GEOMETRIC MORPHOMETRICS, A NEW ANALYTICAL APPROACH TO COMPARISON OF DIGITIZED
IMAGES
I.Ya. Pavlinov. In: Information technologies in biodiversity research. St.Petersburg, 2001. P. 41-64.
INTRODUCTION
The demarcation between shape and size of morphological structures is fundamental for many of biological
investigations. Till recent times, two approaches have been existing and developing in parallel in such investigations,
quantitative analytical and qualitative geometrical.
In the first approach, the exploratory protocol was deduced to «extract» a size component from the overall
form diversity using one or another technique and to obtain, as a «residual», a shape component (Bookstein, 1989a).
Among these techniques were calculation of indices, logarithm recoding, principal component analysis, regression
analysis, etc. Such methods evidently provide an indirect estimates of diversity of shapes which are biologically
sound inasmuch as the respective underlying presumptions are correct. However, the shapes proper, mutual
transformations thereof remain aside. Besides, application of many statistical tests is correct in case of rather large
datasets with normal (or multinormal) distributions of variable which is hold true not frequently.
Somewhat apart stands a quantitative approach which is based on description of the shape diversity by a set
of discrete morphotypes. Calculating their frequencies and comparison of samples by these frequencies provide some
numerical estimate of similarities among the samples. However, two problems there exist. First, variation of shapes
is, a as rule, continuous, so recognition of discrete morphotypes is more or less arbitrary. Besides, particular shapes
are not being compared directly, only some average estimates of among-samples differences result from the analyses.
Unlike the analytical approaches, the geometric one is aimed at comparison of the shapes themselves. In the
modern science it can be traced back to pioneered investigations of d'Arcy Thompson (1917; the last edition is
Thompson, 1992) who was the first to apply transformation grid to illustrate interrelations of various shapes (Fig. 1).
Although his results were reproduces not a once in different manuals on biometrics, it did not won wide recognition.
The cause was simple enough: it didn’t contain any numerical tool, it was pure «figurative» approach allowing only
pairwise visual comparisons. Earlier attempts to unite the method of transformation grid with standard statistical
tools (see review in Bookstein, 1991) had no success: the general cause was that before searching for an appropriate
numerical approach to geometrical analyses of shapes an appropriate approach to description of the shape was
needed. That is, it was first requested to define what are the data and the variables for such an analyses (Bookstein,
1993).
The break-through in this field became an idea of using, as initial variables, not standard linear
measurements but Cartesian coordinates of the landmarks placed on the morphological objects being compared.
Respectively, the differences among the latters by their shapes was determined as the differences among landmark
configurations. This descriptive approach added by a special, so called Procrustes distance gave birth to a new
branch of biometrics which became known as «geometric morphometrics» (Bookstein, 1991).
In short, the latter may be defined as a set of methods of multivariate eigenanalysis of the landmark
coordinates that describe configuration of the morphological objects in the shape space (see below). It is elaborated
as an analytical toolkit allowing to extract size effect from results of investigation of the form of morphological
structures which is achieved by a set of specific algebraic techniques (Rohlf, Marcus, 1993; Pavlinov, 1995). By
these, geometric morphometrics differs significantly from other quantitative approaches to analysis of morphological
shapes such as Fourier analysis or fractal analysis (e.g. Ferson et al., 1985; Hartwig, 1991; Slice, 1993a; Costa,
Cesar, 2000).
Geometric morphometrics was burn just several decades ago when first fundamental ideas was formulated
(Kendall, 1984; Bookstein, 1986). At present, its mathematical apparatus is thought to be developed quite enough in
order to allow resolving many applied problems (Bookstein, 1991, 1996). It is being developed pretty actively:
general essays, collective books and review papers issued that expose adequately its theory, methods and
applications in biological investigations (Rohlf, Bookstein, 1990a; Bookstein, 1991, 1996; Marcus et al., 1993,
1996; Rohlf, 1996, 1999; Dryden, Mardia, 1998; Monteiro, Reis, 1999; Costa, Cesar, 2000). The software for
geometric morphometrics is also quite developed to allow analyses of a lot of biological tasks. There is an Internet
network uniting more than four hundred «geometricians» and accessible by the address
MORPHMET@LISTSERV.CUNY.EDU (supported by L. Marcus), and a site where computer programs are
available (see chapter «Software» below) along with nearly exhaustive bibliography on geometric morphometrics
including more than 500 references (the web-address is http://www.public.asu.edu/jmlynch-geomorph-index.html,
supported by J. Lynch).
Below a short review of basic ideas and methods of geometric morphometrics is provided, along with its
computer programs, a brief glossary, and some results of their use. Taking into consideration that the issue is
addressed to biologists, as well as specialty of the author, the mathematical backgrounds of the approach in question
is omitted.
BASIC FORMALISMS
From a formal standpoint, the object of geometric morphometrics is a set of x-,y- or x-,y-,z- Cartesian
coordinates, for 2- dimensional or 3-dimensional bodies, respectively. The object is described by a pk matrix where
p is number of landmarks and k is its dimensionality as a physical body.
Being digitized, the object can undergone the following transformations reflected in landmark coordinates:
isometric transformations which are translation, rotation, reflection (only position changes), isometric scaling (only
the size changes), and non-isometric stretching/shearing (only the shape changes). It is evident that landmark
coordinates contain initially redundant information about the object (position, size and shape are all incorporated), so
the first task is to elaborate some secondary shape variables which would contain only information about shape
proper. The latter, by definition, is determined by configuration (net location) of the landmarks.
It is to be stressed that these shape variables are initially tensors. This means, first, that each of them is
defined by a pair (or three) of coordinate axes that has no separate sense. Second, these axes are equivalent:
replacing of x by y and vice versa makes no changes in the properties of the objects being compared. This puts
certain limitation on application of geometric morphometrics: if the variables defining axes of a «morphospace» in
which these objects are distributed are not tensors (and thus, in particular, are not equivalent in the above sense), the
objects cannot not seemingly be studied by the approach in question.
Shape transformations studied by geometric morphometrics are decomposed into two components, uniform
(affine, linear) and non-uniform (non-affine, non-linear) (Bookstein, 1991). The first one corresponds to
nonlocalizable transformations: these include total stretching and shearing that are the same at all landmarks. It is
«modeled» by a rectangle turning into a rhomb when parallel lines remain parallel (Fig. 2a). The second component
includes local stretching, shearing, twisting, bending etc that are different at different landmarks and thus localizable.
This component can be represented by non-linear bendings of lines of a rectangle (Fig. 2b).
In order to eliminate the «size factor» from an object description, it is required to express it using landmark
coordinates only. For this, a centroid size is calculated as a sum of squared distances between all landmarks; or,
alternatively, as a square root of sum of squared distances between all landmarks and the object centroid (Bookstein,
1991). This variable is not correlated with any one of shape variables and is used for specimens alignment (see
below).
All procedures of geometric morphometrics implies using a reference object: its landmark configuration
defines geometry of the shape space (see below) and all specimens in the sample are compared relative to it. In some
instances it may be a real or a hypothetical object that depicts, for instance, beginning of a morphological
transformation series. In others, the reference is calculated (by least square analysis or its analogies) as a consensus
(average, mean) configuration so that its overall differences from all specimens in a sample by all (or by some, see
below) landmarks would be minimized.
The total of morphological objects described by landmark coordinates constitutes the figure space of pk
dimensionality (Goodall, 1991; Rohlf, 1996). It is to be stressed that this space has nothing in commong with a
physical space «filed in» with morphological objects. As a matter of fact, it is a special case of phenetic hyperspace,
so it is a mathematical artifact being construed for a given set of digitized objects by application of some
mathematical operations to a set of landmark coordinates.
