Visualisation was aided using wireframe and polygon models, as well as by warping a surface model. A wireframe consists of a few straight lines connecting landmarks in a manner that aids with relating the landmarks back to its biological representation. Figure 3.7 illustrates the wireframe configuration used in the analysis of the baboon crania. Although there have been suggestions of wireframe automation using Delaunay triangulation, it is not ideal as the
wireframe representation should relate to anatomical adjacency, where the linking of landmarks depends on the underlying, unrepresented morphology (Klingenberg, 2009).
The surface model was generated by laser surface scanning of an appropriate specimen. Selection of the ‘appropriate’ specimen was based on the summed scores of the first four principal components, weighted according to the percentage of the total variance accounted for by the respective component. This was used to represent the deviation of each specimen from the mean. A specimen close to the mean for each of the samples was selected and scanned using a NextEngine 3D HD desktop laser scanner (NextEngine, Inc.). The final surface model
representation (Figure 3.15, for the baboon sample) was constructed from the merging of five 360º scans, each composed of 12 independent scans. The landmarks digitized on the physical specimen (Table 3.1) were then located on the surface model using Landmark (ver. 3.6; Institute for Data Analysis and Visualization, University of California, Davis; Wiley et al., 2005) and
108 their coordinates exported as a text document. This particular specimen (Figure 3.15) happened to be a subadult female based on the morphology and size of the permanent canine, as well as the third permanent molars being the only unerupted permanent teeth. This specimen was warped to visualise shape change, but it must be noted that canine morphology, together with tooth number and size, did not play a role in morphological interpretations. Nevertheless, an extension or shortening of the alveolar process of the maxilla may reflect a gain or loss of teeth respectively and interpretations must take this into consideration.
Figure 3.15. Surface model of the specimen used as the basis for visualising shape changes. The
specimen was selected on the basis of its proximity to the mean configuration for the sample.
To depict a particular shape-change a target and reference configuration is chosen. When illustrating the shape changes associated with principal components, for example, the mean shape was selected as the reference configuration while the extreme positive or negative score
109 along the specific principal component would act as the target. Warping of the surface model vertices was performed according to the thin-plate spline procedure for three-dimensions (Bookstein, 1997; as in Mitteroecker et al., 2004). For each representation, the vertices of the surface model are initially warped to the mean shape and then to the target. The reason for the mean shape being the starting point is in order to quantify the shape changes on each
triangulation of the surface model. Each triangulation is thus coloured, on a continuous scale, according to percent surface area change between the mean shape and target shape. Colours chosen were blue, to represent reduced surface area, grey for very little or no change, and red for an increase in surface area. The degree to which colour change is associated with percentage surface area change is set as required and presented with each figure.
An introduction to thin plate splines would be incomplete without at least mentioning
Thompson’s (1917) pioneering work on the transformation of Cartesian coordinates between homologous skeletal elements (Figure 3.16). Unfortunately, it was only in the 1980’s when the mathematical application of this method on biological form was first realised (Bookstein, 1997), possibly as a direct result of computers on the field of mathematics and statistics. But
Thompson’s book was ahead of its time, and although much of our biological understanding and reasoning behind erecting transformation grids has changed (Bookstein, 1996), his publication influenced, if not initiated, much of the work that has brought morphometrics to the point it is now, including Huxley’s (1932) proposal of allometric growth (Thompson, 1961; see editor's notes, pg. 268). This notion of drawing grids to illustrate differences between objects was seen originally in the very early work of Albrecht Dürer and used to illustrate geometric variation of the proportions of the human head (as acknowledged in Thompson, 1961).
110 Figure 3.16. Cartesian transformation grids of Thompson (1917). He had used the baboon, on the far
right, to depict an extreme of the Cartesian transformation between the human and chimpanzee crania.
Bookstein (1996) established the mathematical procedures for estimating unknown points between sampled data points using principles of interpolation from the fields of physics and engineering. His work resulted in what is referred to as thin plate splines, a method of interpolating the position of a Cartesian grid according to the global and local shape changes between two configurations (Figure 3.17). I will not delve into the mathematics of mapping the Cartesian grid to the target, but I will emphasise that its application can be extended to three- dimensions and be used to map surface models from a reference configuration to a target. This makes logical sense when one considers that a surface model is a suite of points (vertices) in three-dimensional Cartesian space on which polygons are drawn. The thin plate spline
determines the new position of these points, depending on the relative positions of the landmarks of the reference configuration and the final position of the landmarks in the target configuration. For greater mathematical detail and discussion, see Bookstein (1997).
111 Figure 3.17. Thin plate spline transformation of the vertices of a Cartesian grid. The central and
lateral landmarks of the reference configuration on the left are shifted slightly down and up, respectively, to form the target on the right. Modelled from an example in Bookstein (1997) using Microsoft Excel 2007.