usually distinguished: traditional morphometrics, landmark-based morphometrics and outline-based morphometrics.
"Traditional" morphometrics Traditional morphometrics analyzes lengths, widths, masses, angles, ratios and areas. In general, traditional morphometric data are measurements of size. A drawback of using many measurements of size is that most will be highly correlated; as a result, there are few independent variables despite the many measurements. For instance, tibia length will vary with femur length and also with humerus and ulna length and even with measurements of the head. Traditional morphometric data are nonetheless useful when either absolute or relative sizes are of particular interest, such as in studies of growth. These data are also useful when size measurements are of theoretical importance such as body mass and limb cross-sectional area and length in studies of functional morphology. However, these measurements have one important limitation: they contain little information about the spatial distribution of shape changes across the organism. They are also useful when determining the extent to which certain pollutants have affected an individual. These indices include the hepatosomatic index,
gonadosomatic index and also the condition factors (shakumbila, 2014).
Landmark-based geometric morphometrics '' 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
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
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. 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.
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 or three dimensions. 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. 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 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. Diffeomorphic registration, introduced in the 90s, is now an important player with existing code bases organized around ANTS, DARTEL, DEMONS,
LDDMM, StationaryLDDMM 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) framework for shape comparison. On such deformations is the right invariant metric of
Computational Anatomy which generalizes the metric of non-compressible Eulerian flows but to include the Sobolev norm ensuring smoothness of the flows, metrics have now been defined associated to Hamiltonian controls of diffeomorphic flows.
Outline analysis performed on an outline analysis of some
thelodont denticles.
Outline analysis is another approach to analyzing shape. What distinguishes outline analysis is that coefficients of mathematical functions are fitted to points sampled along the outline. There are a number of ways of quantifying an outline. Older techniques such as the "fit to a polynomial curve" and Principal components quantitative analysis have been superseded by the two main modern approaches:
eigenshape analysis, and
elliptic Fourier analysis (EFA), using hand- or computer-traced outlines. The former involves fitting a preset number of semilandmarks at equal intervals around the outline of a shape, recording the deviation of each step from semilandmark to semilandmark from what the angle of that step would be were the object a simple circle. The latter defines the outline as the sum of the minimum number of ellipses required to mimic the shape. Both methods have their weaknesses; the most dangerous (and easily overcome) is their susceptibility to noise in the outline. Likewise, neither compares homologous points, and global change is always given more weight than local variation (which may have large biological consequences). Eigenshape analysis requires an equivalent starting point to be set for each specimen, which can be a source of error EFA also suffers from redundancy in that not all variables are independent. One criticism of outline-based methods is that they disregard homology – a famous example of this disregard being the ability of outline-based methods to compare a
scapula to a potato chip. Such a comparison which would not be possible if the data were restricted to biologically homologous points. An argument against that critique is that, if landmark approaches to morphometrics can be used to test biological hypotheses in the absence of homology data, it is inappropriate to fault outline-based approaches for enabling the same types of studies. ==Analyzing data==