The procedure of geometric morphometrics starts with the objects being centered on the origin (their
centroids are superimposed by translations). Then their centroid sizes are scaled to unit due to which the objects
become aligned and the «size factor» is eliminated from all subsequent comparisons. The alignment is fulfiled by
minimizing differences among respective centroid sized calculated using all landmarks (procrustes method) or a pair
of priory defined landmarks (baseline method).
As a result of these manipulations, the above figure space turns into the shape space, or the Kendall’s shape
space, a keystone of the entire geometric morphometrics. Its metrics is called Procrustes metrics as it is formed
basically by the Procrustes distance which is a specific analogy of Euclidian distance (see below). Coordinate axes
of a Kedall’s space could be assumed tensros. Its dimensionality is pk–k–k(k–1): for 2-dimensional objects its has
2p–4 dimensions, while for 3-dimensional objects its has 3p–7 dimensions. This space is fundamentally nonEuclidean: its geometry, for a simplest case of a plane objects described by three landmarks, can be visualized by a
hypersphere surface (Fig. 3). The morphological objects (shapes) are points on this surface. For more complex
morphological objects, the geometry of the shape space appears to be substantially more complicated and cannot be
so easily visualized (Goodall, 1991).
Metric properties of this space allow to assess differences among shapes as a distances between them. The
basic dissimilarity measure is the above Procrustes distance. It was first suggested as early as in 60-ies (e.g. Sneath,
1967), but it was geometric morphometrics that estimated its true worth. There are several versions of this distance
with pretty simple interrelations (Dryden, Mardia, 1998; Rohlf, 1999a). The angle Procrustes distance  is defined
as an angle (in radians) between radii connecting two points on the hyperspace surface to its center (see Fig. 3), its
numerical expression is arccosine of the square root of sum of squared distances of landmark coordinates of aligned
objects (Bookstein, 1993). It can be defined also as the geodesic distance as it provides estimation of distance by the
hyperspace surface. The chord Procrustes distance dp is defined as length of the chord connecting the same points
and is calculated as the square root of sum of squared distances of landmark coordinates. These two distances are
related by the formula =2sin-1(dp /2). As the Euclidean distance, the Procrustes distance is a metrics and thus can be
used in cluster and ordination standard routines (see «Application of standard routines» below).
Non-Euclidean geometry of the shape space makes it difficult to use multivariate methods based on
assumption of orthonormality of covariance matrix and on linear combinations of variables. It is evident that the
more dissimilar are particular shapes, the more distantly are situated the respective points on the hyperspace, and
hence the more prominent are non-linear effects. Therefore the statistical methods adequately describe shapes
differences (distribution of points on the hyperspace) if the latters are «small» (Bookstein, 1991).
To avoid non-linearity effects, the Kendall’s space is approximated by a tangent space that has a Euclidian
geometry. It is visualized by a hyperplane tangent to the Kendall’s hypervolume (see Fig. 3). Respective tangent
point to sphere point defines position of an consensus (for the given dataset) landmark configuration. The shapes can
be defined now as projections of the initial points onto this hyperplane, so all shape variations are imbedded without
significant loss of information. Their dissimilarities are now evaluated as distances in the tangent space, which are
Euclidean distances de. It is evident that de and dp are related by monotonous linear function, so that the scale of
dissimilarities is not much disturbed, especially if they are not especially large.
Math properties of the shape space allow to postulate that the differences among morphological objects
actually studied by biologists are generally «small»: respective points are distributed in the environments of the
tangent point (Rohlf, 1996, 1999). Empirically, the above disturbance is estimated by comparison of two distances
calculated for the given dataset, Procrustes (in form of geodesic on the hypervolume) and Euclidean (on the
hyperplane) using programm TPSmall. The linearity assumption is met in nearly all the cases studied, except
comparison of an object with its reflection (Rohlf, 1999).
It is to be stress that geometry of both Kendall’s and tangent spaces is defined not by interrelation among
the specimens studied (in particular, by their landmark covariation) but by the references configuration only.
Figuratively speaking, the geometry of the shape space is first established using this configuration and then the
objects are projected on its hypersurface. This means, among others, that its geometry (in particular, orthonormality)
is pre-established by Procrustes metrics prior to any analyses of the data and does not depend on interrelations
among objects themselves (unlike what is presumed by factor analysis) (Bookstein, 1996).
It follows from the immediate above that the reference configuration plays very important role in
explorations using methods of geometric morphometrics. Defining reference one or another way and/or changing its
landmark coordinates means changing geometry of the shape space. Therefore it seems reasonable to use the
consensus configuration as a reference: it corresponds to the tangent point and is situated in an average position
relative to distribution of the objects studied, so alignment of the latters relative to the reference thus defined causes
less disturbance of initial (on the Kendall’s space surface) similarity relations among the shapes as compared to any
«peripheral» configuration (Rohlf, 1996). Besides, the properties of the shape space become dependent, at least in
part, on geometry of the objects for which consensus reference is calculated.
Such a way of construing the shape spaces implies an important property thereof: they turn to be, that is to
say, «local». This means that each shape space based on a particular reference configuration appeares to be
«isolated» from all other shape spaces based on any other references. This is a pure mathematical property followed
from the non-monotonicity theorem (N. MacLeod, in litt.), and it makes such spaces a kind of «close axiomatic
systems», so some «translators» are requested for their mutual interpretation. This problem only begins to be
recognized, but hardly yet completely. As a matter of fact, incommensurability of results obtained for different shape
spaces (operationally, for different datasets) contradicts to the vary nature of the comparative method without which
not any biological research is even possible. Thus, the problem in question requests some general resolution without
which possibilities of geometric morphometrics appeares to be quite «local».
It follows from fundamental properties of the shape space that methods of geometric morphometrics do not
generally bounded by distribution laws. Therefore most of limitations related to those laws are not in effect for them.
As a consequence, this makes unnecessary calculating confidence intervals for the shape variables and, subsequently,
estimation of statistical significance of the results (but see chapter «Quantitative shape comparisons» below).
THE DATA
The research field of geometric morphometrics is a diversity of morphological structures, that is of physical
bodies on which landmarks can be reasonably placed and Cartesian coordinates thereof can be taken. This diversity
may be uncertain individual variation; differences among any discrete groups — taxa, sexes, ages, insect casts,
biomorphs, ecotones and others; series of postures or locomotory phases (MacLeod, Rose, 1993); fluctuating
asymmetry (Klingenber, McIntyre, 1998); etc.
The morphological structure is actually 3-dimensional. However, some technical limitations, precision of
digitizing devices in particular, make acquiring of 3D data for small (up to several centimeters) objects at present
rather problematic (Dean, 1996). Therefore the more common practice now is to work with 2D objects: these may be
either 2D projections of original 3D objects or such plane structures for which third dimension has no special
biological meaning. The latters are exemplified by insect wing, plant leaf, chewing surface of vole tooth, etc.
Such a reduction may lead to loss of some relevant information. As a half-measure, a method of «pseudo3D» presentation of an object is developed which is based on analysis of a set of 2D projections placed at strictly
fixed angles (Fadda et al., 1997).
