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3D Surface Analysis:
POLLY & MACLEOD

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Abstract
Introduction
Material and Methods
Results and Discussion
Conclusions
Acknowledgements
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MATERIALS AND METHODS

Bones

The dorsal side of the calcaneum of 10 living species of Carnivora were scanned in three dimensions (see below, Table 1). These same scans were used in a previous study on the evolution of locomotor morphology in living Carnivora (Polly 2008).

Each bone was assigned a locomotor type following Van Valkenburgh (1985) and Taylor (1976). Terrestrial: animals that spend most of their time on the ground (e.g., dogs and hyenas). Scansorial: animals that spend considerable time on the ground, but are also good climbers (e.g., most felids). Arboreal: animals that spend most of their time in trees (e.g., olingos, red pandas). Natatorial: animals that spend time in both the water and on land (e.g., otters).

Stance, or the position of the heel during normal locomotion (Clevedon Brown and Yalden 1973; Gambaryan 1974; Hildebrand 1980), was recorded for each species using the following categories. Plantigrade: animals that walk with their heels touching the ground (e.g., red pandas). Semidigitigrade: animals that often keep their heels elevated during locomotion (e.g., many mustelids). Digitigrade: animals always have their heels elevated during normal locomotion, using the metatarsus as an additional limb segment (e.g., dogs, felids). The number of toes on the hind foot was also recorded as four or five.

Four fossil calcanea were also scanned in dorsal view (Table 2). These taxa were characterized with the same categories using the same criteria as the living taxa (b). These characterizations were used to assess the accuracy with which eigensurface could be used to assign characterizations using the modern calcanea (see below).

Scans

Calcanea from the living species were scanned at Queen Mary, University of London, in 2005 using a Roland Picza Pix-4 3D pin scanner. This scanner drops a pin onto the surface of the bone and records the three-dimensional x,y,z coordinates of that surface point. The scanner moves across the surface of the object dropping the pin in a grid pattern. Scans of the fossil calcanea were made at The Natural History Museum, London, in 2006 using a Konica-Minolta VIVID 910 laser scanner. This scanner uses reflections of laser-beam light to determine the distance between the surface of a specimen and the scanner. By scanning the laser beam across the field of view, a 3D map of the specimen's surfaces is recorded as a set of x,y,z coordinate points. This point cloud can then be operated on using specialized 3D imaging software (e.g., used to construct a virtual facsimile of the specimen, interpolated to optimize representation of the surface, edited to fill holes or eliminate scanning artefacts, resampled to reduce redundancy). All scans were exported as ASCII x,y,z point clouds using the commercial software bundled with the respective scanners.

Eigensurface Analysis

Eigensurface is the analysis of a standardized set of x,y,z coordinate points representing the nodes of a grid. These points represent an interpolated subsample of the point cloud of points from a three-dimensional scan of an object's surface. First the x,y,z coordinate datasets representing the processed object scans were rotated to a topologically 'homologous' orientation and then a gridding algorithm was applied. All scans were translated, scaled, and aligned using Procrustes (GLS) superimposition (Rohlf 1990) based on five landmark points taken from among those represented by the scan surface (Figure 3.1). The Procrustes fit was optimized using only the five landmarks, but the entire surface scan was carried with those points as they were superimposed on the sample mean shape. This operation transforms the raw x,y,z coordinates by removing differences in size, position, and rotation. The procedure for orienting the scan data was nearly identical to the one described by Wiley et al. (2005), but those authors used the scan data for illustrative purposes rather than further analysis.

Next an analytical surface grid was extracted from the Procrustes-oriented points. This grid was created by 'sectioning' each scan along its first principal component (the major axis of the object) into a fixed number of rows (Figure 3.2). Each row was then divided along the second principal component to create square sections, each of which contains one or more surface points (the exact number will depend on the scanning density and the number of sections). This procedure was repeated for each of the scans. Depending on the shape of the original object, the same row may have more sections on one object than on another. A standardized number of sections was determined by taking the sample median number in each row for the entire sample of objects. (See MacLeod 2008 for an alternative eigensurface gridding procedure.) The standardized sections were then applied to all the specimens. The analytical surface grids were then created by placing one 3D point in each section using the median x,y,z coordinate of the original scan points in that section. For the fossil carnivore analysis each calcaneum was gridded into 50 rows along the first principal component, which yielded a total of 2,190 nodal grid points.

All points in the analytical surface grids were then superimposed using the Procrustes (GLS) method to remove any residual differences in size, translation, and rotation. The rest of the procedure is identical to standard methods for comparing geometric landmark points (e.g., Rohlf 1993; Dryden and Mardia 1998; interested readers should refer to these works for equations and algorithms to carry out the following steps). The mean surface shape, or consensus, was subtracted from the Procrustes superimposed data to produce Procrustes residuals. A principal components analysis (PCA) was carried out on the covariance matrix of these residuals to decompose this shape similarity matrix into its respective eigenvalues and eigenvectors. We used singular value decomposition (SVD) to calculate the PC eigenvalues and eigenvectors because standard eigen-decomposition algorithms cannot be used on matrices of reduced rank, which result from the loss of dimensionality through Procrustes superimposition. To make the computation less intensive, a 'Q-mode' covariance matrix of the objects was calculated from the transpose of the Procrustes residuals to produce matrices of object eigenvectors and eigenvalues, U and V, respectively. Eigenvectors for the variables were calculated as U.R, where R is the matrix of Procrustes residuals. Principal component scores for the objects were calculated by scaling the transpose of U by the square root of V. The PC object scores were used as shape coordinates for subsequent analysis, and the eigenvectors for the variables defined the coordinate system for the shape space. PC scores were used as shape coordinates because their number is equal to the real degrees of freedom in the data set and because each PC is mathematically orthogonal (there is no correlation between scores on one PC and another), which simplifies statistical analysis (Rohlf 1993; Dryden and Mardia 1998).

Three-dimensional shape models were constructed for the first three PCs to illustrate the shape variation described by each. Models were constructed using the standard geometric morphometric procedure of multiplying the point in shape space to be modelled by the eigenvector associated with that PC and adding the result to the mean shape (Rohlf 1993). The same procedure was used for 2D eigenshape results (see MacLeod (1999) and 3D eigensurface results (see Polly 2008).

The PC score shape coordinates are quantitative representations of the variance in shape of the calcaneum surface; collectively the variance in the PC scores and the Euclidean distances between specimens in PC space preserve the original variance and distances between the gridded scans. All PC axes were retained for evaluating shape similarity. All transformations were performed in Mathematica 5.0.

Functional Analysis

The average calcaneum shape was calculated for each locomotor mode, stance type, and digit number using only the scores of the living species. The fossil calcanea were categorized in each of these three categories by finding which of the averages it was closest to. Closeness of fit was measured as the Procrustes distance, which is the square root of the sum-of-squared distances between the superimposed landmarks or between the vectors of PC scores for two objects and is a Euclidean measure of distance similar to those used in many cluster analyses.

 

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3D Surface Analysis
Plain-Language & Multilingual  Abstracts | Abstract | Introduction | Materials and Methods
Results and Discussion | Conclusions | Acknowledgements | References
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