La Morfometria GeometricaLa Morfometria Geometrica

Dr Corrado CostaDr Corrado Costa

Università di Roma “La Sapienza” – Zoogeografia – 30 Aprile 2014

FORM = SIZE + SHAPE

DEFINITION: In morphometrics, we represent the form of an object by a point in a

space of form variables, which are measurements of a geometric object that are

unchanged by translations and rotations. If you allow for reflections, forms stand for all

the figures that have all the same interlandmark distances. A form is usually represented

by one of its figures at some specified location and in some specified orientation. When

represented in this way, location and orientation are said to have been "removed."

DEFINITION: Shape is the geometric properties of a configuration of points that are DEFINITION: Shape is the geometric properties of a configuration of points that are

invariant to changes in translation, rotation, and scale. In morphometrics, we represent

the shape of an object by a point in a space of shape variables, which are measurements

of a geometric object that are unchanged under similarity transformations. For data that

are configurations of landmarks, there is also a representation of shapes per se, without

any nuisance parameters (position, rotation, scale), as single points in a space,Kendall’s

shape space, with a geometry given by Procrustes distance. Other sorts of shapes (e.g.,

those of outlines, surfaces, or functions) correspond to quite different statistical space.

FORM = SIZE + SHAPE

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DEFINITION: A specific point on a biological form or image of a form located

according to some rule. Landmarks with the same name, homologues in the

purely semantic sense, are presumed to correspond in some sensible way over

the forms of a data set

Landmarks

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Type I landmark - A mathematical point whose claimed homology from case to case is supported

by the strongest evidence, such as a local pattern of juxtaposition of tissue types or a small patch

of some unusual histology.

Type II landmark - A mathematical point whose claimed homology from case to case is

supported only by geometric, not histological, evidence: for instance, the sharpest curvature of a

tooth.

Type III landmark - A landmark having at least one deficient coordinate, for instance, either end

of a longest diameter, or the bottom of a concavity. Type III landmarks characterize more than one

region of the form. The multivariate machinery of geometric morphometrics permits them to be

Landmarks

region of the form. The multivariate machinery of geometric morphometrics permits them to be

treated as landmark points in some analyses, but the deficiency they embody must be kept in mind

in the course of any geometric or biological interpretation

Landmarks

DEFINITION: Procrustes methods - A term for least-squares methods for estimating nuisance

parameters of the Euclidean similarity transformations. The adjective "Procrustes" refers to the

Greek giant who would stretch or shorten victims to fit a bed and was first used in the context of

superimposition methods by Hurley and Cattell, 1962, The Procrustes program: producing a direct

rotation to test an hypothesized factor structure, Behav. Sci. 7:258-262. Modern workers have often

cited Mosier (1939), a psychometrician, as the earliest known developer of these methods.

However, Cole (1996) reports that Franz Boas in 1905 suggested the "method of least differences"

(ordinary Procrustes analysis) as a means of comparing homologous points to address obvious

problems with the standard point-line registrations (Boas, 1905). Cole further points out that one of

Boas' students extended the method to the construction of mean configurations from the

superimposition of multiple specimens using either the standard registrations of Boas' method

Procrustes Methods

superimposition of multiple specimens using either the standard registrations of Boas' method

(Phelps, 1932). The latter being essentially a Generalized Procrustes Analysis.

DEFINITION: least-squares estimates - Parameter estimates that minimize the sum of squared

differences between observed and predicted sample values.

References:

Cole, T. M. 1996. Historical note: early anthropological contributions to "geometric morphometrics." Amer. J. Phys. Anthropol.

101:291-296.

Boas, F. 1905. The horizontal plane of the skull and the general problem of the comparision of variable forms. Science, 21:862-

863.

Phelps, E. M. 1932. A critique of the principle of the horizontal plane of the skull. Amer. J. Phys. Anthropol., 17:71-98.

Mosier, 1939, Determining a simple structure when loadings for certain tests are known, Psychometrika 4:149-162.

Coordinates to Distances

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Centroid Size

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DEFINITION: Centroid size is the square root of the sum of squared distances from the

landmarks to the centroid of the landmarks. In the absence of allometry, it is the only size

measure that is uncorrelated with all shape variables.

Procrustes Distances

Procrustes distance between two aligned shapes

where X, Y have been scaled, translated, and oriented so as to minimize dxy

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Consensus Configuration

DEFINITION: Consensus Configuration - A single set of landmarks intended to represent

the central tendency of an observed sample for the production of superimpositions, of a

weight matrix, or some other morphometric purpose. Often a consensus configuration is

computed to optimize some measure of fit to the full sample: in particular, the Procrustes

mean shape is computed to minimize the sum of squared Procrustes distances from the the

consensus landmarks to those of the sample.

Figura Consenso

Example

Bookstein Shape Coordinates

Translation

Rotation

Scaling

Shape Difference Model

( ) iiiii HExx τρ 10 ++=Describes differences in co- ordinates as being due to scale, rotation, translation, and

actual shape difference, E i

Procrustes Distance: The square root of the sum of squared

elements of E i (approximately)

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Thin-Plate Spline

After the correct operations of superimposition each object will correspond to an object in a space that correspond to

the Kendall’s shape space; this space in non Euclidean (non metric measurement), the distances are named

Procrustes distances.

DEFINITION: Thin-plate spline - In continuum mechanics, a thin-plate spline models the form taken by a metal plate

that is constrained at some combination of points and lines and otherwise free to adopt the form that minimizes bending

energy. (The extent of bending is taken as so small that elastic energy - stretches and shrinks in the plane of the original

plate - can be neglected.) One particular version of this problem - an infinite, uniform plate constrained only by

displacements at a set of discrete points - can be solved algebraically by a simple matrix inversion. In that form, the

technique is a convenient general approach to the problem of surface interpolation for computer graphics and

computer-aided design. In morphometrics, the same interpolation (applied once for each Cartesian coordinate) provides

a unique solution to the construction of D'Arcy Thompson-type deformation grids for data in the form of two landmark

configurations. configurations.

Thin-Plate Spline EquationBeing a non Euclidean Space, is not possible to apply the traditional Multivariate Statistics methods.

This space, through the Thin-Plate Spline interpolation function, is transformed in an Euclidean (metric)

space. This interpolation function has many advantages like the possibility to visualize the deformation

through the splines.

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After the application of the function, every individual will have a new set variables in a matrix named

weight matrix.

The method decompose the morphologica variability in a Non-uniform Component and in an Uniform

Component. Every component could ve visualized through the splines and statistically analyzed. This

analysis applied to morphometric data is named Relative Warp Analysis.

Thin-Plate Spline

U2

After the application of the function, every individual will have a new set variables in a matrix

named weight matrix.

The method decompose the morphologica variability in a Non-uniform Component and in an

Uniform Component. Every component could ve visualized through the splines and statistically

analyzed. This analysis applied to morphometric data is named Relative Warp Analysis.

Stretching

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Bending

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Warps

• Principal warps - eigenvectors of the bending energy matrix. Solely determined by the

reference configuration.

• Partial warps - sets of principal warps forming a set of basis vectors for shape tangent

space. Used to characterize individual specimens (partial warp scores).

• Relative warps - weighted PCA of partial warp scores