At present, the question of «nature» of the objects to which geometric morphometric methods could be
applied is nearly uninvestigated. It is accepted by a default that they are to be «classic» morphological entities such
as a skull. However, as far as the analysis is applied to not an object itself but to it digitized screen image, the above
question is not as simple as it may look. Its general decision being unavailable, two particular but significant
restrictions are to be indicated. First: the variables defining a «morphospace» are to be tensors in order to the latter
could be treated as a shape space (see above chapter «Basic formalisms»). Consequently, if this condition is not met
then an object could not probably be studied by means of geometric morphometrics. Second, followed from the first:
as far as alignment of the objects involves their rotation, then if such an operation changes position of these objects
relative to the non-equivalent axes of a respective «morphospace», the vary alignment becomes nonsense, so the
entire approach does.
The sample of specimens to be studied by geometric morphometric methods is construed under the
following conditions.
The morphological structures are strictly comparable if they are projected onto the same shape space. This
means that all the specimens, even if they belong to different groups of interest, are to be included into the same
sample (excluding some special cases, see below) for which a common reference is to be defined, be it calculated
average (consensus) or a real/hypothetical object. For instance, if differences among sexes or local populations will
be studied by means of dispersion or discriminant analyses, all relevant specimens are to be included into the same
dataset which undergone geometric morphometric procedure. If the purposes of a research project presumes analysis
of shape variables separately in each of such groups, for the latters to be commensurable the next method should be
followed. All the specimens are first united in the common dataset for which a common reference configuration is
defined. Thereafter, each group is studied separately but relative to the same reference: this presumably means that
they all are dirstributed in the same shape space.
The methods of geometric morphometrics are based on simultaneous analysis of configuration of the
landmarks located on the morphological structure under investigation. Therefore the most «appropriate» are solid or
rigidly articulated structures with least degrees of freedom (Adams, 1999): an insect wing, a plant leaf, an axial skull
projection are examples. Unlike these structures, the bat wing is less «suitable» for geometric morphometrics: its
elements are very mobile relative to each other, so defining «standard» position provides certain problems (Birch,
1997).
As to the disarticulated structures, landmark coordinates of only those can be combined into the same
dataset for which such a combination has a biological meaning. For instance, it is possible to unite in a dataset the
landmarks placed on the common projection of the axial skull and the jaw, preferably with occluded toothraws
corresponding to a standard position. Contrary to this, there is no sense to unite in the same dataset configurations of
the skull and, say, the scapula. A separate dataset should be created for each of such structures which should be
analyzed separately: they can be compared subsequently by canonical or correlation analysis (see the next chapter).
It is possible to calculate consesnsus configuration for any set of specimens in a dataset. For instance, if
«averaged» differences among sexes or species are of interest, a separate consensus is to be calculated for each of the
group in question, and then all these configurations are to be united in a new dataset. It allows to make the
differences more clear-cut. Analysis of the gnathosoma shape in several tick species of the genus Ixodes (Voltzit,
Pavlinov, 1994) indicates that the differences among adult males are more conspicuous when consensus
configurations, and not separate individuals, are compared (Fig. 4).
As far as geometric morphometrics is based on pairwise comparisons of the objects and does not take into
consideration their distributions, the sample size — number of both specimens and landmarks — is irrelevant in most
study cases. However, if standard statistical routines (such as dispersion analysis) are supposed to apply, the sample
size is to be large enough. Moreover, if these routines include covariance matrix inversion, the number of specimens
should no less than 4 times exceeds the number of landmarks (Bookstein, 1996).
The above «reasonability» of landmark placing means, before all, that the landmarks on one object should
correspond unambiguously to the landmarks on another object. This means that the corresponding landmarks should
be fixed at the points on the object surface that are in a certain sense «the same». There are several bases on which
landmark correspondences can be established, according to which several types of landmarks can be recognized
(Bookstein, 1990a; Slice et al., 1996; MacLeod, 2001). The landmarks of type I are fixed in accordance to the
classical criteria of homology (special quality, befor all). The landmarks positioned at place of certain tendon
attachment to the bone or of fusion of certain veins on the insect wing are examples. For the landmarks of type II not
only strictly biological but also geometrical criteria are taken into consideration: for instance, certain points of
maximal curvature of a contour line of a vole dental crown or an oak leaf. Landmarks of type III are defined
exclusively geometrically: they are placed at extreme points of a curve (for instance, depicted by the ends of
diameter). Only corresponding landmarks of type I and, in part, of type II could be treated as homologous; however,
the landmarks of type III are hardly homologous, they are simply equivalent (MacLeod, 2001).
If exact placing of landmarks is impossible due to absence of any strict «bindings», outline points, or
semilandmarks could be applied (Bookstein, 1997; Pavlinov, 2000a; MacLeod, 2001). They are distributed evenly
according to certain algorithm along a contour lacking any unambiguously corresponding («homologous») inflection
points. In such a case, the specimens are compared not by particular landmarks but by the entire sequence of
semilandmarks. Respectively, equivalency is established not between particular landmarks but between the sequence
of semilandmarks corresponding to entire contour curvature (Fig. 5). This equivalency is warranted by two factors:
number of semilandmarks is to be the same for all the contours compared, and terminal semilandmarks of a sequence
should be fixed at strictly defined positions (they should correspond to true landmarks of types I or II).
Efficiency of semilandmarks in analyses of shapes is still questionable (Sampson et al., 1996; N. MacLeod,
in litt.). The problem is that the semilandmarks which sequence was generated by a single algorithm are mutually
correlated; besides, they should be placed close to each other in order to describe outline adequately. This puts
certain limitations on applicability of methods of geometric morphometrics. First, use of semilandmarks provides
somewhat biased estimate of shape diversity (it diminishes variability). Second, their mutual closeness makes it
possible to apply Procrustes fit only, while resistant fit method cannot be probably used (see on the fitting methods
below) (J. Rohlf, in litt.). (By the way, standard landmarks are also being preferably placed evenly and without local
concentration for the same reason). Thus, the semilandmarks are of subsidiary significance only.
In a real situation, a combination of both landmarks of various types and semilandmarks is often applied.
For instance, in description of lateral projection of muroid rodent skull (Fig. 6a), the points 9, 11, 17 correspond to
landmarks of type I, the points 6, 7 — to landmarks of type II, the points 8, 13 — to landmarks of type III, and the
points 1-5 — to semilandmarks.
All shape transformation in both Kendall’s and tangent spaces are continuous. Therefore all qualitative
modifications causing change in topology of a morphological structure are at the moment inaccessible to geometric
morphometrics. In particular, it is impossible to describe and investigate analytically by its methods any differences
caused by appearance/disappearance of structural elements, such as perforations, accessory bones on the skull or
tubercles on a tooth, veins on insect wing or on a leaf. There is a promising idea opening a possibility to apply some
apparatus similar to Tom’s catastrophe theory: it implies description of quantitative shape transformations as breaks
(«creases») in the surface of the Kendall’s space (Bookstein, 2000, 2001).
As long as such methods are just being elaborated, some «roundabout maneuvers» could be applied. For
example, for a large sample, an absent structure could be indicated by a kind of «virtual landmark» which
coordinates are calculated as averaged coordinates of respective real landmarks for entire sample (Marcus et al.,
2000; Pavlinov, 2001). If fusions and divisions of some structure are involved in shape transformations, they could
be described at least in come cases by fusions and divisions of respective landmarks, their total number remaining
unchanged (Pavlinov, 2001).
A specific methodological problem is provided by bilateral symmetrical structures (Bookstein, 1996). When
the objects with various magnitude of asymmetry are compared, their symmetry axes would also be rotated to reach
the best fit by the least square method. Therefore, if asymmetry is not an immanent property of the morphological
structure (as in a dolphin skull) or a specific subject of investigation (as a fluctuating asymmetry), it is possible to
work with one of the bilateral object’s sides. In such a case, the landmarks are initially placed on one side only, or
the same set of landmarks is first defined for both sides and coordinates of each equivalent pair are then averaged.
