Geometric morphometrics
A survey in seven questions
Thin plate spline of a midsagittal profil with landmarks 
Q 1. What do we use geometric morphometrics for?
A 1. We need geometric morphometrics for answering questions about how parts of a body vary or how they respond to processes like growth, evolution, or injury, or about how we control these parts of a body (like by nutrition or surgery or animal husbandry) or how we react to parts of a body (like by calling a face beautiful or sexy). All these processes can be understood or the questions answered by talking about how points are "pushed around," and geometric morphometrics is the best way to talk about the patterns for what the points are doing during all these processes  and about the exceptions to these patterns, like birth defects. Geometric morphometrics combines rich data sets from modern medical imaging devices looking at organisms with strict rules for how to talk about the differences in the size and shape of the organisms being studied. It is very useful in every single branch of anthropology, and also in many other fields like cognitive psychology and medicine.
To talk about geometric morphometrics we need to introduce three
other words first  see the next two questions.
Q 2. What is morphometrics?
A 2. Morphometrics is a collection of
tools for treating information about the geometry of organisms by statistical methods  talking
about their variation and the uncertainty of conclusions we draw about
them.
Q 3. Oh, yes. What is statistics? what is geometry?
A 3. Statistics consists of methods
for learning from measurements under conditions of uncertainty (this
can mean uncertainty about measurements or uncertainty about the
theories that are supposed to explain the measurements).
Geometry is what you
studied in Gymnasium or in high school: the rules for making sense of
data about positions, distances, and angles in our ordinary
threedimensional world. You remember that some of geometry is about
ways of measuring distances or angles, while another part of geometry
is about the "theorems" that tell you how the measures have to come out
sometimes.
Q 4. OK, now what is geometric morphometrics?
A 4. GEOMETRIC MORPHOMETRICS is the statistical analysis of a
particular
kind of geometric
information about organisms: information about exactly where the parts
of the organism are with respect to each other. Geometric morphometric
tools combine morphometrics (geometry of organisms), computer science,
and modern engineering to focus on information from the actual
locations of points on or inside organisms. We need the engineering to
get this information from complicated threedimensional objects like
fossils or faces, and we need computer science to process the huge
amount of information that we get from a the surface or the inside of
more than one whole organism.
Q 5. What information does geometric morphometrics use?
A 5. In geometric morphometrics, information comes from
mathematical points (the tiniest possible locations) taken a few at a
time or else many at a time. The simplest kind of data is from just a
list of points, like "the tip of the nose and the center of the earhole
and the center of the forehead." More complicated data can come from
the outer surface of a form, an internal boundary surface between two
of its parts, or a curve (like a nerve fiber) that wanders inside the
organism, and this more complicated information can be studied by the
same methods that work for the lists of points.
Q 6. What is so special about geometric morphometrics?
A 6. There is a mathematical problem here that is much more
difficult than you might think  its solution was worked out only about
30 years ago: how to talk about the locations of all these points and
parts in a way that doesn't lead to arguments. One trick is not to talk
about position, actually, but only about the distances between
different positions of the same landmark point.
There is also a second trick: to study the shape of a form and its
size separately, and combine them only at the end. When we average a
set of forms, for instance, or explain them by age or genes, we are
actually averaging their size and their shape separately, or predicting
(explaining) their size and shape separately. At the end, we go back to
computer science for another important tool, the display of shape
differences as deformations. These animated graphics use a
surprising range of your own visual abilities to turn the statistical
findings back into patterns you can understand on the computer screen
or in the virtual space of a 3D display right in front of you.
Q 7. What are some other words I will run into?
A 7. Here are some more important terms in geometric morphometrics:
landmarks  the named points, curves, or surfaces whose
locations are tracked
Procrustes coordinates  numbers for the shape of a sample of sets of landmarks
Thinplate spline  a formula for turning the movement of a set of landmarks into a graphic that looks like a deformation
Shape regression  a statistical method for finding how a shape depends on factors that predict it (evolution) or that it predicts (attractiveness)
Allometry  the dependence of shape on size (not specific to
geometric morphometrics, but across all of biology)
Now go back and read the answer to question 1 again.
Thin plate splines 
Procrustes superimposition 
Semilandmarks 
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 ...

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
...

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.