Methods of digitizing morphological structures depends are diverse enough and depend on size (large —
small) and configuration (3D — 2D) of the objects themselves, on the purposes of investigation (whether it allows to
«reduce» a 3D object to 2D one), as well as on availability of respective hardware (see a little bit outdated review:
Becerra et al., 1993). It is recommended to take coordinates on the same object more than once and then to use their
average values. For many purposes, use of screen digitizers is most efficient; for semilandmarks it is probably the
only possible. Among most popular are softwares written for geometric morphometrics directly: these are Morpheus
et al., WinDig and especially TPSdig.
Cartesian coordinates of landmarks or semilandmarks are most often used in geometric morphometrics.
However, if it is desirable for any reason to fix some preferred «starting position» in the shape description, so called
Bookstein (or two-point) shape coordinates are calculated (Bookstein, 1991). For this, two landmarks are first fixed
to define a baseline with respective coordinates (0,0) and (1,0). Other landmarks are re-defined as vertices of
triangles with the common base defined by the baseline. Their coordinates are re-calculated respectively which leads
by implication to alignment of the objects.
The least square based methods are suggested to use to replace missing coordinate data by respective
averages (Slice, 1993c). Due to this, many paleontological data could be studied by geometric morphometrics.
QUANTITATIVE SHAPE COMPARISONS
As the morphological objects are generally incomparable by original coordinates, the numerical geometric
morphometric analyses begins with their replacement by some shape variables which values are landmark
coordinates in the shape space.
For this, the specimens (more exactly, their digitized images) are first aligned: this action eliminates their
differences due to translations, scaling and rotations but not reflections. The most usually applied are the standard
least square methods (in particular, Procrustes fit method) and specific for geometric morphometrics resistant fit
method. They can be imagined as rigid rotation of one object relative to another so that their differences become
minimized, either by all landmarks or by those which differences are not exceptionally large (Rohlf, 1990; Rohlf,
Slice, 1990; Bookstein, 1991).
Some differences remain among specimens after their alignment indicated by dispersion of landmarks
around reference configuration (see Fig. 6b). They correspond, by definition, to differences of landmark
configurations in the tangent space, that is to dissimilarity of the shapes. This dispersion around each of the
landmarks is expressed by so called Procrustes residuals which are considered as special case of shape variables,
Procrustes coordinates (Rohlf, 1999). Their sum gives overall Procrustes distance among the specimens.
There are two methods of comparison of shapes in geometric morphometrics (Rohlf, 1996). One of them is
based on the above least square method, it is most efficient if overall similarity depends largely on few landmarks.
Another one is the thin-plate spline analysis, it works best if the similarity depends on many of landmarks and so just
weakly localizable.
The least square method in geometric morphometrics is represented by the superimposition method which
is, as a matter of fact, extension of Procrustes or resistant fit rigid alignment. It is resulted in a Procrustes distance
matrix to which other appropriate multivariate routines can be applied directly (see chapter «Application of standard
routines» below). In particular, this matrix can be decomposed into eigenvectors by principal component analysis:
this gives a new set of shape variables which are now coordinates in the space of principal components.
Significantly different is thin-plate spline analysis based on analogy of a 2D morphological object to a thin
homogenous deformable metallic plate (Bookstein, 1989b, 1991). Accordingly to its methodology, one specimens is
fit to another by its non-rigid «stretching/shearing», and numerical estimate of degree of such a smooth deformation
is bending energy coefficient. Unlike Procrustes distance, the latter cannot be interpreted as a overall dissimilarity
estimate: it characterizes «localizability» of transformation which turns the reference to (fit) into particular
specimens under investigation. It is fulfilled by softwares Morpheus et al. and especially TPSrelw. Its application
produces several kinds of shape variables called the warps.
The thin-plate spline analysis begins with decomposing the bending energy matrix into orthogonal
eigenvectors which are called principal warps (Bookstein, 1989, 1990). Their geometry is defined by reference
configuration exclusively and describes possible directions of transformations of the latter in the tangent space. The
eigenvalues decreasing sequentially from the first to the last are determined by scale (localizability) of reference
transformations: the higher is a particular eigenvalue, the more localizable is respective transformation. Three last
(zeroth) principal warps correspond to uniform shape component transformations. If an investigation is aimed at nonaffine (localizable) transformations, these zeroth warps could be excluded from subsequent analyses. The principal
warps cannot be regarded each as bearing some specific biological meaning. In a sense, they could be considered as
an analogy of Fourier harmonics which refer to different spatial scales (Rohlf, 1998).
In order to express each specimen in term of thin-plate spline parameters (that is, to fit it to the tangent
space), a new set of shape variables is yielded called partial warps (Bookstein, 1989). They are calculated as
coordinates of projections of aligned specimens on each of the eigenvectors. They indicate for each specimen how
much of each principal warp is needed to transform the reference into that specimen (Rohlf, 1993). From this it
follows that geometry of partial warps is determined exclusively by geometry of principal warps and subsequently
does not depend on landmark covariation in the sample studied. Therefore this geometry is unstable: change of
reference configuration implies change of both the geometry itself and partial warp values. The partial warps are
intercorrelated and thus each one of them cannot be used as a separate biological trait (Rohlf, 1998).
Distribution of specimens along the axes of partial warps illustrates the spatial scale at which the main
differences among them are observed. For instance, analysis of distribution of sex and age groups of Ixodes ticks
(see Fig. 4) in the space of 12 partial warps indicates the following. Adult males in total are most separable by 11th
and 5th warps and by uniform component (Fig. 7): it means that they differ from remainders mainly by total shape of
their gnathosoma. In addition, the male group № 8 takes quite isolated position by 3d partial warp, so its specifics is
expressed significantly by small-scaled differences.
To eliminate limitations intrinsic to both principal and partial warps, relative warps are calculated by
principal component analysis as orthogonal eigenvectors of covariance matrix of partial warps (Bookstein, 1991;
Rohlf, 1993). Dimensionality of a new shape space defined by relative warps is nmin(n–1, p(or p–3)), that is it
depends basically on the number of objects and not of landmarks. Distribution of specimens in a newly established
shape space is defined by their projections onto relative warps. Their eigenvalues have the same meaning as in
standard PCA: they reflect amount of total variance «explained» by respective relative warps and decrease from the
first to the last one. Subsequently, analysis of relative warps implies reduction of shape space dimensionality and
their orthogonality allows to treat them as separate traits.
The parameter  is used in calculations of relative warps which allows to assign different or equal
«weights» to differently scaled shape transformations (that is, to different principal warps) (Rohlf, 1993). If >0
then «zeroth» principal warps are excluded from the analysis and ordination of specimens is made on the basis of
bending energy matrix. The value of =1 yeilds the relative warp analysis in which the principal warps are weighted
inversely by the square root of their eigenvalues. As a consequence, the large-scaled variation is given more weight
than small-scaled one. If =0, then all principal warps are included in the analisys and all are given the same weight,
thus relative warp analysis is actually carried out on the basis of the Procrustes fitting and not on the thin-plate spline
analysis. This value of =0 is to be accepted if landmarks (and especially semilandmarks) are situated close to each
other.
It is essential that if =0 then distances among specimens in the relative warp space are the same as the
distances by original landmark coordinates (Rohlf, 1993). It means that distribution of the specimens in that space
reflects directly the structure of dissimilarity relationes defined by Euclidian metrics. Therefore, zero value of  is
generaly preferable. However, if the research is aimed at analysis of allometric growth gradients, then =1 should be
adopted, because allometric relations involve usually large-scaled shape transformations.
In order to «localize» particular relative warps they have to be «tied» with particular landmarks. For this, a
matrix of loadings (decomposed in Cartezian axes) of each landmark onto each relative warp is calculated (Table 1).
As in PCA, the loadings indicate how much is input of the given landmark into transformations of the reference
along relative warps as the axes of respective shape space. Besides, because of orthogonality of relative warps,
distributions of landmark loadings among these warps make it possible to discuss which components of shape
transformations are mutualy correlated or, contrary, independent.
It is seen from the Table 1 that in case of the axial skull of muroid rodents (see Fig. 6), the highest loading
in the first relative warp is provided by landmarks 11, 17, 18, while in the second relative warp it is provided by
landmarks 14, 15, 19. This means that changes in position of proximal part of the toothraw and the masseteric plate
are most correlated with each other, on one hand, and least correlated with displacement of the middle portion of the
zygoma and condylus, on another hand. Input of the unform shape component is estimated similar way. It is clear
from the same Table that explaned variance increases after elimination of this component from the analysis. Thus,
one may conclude that small-scaled shape transformations prevail over large-scaled ones.
Localizability of these transformations can also be deduced from distributions of explaned variances among
relative warps. As it is seen from the same Table, proportion of variances explaned by respective 1st relative warps
is noticibly higher for the axial skull than for the mandible. Together with distributions of landmarks loadings within
each relative warps, it means that non-affine shape transformations are more promimnent and more localizable on the
axial skull as comapred to the mandible.
In verifying hypothesis of non-random differences of samples or correlation of structures several statistical
tests can be applied that are adapted to specifics of shape variables, before all to mutual correlation of partial warps.
Overall differences among samples are tested by Goodall’s F criterium (based on analysis of Procrustes distances) or
Hotelling’s T2 criterium (based on analysis of Procrustes residuals) (Goodall, 1991; Dryden, Mardia, 1998). Some
randomization techniques are also employed — permutation, bootstreping, etc. (Dryden, Mardia, 1998).
In comparing several groups by entire set of shape variables a multivariate dispersion analysis could be
applied (MANOVA, available in TPSregr and APS.). For example, in analysis of effects of taxonomic allocation and
trophic specialization on the skull shape in muroid rodents, the following results was obtained. Respective Goodall’s
F values are equal to, respectively: for the axial skull — 1.86 and 5.43 (df 38, 608; p=0,002 и р0.001), and for the
mandible — 3.12 и 2.79 (df 22, 352; р0.001). It is seen that for the axial skull, taxonomic allocation is less (the
least, actually) significant than trophic specialization, while for the mandible the ratio is opposite. If particular shape
variables are of interest then standard statistical routines are employed (see chapter «Application of standard
routines» below).
Covariation of the shapes which landmark coordinates cannot be united into the same dataset (see previous
chapter) can be investigated in several ways. One of them is based on a method similar to canonical correlation
analysis (available in the TPSpls) which allows also to explore effect of any quantitative variable (size, climate factor
etc) on the shape transformation. For instance, correlation of the first vector for axial skull shape with that for
mandible is high, while its correlation with the body size is low (Fig. 8): correlation coefficients are 0.96 and 0.62,
respectively.
Another method is to anaylize correlation between relative warps calculated for different shapes (or between
relative warps and other variables). This approach is of interest, as it permits to «localize» correlations of those
shapes. For examples, in the above muroid rodents most correlated appeared to be first relative warps of both axial
skull and mandible. Looking at the figures in the Table 1 and at landmark locations on Fig. 6, one may conclude that
changes in toothraw and masseteric plate (landmarks 11, 17, 18) are most correlated with changes in configuration of
the base of proximal part of the mandible (landmarks 4, 6, 11).
Among most problematic in geometric morphometrics remains assessment of diveristy (dispersion) of shape
variables. At present, this methodology lacks any statistics similar to standard coefficient of variation. It is
theoreticaly possible to evaluate overall shape variability using Procrustes distance; however, two unclear points
appear. On the one hand, developers of geometric morphometrics used to stress that most interesting are «local» and
not «total» shape differences, so for them it does not make much sense to apply any generalized indices of
similarity/dissimilarity like «taxonomic distances» (Bookstein, 1996; Rohlf, 2001a; N. MacLeod, in litt.). On the
other hand, statistical properties of Procrustes distance are not properly studied yet: it is established empiricaly
(personal unpublished data) that its values positively correlated with the number of landmarks, but the nature of this
correlation is not clarified and, consequently, no corrections factor is figured out.
GRAPHIC METHODS
As it was noticed in the introductory chapter, geometric morphometric was born mainly as a «technical
device» for analytical resolution of geometric tasks exposed by D’Arcy Thompson. And developers of this
methodology used to stress that its most prominent edvantage is not the «figures» but the geometric images. At the
same time, it is advised to keep in mind that these images are deduced from approximate algorithms and would not
be taken to literally: they are rather visual «metaphors» of shape transformations than exact mappings of them
(Bookstein, 1996).
The simplest way to visualize variabilty of shapes is decomposing residuals obtained by resistant fit method
into eigenvectors for each landamrk separately. They characterize direction and amplitude of variation at each
landmark not taking into account landamrk covariation. The most obvious and most easily interpretable
representation is a set of ellipsoids depicting boundaries of confidence intervals for dispersion of landmark
coordinates for a given dataset (see Fig. 6b). This approach provided by the software GRF is actually univariant and
therefore is not especially popular among recent researchers.
More sofisticated are methods based on the thin-plate spline analysis: they embrace correlation between
lanmarks and thus are multivariate. The resulting graphic images illustrate partial and relative warps in two ways.
One of them is transformation grid, another is a set of vectors (Fig. 9, 10). The both are available in the programs
TPSsplin, TPSpls, TPSrelw, TPSregr, Morpheus et al.
Transformation grid is initially orthogonal (Fig. 9a). The more are differences among shapes in the
vicinities of a landmark, the more is deformation of respective fragment of the grid (Fig. 9b). Smoothness of the
function corresponding to bending energy coefficient allows to define grid configuration between any pair of
landmarks, so it might be of arbitrary density. Contrary to this, vectors are tightly bounded to the landmarks (Fig.
9c). They have to be analyzed in their totality, as any one vector has no independent meaning. As dissimilarity
relation defined by bending energy is not symmetric, graphic representation of superimposition of A onto B is not the
same as B onto A (fig. 10).
In relative warp analysis, the graphic representation allows to localize most expressed shape changes
corresponding to a particular warp. Generally, degree of grid transformation arownd a landmark (or vector length)
corresponds to loading of this landmark into respective relative warp.
Graphic representation of shape differences avails both for a pairwise comparison (see Fig. 10) and for a
multi-object sample. In the latter case, of importance is a possibilty to visualize potential transformations of
reference configuration in the space of relative warps. This approach allows to demonstrate graphically, how the
shape of the specimens is transformed along particular relative warps in either «positive» or «negative» directions
from the zeroth point corresponding to consensus configuration (Fig. 11).
Graphic approach can serves as an important addition to numerical multivariate methods (like MANOVA)
of analisys of among-group dissimilarities. For this, consensus configurations are calculated for each of the groups
under comparison (see chapter «Data» above) and superimposed upon each other. Configurations of respective
transformation grids show where the differences revealed statistically are localized.
An interesting possibility to trace shape transformations among taxa is provided by program TPStree, given
that there is a dendrogram with a priory defined topology that reflects certain (for instance, phylogenetic) relations
among those taxa. For each fragment of this dendrogram, it is possible to get a grid which transformations
correspond to differences among average (consensus, not ancestral!) configuration of respective group of taxa and a
hypothetical one placed at the base of dendrogram.
APPLICATION OF STANDARD ROUTINES
Geometric morphometrics deals with shape deformations proper, so its «ultimate aim», strictly speaking, is
limited to Procrustes distances and partial warp scores that indicate to what extent and how one shape differs from
other(s). However, if the research is managed to go further, other numerical techniques are to be applied borrowed
from the standard statistics.
Some of these techniques are built in the geometric morphometric toolkit: among them are PCA, canonical
correlation, regression and dispersion analyses (considered in short above). At the same time, geometric
morphometrics still lacks some commonly used standard methods — for instance, cluster analysis, multidimensional
scaling, stepwise discriminant analysis etc. Their applications are in part limited by properties of shape variables
distribution that should certainly be taken into consideration when a research project is planned (Bookstein, 1996;
Rohlf, 1998, 2001a). Most of these properties were mentioned above; to sum up, the following points should be
stressed.
Before all, it should be reminded that any comparison of different groups of specimens by shape variables is
valid only if the specimens belonging to those groups are all included in a common dataset. Numerical results
obtained independently for different datasets by means of geometric morphometric tools are not strictly
commensurable.
Procrustes distance is a metric and may be dealt with as the Euclidean distance — for instance, it may be
used in multidimensional scaling (Gower, 1975) or in cluster analysis. In contrast to it, bending energy coefficient is
neither metrics nor ultrametrics, so the bending energy matrix should not be studied by standard methods.
Partial warps are mutually correlated and therefore reflect shape changes in their total only. Any one of
them taken in isolation does not bear any sensible biological information and thus should not be analyzed separately
from others. Besides, this shape variable, as a matter of fact, is not a vector but a tensor as it is defined by a pair (or a
triplet) of x,y(z) coordinates. Therefore it is not recommended to explore npk matrix of partial warp scores by any
of univariance methods or by the multivariance methods assuming orthonormalitry (such as standard factor analysis).
To this matrix, applied could be such methods as multiple regression, multiple analyses of variance and covariance
(MANOVA, MANCOVA), complete discriminant analysis (without selection of variables); cluster analysis of
Euclidean distances calculated for the entire set of partial warps; also applicable are such multivariate estimates as
Hotelling’s Т2, Mahalanobis’ D2 , Wilkson’s  etc.
Unlike partial warps, relative warps are not correlated with each other, so it is possible to apply to them not
only all the above methods but univariance ones and a stepwise discriminant analysis, as well. The only reservation
should be made that relative warps (as principal components in general) are a kind of «mathematical artifacts»,
therefore analysis of differences among samples by each warp using, say, the Student’s T criterium makes no special
sense.
As in the case of partial warps, differences by relative warps can be evaluated numerically by Euclidean
distance, which produces the same estimates as the Procrustes distance. This technique is of special meaning, as
there some tasks exist that require standardization of the data, and such a standardization meets least objections just
in the case of relative warps. For instance, it is possible to study structure of shape diversity by means of
multidimensional scaling. For this, distributions of stresses calculated at each step of iterative procedure of
dimensionality reduction are compared for the distance matrices calculated for real and random data (Puzachenko,
2001). But because the stress values are directly proportional to the distance values, the both datasets are to be
equally standardized, and this could most easy be done for the original variables rather then for calculated distances.
Stress distributions obtained for two Euclidean distance matrices are shown on Fig. 12. One of these
matrices was calculated for relative warps scores of 60 third upper molars of the rock vole genus Alticola (some are
drawn on Fig. 5); another one was calculated for a set of variables generated by a random number calculus; in both
cases the data were standardized in the interval [0,1] (SYSTAT package was used for standardization and
randomization and Statistica for Windows package was used for multidimensional scaling). As it is seen, two
distributions appeared to be nearly the same which proves basically random distribution of the tooth shapes in a
shape space. One can conclude from this that there is no sign of discreteness in the shape differences thus estimated
and to argue (at least for this particular case) against the vary possibility of recognizing «discrete» morphotypes.
Analysis of effect of taxonomic allocation and trophic specialization on the skull shape in muroid rodents
using Fisher’s F estimation of relative warps (ANOVA in Statistica for Windows) gave the following results. The
highest correlation with both factors was obtained, as it had to be anticipated, for 1st relative warp which explained
the largest portion of a total variance. As to the ratio of the two factor effects, the present results differ to some
degree from those obtained using Goodall’s test (see chapter «Numerical shape comparisons» above). The both
morphological structures now appeared to be most dependent on taxonomic allocation rather than on trophic
preferences: Fisher’s F is equal to, respectively, 30.38 and 12.81 for axial skull and 16.73 and 4.01 for mandible
(trophic specialization was more significant for the axial skull by Goodall’s F). It is probable that 1st relative warp
extracted by thin-plate spline analysis does not yet incorporate some details of shape differences which is contained
in a complete set of Procrustes residuals.
A possibility of incorporating numerical results of geometric morphometric analyses in cladistics, which
methods became now quite «routinal», constitutes a special problem. Several attempts based on partial warps
(Zelditch et al., 1995; Naylor, 1996; Zelditch, Fink, 1998) met rather sharp criticism (Adams, Rosenberg, 1998;
Rohlf, 1998, 2001a; MacLeod, 2001). The causes are evident enough. Partial warps are «mathematical artifacts»:
they are linear combinations of original landmark coordinates calculated according to certain algorithm. Therefore,
unlike the original landmarks, they could not be treated as «homologous» traits. In this respect, relative warps have
some benefits over partial ones, as they could be «tied» to particular landmarks (through their «weights», see above)
and thus could be used in cladistic analysis as the traits (MacLeod, 2001). However, another problem emerges: the
both shape variables are strictly continuous, this property following not from peculiarities of morphological
structures themselves but from smoothness of the interpolation function corresponding to bending energy coefficient.
It is evident that bringing any «discreteness» into their distribution, as it is requested by parsimony methods in
cladistics, contradicts to the nature of these variables.
James Rohlf (1998, 2001a) suggests that most accurate, in respect to phylogenetic interpretation of
numerical data resulted from geometric morphometrics, is a maximum likelihood approach (in sense of Felsenstein,
1973, 1988, 2001) applied to Procrustes distance. His opinion is based on an idea that, for the both, the initial
statistical model is a Brownian motion around central momentum. However, at least as far as complex morphological
structures are concerned, such a model has probably no biological meaning. At best, it could be used to formulate a
kind of a null-hypothesis, not a working one.
On the other hand, if there is a cladogram deduced from some other data source, it is possible to «fit» to it
the shape transformations resulted from geometric morphometrics (see chapter «Graphic comparisons» above).
Besides, one could wish to apply contrasts method to determine a «phylogenetic load» into the shape diversity and
thus to discuss more thoroughly possible causes of historical transformations of that shape (e.g. David, Laurin, 1996;
Schaefer, Lauder, 1996).
THE SOFTWARE
Numerical methods of geometric morphometrics are not yet built in standard statistical packages such as
SYSTAT, SPSS, SAS, Statistica and others. However, they are available in a number of more special programs many
of which were already mentioned above. Most of them are designed for PC platform (with operational systems DOS,
Windows, less frequently OS/2 and UNIX), and few are for Apple Mac. PC-designed programs work under DOS in
earlier versions and under Windows (or occasionally under other systems) in their more recent upgrades. The
landmark coordinate data are kept in standard text files in ASCII codes, sometimes in electronic table format. Nearly
all these programs are free-ware (besides Morphologika and the last version of NTSYSpc which are commercial
products) available through the Internet site by address http://life.bio.sunysb.edu/morph/ (supported by F.J. Rohlf).
Below is given a brief compendium of these programs (listed alphabetically). The outdated program
products, for which more recent upgrades or more advanced softs are available, are not reviewed.
APS (Penin, 2000) explores a multi-specimen sample, extracts principal components (relative warps) based
on procrustes residuals; compares two subsamples by discriminant analysis of procrustes residuals; fulfils regression
of each of the relative warps against centroid size. Graphical representation of results is a scatter-plot of specimens
in the space of those principal components.
GRF (Rohlf, Slice, 1991) and GRF-nd (Slice, 1993b) are DOS applications working with 2D (GRF) or nD
(GRF-nd) objects. GRF-nd supports datasets with partially missing landmark coordinates. The specimens are fitted
by least square or resistant-fit methods, Procrustes residuals are decomposed by PCA into eigenvectors, dispersion
limits and direction are displaced graphically as ellipsoids or vectors.
Morpheus et al. (Slice, 1993c) is an advanced version of GRF-nd designed for PC (under Windows, OS/2,
UNIX) and Apple Mac. Undertakes Procrustes analysis (pairwise comparison of specimens by all landmarks),
transformation grid provides graphical representation of results. Fulfils Fourier analysis, as well. Allows to save
coordinates of a consensus configuration, Fourier harmonic coefficients.
Morphologika (O’Higgins, Jones, 1998) is a beautifully designed Windows application supporting both 2D
and 3D coordinate data; allows to select specimens and landmarks. Analytical facilities are limited, however: it
makes Procrustes fit, calculates principal components (relative warps) of Procrustes residuals, displays both scatterplot of specimens and transformation grid configurations in the relative warps space.
NTSYS-pc (Rohlf, 2000a) of 2.0 and higher versions is a Windows application in which some geometric
morphometric tools are built-in. Allows to analyze both 2D and 3D objects (unlike TPS series of the same author),
calculates principal and partial warps.
Shape (Cavalcanti, 1996) calculates and saves in a data file the Bookstein coordinates relative to a priory
user-defined baseline.
TPS is a series of programs issued by F.J. Rohlf. Their earlier versions were designed for DOS (are not
considered here), more recent are Windows applications designed in both 16 and 32 bit versions. They allow to work
with 2D objects only. Many of them have built-in screen graphic editor (manipulations with colours, size, labeling)
and save the images in graphic format files. The latest versions constitute a most powerful toolkit for geometric
morphometrics, all with rather detailed helpers.
TPSdig (Rohlf, 2001b) is a screen digitizer, operates with the files of most standard raster formats; also
permits loading some multimedia video files and capturing images from them; simple enhancement operations with
the images are possible. Allows to locate, maintain and edit on the screen landmark and outlines coordinates from
digitized images and to save them in a text file. When the original image file is also saved in a directory indicated in
the data file, loading the latter yields appearance of the image with the landmarks previously collected.
TPSpls (Rohlf, 1998a) computes canonical correlations among two shapes or among a shape and a nonshape variable. Most of the facilities are similar to those in TPSregr (see below).
TPSpower (Rohlf, 1999d) is an auxiliary program. Given an estimate of the means of two populations, the
expected amount of variability at the landmarks, and a sample size, it computes the statistical power expected using
various statistical methods.
TPSregr (Rohlf, 1998b) undertakes multiple regression and dispersion analyses of partial warps using
linear models. In the first case, one or several continuous independent variables (such as size, temperature etc) are
used, in the second case they are discrete (group belonging). Displaces shape changes predicted by regression model
in form of transformation grid changes. Provides statistical estimates of congruence of independent and shape
variables using standard criteria or permutation approach. Displays empirical distribution of specimens along
regression line; permits saving this picture in a graphic file, as well as regression coefficients and Procrustes
residuals.
TPSrelw (Rohlf, 1998c) calculates principal, partial, and relative warps. Calculates and optionally saves in
a file the landmark coordinates of both references configuration and of each specimen in a sample after their
alignment, partial and relative warps scores, and landmark loadings. Varying parameter  values and
inclusion/exclusion of uniform shape component is possible. Calculates SS for each of relative warp. Displays shape
changes as deformations grids or vectors, scatter plot of specimens in the spaces of partial and relative warps,
permits saving all screen images in graphic files.
TPSsmall (Rohlf, 1998d) is used to determine whether the amount of variation in shape in a data set is
small enough to permit statistical analyses using linear model.
TPSsplin (Rohlf, 1997) compares two shapes based on thin-plate spline analysis, displays results as
transformation grids or vectors (facilities are the same as in TPSrelw). Also optionally calculates and saves in a file
Procrustes and geodesic distances.
TPStree (Rohlf, 2000b) makes it possible to trace shape changes on a user-defined hierarchical tree (both
ultrametric and additive). Reads tree description from a NEXUS file. Displays the tree and transformation grid which
configuration changes accordingly to cursor position on the tree. For each position (corresponding to a hypothetical
object), grid configuration, landmark coordinates and partial warp scores can be saved in respective files.
TPSutil (Rohlf, 2000c) is a small program to edit data files: permits combining several files into one, to
select and deselect landmarks, and to randomize specimen list.
WinDig (Lovi, 1996) is a rather simple screen digitizer with some facilities permitting editing screen
images.
Besides the above programs, some geometric morphometric routines are available from standard statistic
packages, such as Matlab, SAS, if respective program languages is used (Marcus, 1993).
A BRIEF GLOSSARY
The below glossary is a brief version of the one published by Slice et al. (1996). Its updated version can be
found in the Internet site by already indicated address http://life.bio.sunysb.edu/morph/.
 — a «weighting» parameter used in relative warp analysis to give different (if not equal to zero) or same
(if zeroth) «weights» to different principal warps.
Affine transformation — a linear transformation of a shape in which parallel lines remain parallel (turns
square into parallelogram). Equivalent to uniform transformation.
Alignment — an isometric transformation of the objects by which their centoids are scaled to unit in
respect to that of reference configuration.
Baseline — an imaginary line connecting the pair of landmarks that are assigned to fixed locations (0,0)
and (1,0) in Cartesian coordinate system to define a basis for calculating Bookstein shape coordinates.
Bending energy — a metaphor borrowed from the mechanics of thin metal plates. It is the (idealized)
energy that would be required to bend the metal plate so that the landmarks were change their position appropriately.
The bending energy of an affine transformation is zero since it corresponds to a tilting of the plate without any
bending. The value obtained for the bending energy corresponding to a given displacement is inversely proportional
to scale of shape transformations. Such quantity should not be interpreted as a measure of dissimilarity (e.g.,
taxonomic distance) between two forms.
Bookstein shape coordinates, two-point shape coordinates — for the 2D data, recalculated coordinates of
landmarks defined as vertices of triangles which bases coincide with a baseline.
Centroid size — the size measure used in geometric morphometrics to scale a configuration of landmarks
so it can be plotted as a point in the shape space. It is the square root of the sum of squared distances of a set of
landmarks from their centroid, or, equivalently, the square root of the sum of the variances of the landmarks about
that centroid in x- and y- directions.
Consensus configuration — a set of landmarks representing the central momentum of the sample studied.
It is often computed to optimize some measure of fit: in particular, in the Procrustes fit a mean shape is computed to
minimize the sum of squared Procrustes distances among specimens in the sample.
Kendall's space — the basic non-linear interpretation of the shape space in geometric morphometrics.
Informally, it is represented by a sphere on which surface the points corresponding to particular shapes are
distributed. It is defined by Procrustes metrics and provides a complete geometric setting for analyses of the shapes.
Most multivariate methods of geometric morphometrics are linearizations of statistical analyses of distances and
directions in this underlying space.
Landmark — a specific point on a morphological object or on its digitized image located according to
certain rule. Position of landmarks is defined based either on the classical homology criteria (landmarks of type I) or
on geometric properties of the object (landmarks of types II and III) (see also semilandmarks).
Morphospace — any hyperspace in which the objects being compared are distributed. The objects are
characterized by any arbitrary (not obligatory equivalent) variables that define axes of this space.
Partial warps — an auxiliary structure for the interpretation of shape changes. Geometrically, partial warps
constitute an orthonormal basis for a tangent space. Algebraically, they are eigenvectors of the bending energy
matrix describing a shape deformation along each coordinate axis. Except for the very largest-scale and for uniform
shape component, partial warps are (approximately) localizable and have (an approximate) scale. The partial warp
scores are projections of the shapes on principal warps calculated from an orthogonal rotation of the full set of
Procrustes residuals.
Principal warps — eigenfunctions of the bending energy matrix interpreted as actual warped surfaces
(thin-plate splines) over the original landmark configuration. Principal warps together describe complete
decomposition of a sample of shapes relative to the sample reference shape. Together with the uniform shape
component, supply an orthonormal basis for a space that is tangent to Kendall's space.
Procrustes distance — a measure of dissimilarities among shapes by landmark coordinates. Calculated as
the square root of the sum of squared differences between the positions of the landmarks in two optimally (by leastsquares) aligned and superimposed configurations at centroid size.
Procrustes fit — a set of least-squares methods for estimating dissimilarities among shapes by landmark
coordinates. The adjective «Procrustes» refers to the Greek giant who used to stretch or shorten victims to «fit» his
bed.
Procrustes residuals — the set of vectors connecting the landmarks of a specimen to corresponding
landmarks in the reference configuration after a Procrustes fit. The sum of squared lengths of these vectors is
(approximately) the squared Procrustes distance between the specimen and the reference.
Reference configuration, reference — a configuration of landmarks to which data are fit. It may be
another specimen in the sample or the average (consensus) configuration for a sample. The reference configuration
corresponds to the point of tangency of the linear tangent space to spheric Kendall’s space. The average
configuration is preferable as the reference in order to minimize distortions caused by this linear approximation.
Relative warps — principal components of a distribution of shapes in a tangent space. Each relative warp,
as a direction of shape change around the reference, can be drawn out as a thin-plate spline transformation. In a
relative warps analysis, the parameter  is used to «weight» shape variation by the geometric scale of shape
differences. Relative warps can be computed from partial warps or from Procrustes residuals.
Resistant fit — a superimposition methods that use median-based estimate of fitting parameters rather than
least-squares estimates. It takes into account landmarks that provide small differences among shapes and thus is less
sensitive to extreme values than those of comparable procrustes fit methods. However, resistant-fit methods lack the
well-developed distributional theory.
Semilandmark — an element of a single sequence of points along an outline that is generated by a single
algorithm.
Shape — in geometric morphometrics, a geometric properties of a configuration of points that are invariant
to changes in translation, rotation, and scale. The shape of an object is represented by a configurations of landmarks,
as a single point in a shape space defined by a set of shape variables. In morphometrics there are also other sorts of
shapes (e.g., those of outlines, surfaces, or functions) correspond to quite different statistical spaces.
Shape space — a fundamental algebraic construction in geometric morphometrics. Each point in this space
represents a configuration of landmarks irrespective of size, position, and orientation, which is the shape by
definition. In the shape space, scatters of points correspond to scatters of entire landmark configurations, not merely
scatters of single landmarks.
Shape variable — any measure of the geometry of a biological form or of its digitized image that does not
change under translations, rotations, and changes of geometric scale. Shape variables include angles, ratios, and any
of the sets of shape coordinates that arise in geometric morphometrics (Procrustes residuals, principal, partial and
relative warps etc).
Superimposition — transformation of one or more shape to achieve some geometric relationship to another
shape. These transformations can be computed by matching two a priori defined landmarks (Bookstein coordinates),
few landmarks (resistant fit), or by least-squares optimization of residuals at all landmarks (Procrustes fit).
Tangent space — a linear space (plane) that is tangent to Kendall's space at a point corresponding to the
shape of a reference configuration. If variation in shape is small then Euclidean distances in the tangent space can be
used to approximate Procrustes distances in the Kendall's space. Since the tangent space is linear, it is possible to
apply conventional statistical methods to study variation in shape.
Uniform shape component — a part of the difference in shape between a set of configurations that can be
modeled by an affine transformation. Together with the partial warps, the uniform component supplies an
orthonormal basis for shape space.
LITERATURE CITED
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959-970.
Adams D.C., Rosenberg M.C. Partial warps, ontogeny, and phylogeny: a comment on Zelditch and Fink (1995) //
Sys. Biol. 1998. V. 47. № 1. P. 168-173.
Becerra J.M., Bella E., Garcia-Valdecasas A. Building your own machine image system for morphometric analysis: a
user point of view // Eds Marcus L.F., Bello E., Garcia-Valdesas A. Contributions to morphometrics.
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Table 1
Numerical estimates (loadings, explaned variance) of 1st and 2d relative warps (RW1, RW2) calculated for axial
skull and mandible of muroid rodents (after Pavlinov, 2000b, modified)
Loadings
Landmark number
Axial skull
Mandible
RW1
RW2
RW1
RW2
x
y
x
y
x
y
x
y
1
16.1
-28.5
27.4
-27.8
-19.2
0.2
4.0
-1.2
2
-10.9
-0.6
-8.0
-6.8
1.1
2.3
-15.8
5.4
3
-6.5
11.8
-19.0
11.5
23.0
4.2
-24.9
1.2
4
-18.7
6.1
-9.4
-13.0
-33.5
3.2
-0.2
12.2
5
-16.7
-8.0
3.2
-1.5
-12.0
36. 2
24.9
20.4
6
-25.8
-8.9
-8.2
-8.6
8.8
-41.7
-38.8
-28.9
7
11.9
-2.7
-0.3
-1.7
7.3
-10.9
1.9
-1.7
8
43.7
-3.9
47.2
28.0
0.2
1.2
25.6
32.9
9
-9.7
49.3
-62.5
58.3
4.3
8.0
-29.9
-28.6
10
47.4
-3.8
17.9
-29.9
-26.6
1.8
31.8
13.9
11
128.9
115.2
-41.3
-4.5
39.3
1.8
-21.4
-11.6
12
48.4
-55.3
0.6
-16.5
0.4
-1.6
-18.7
5.6
13
21.9
-6.5
-41.1
-21.9
6.5
-5.1
61.5
-19.7
14
-64.4
37.0
192.5
121.9
15
30.0
-32.8
-144.6
-104.1
16
-7.5
18.0
-7.3
39.5
17
107.2
-21.4
-57.7
-117.0
18
119.8
41.4
2.9
180.3
19
44.3
-92.5
128.3
-60.3
20
6.3
-74.1
0.6
53.3
21
25.9
27.0
-2.3
-9.5
Explained
Uni+ 43.23
12.15
32.88
21.25
variance,
Uni–
46.07
12.57
32.61
22.58
%
Comment. Landmark ns see on Fig. 6. Uni+ — uniform included, Uni– — uniform excluded.