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. 2015 Aug 19;2(8):150302.
doi: 10.1098/rsos.150302. eCollection 2015 Aug.

An advanced shape-fitting algorithm applied to quadrupedal mammals: improving volumetric mass estimates

Charlotte A Brassey  1 , James D Gardiner  2
Affiliations

An advanced shape-fitting algorithm applied to quadrupedal mammals: improving volumetric mass estimates

Charlotte A Brassey et al. R Soc Open Sci. .

Abstract

Body mass is a fundamental physical property of an individual and has enormous bearing upon ecology and physiology. Generating reliable estimates for body mass is therefore a necessary step in many palaeontological studies. Whilst early reconstructions of mass in extinct species relied upon isolated skeletal elements, volumetric techniques are increasingly applied to fossils when skeletal completeness allows. We apply a new 'alpha shapes' (α-shapes) algorithm to volumetric mass estimation in quadrupedal mammals. α-shapes are defined by: (i) the underlying skeletal structure to which they are fitted; and (ii) the value α, determining the refinement of fit. For a given skeleton, a range of α-shapes may be fitted around the individual, spanning from very coarse to very fine. We fit α-shapes to three-dimensional models of extant mammals and calculate volumes, which are regressed against mass to generate predictive equations. Our optimal model is characterized by a high correlation coefficient and mean square error (r (2)=0.975, m.s.e.=0.025). When applied to the woolly mammoth (Mammuthus primigenius) and giant ground sloth (Megatherium americanum), we reconstruct masses of 3635 and 3706 kg, respectively. We consider α-shapes an improvement upon previous techniques as resulting volumes are less sensitive to uncertainties in skeletal reconstructions, and do not require manual separation of body segments from skeletons.

Keywords: Mammuthus; Megatherium; body mass; fossil; volumetric; α-shapes.

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Figures

Figure 1.
Figure 1.
A ‘family’ of α-shapes for a given two-dimensional dataset. (a) The special case ‘convex hull’ α-shape calculated when α is infinite; (b−d) α-shapes calculated as the radius α decreases relative to the point spacing of the dataset; (e) original ‘solid’ geometric shape upon which point cloud (a−d) was derived. The suite of α-shapes ranges from ‘crude’ to ‘fine’ representations of the two-dimensional dataset.
Figure 2.
Figure 2.
Variation in r2 against the refinement coefficient (k) for a series of downsampled skeleton datasets. r2 values derived from the natural-log transformed OLS regression of body mass (kg) against αvol (m3). Grey shaded area represents narrow range in which optimal k-values (as defined by highest r2) occur for all datasets. Inset: four α-shapes fitted around the skeleton of Camelus comprising 500 000 points, illustrating increasing refinement coefficients from A to D. Arrows indicate position of each α-shape on the curve of r2 against k.
Figure 3.
Figure 3.
The relationship between α-shape volume and body mass for articulated skeletons comprising 500 000 points, for the ‘optimal’ k-value of 0.427. Extant taxa, filled circles; Ma. primigenius, open square; Me. americanum, open diamond.
Figure 4.
Figure 4.
Change in volume relative to the original data (500 000 points per skeleton) with downsampling for three example skeletons. Coloured bands represent the standard deviation (10 repeats) either side of the averages (black lines).
Figure 5.
Figure 5.
(a) Mammoth (Ma. primigenius) and (b) sloth (Me. americanum) α-shapes illustrating the fit achieved when refinement coefficient k=0.4274 (number of points=500 000). At this refinement coefficient, α-shapes join the right and left appendages within the fore- and hindlimbs, while leaving the forelimbs separate from the hind limbs. The skull of Me. americanum is also joined to the fore limbs.
Figure 6.
Figure 6.
Camel ribcages at two refinement coefficients: (a) k=0.157 and (b) k= 0.427. These correspond to the two peaks in r2 for the regression fits (figure 2b,c). At lower values of k, the α-shape passes internal to the ribcage and has an overall tight fit. At the higher coefficient (which yields the best-performing predictive equation), the α-shape does not pass within the ribcage and has a coarser fit to the point cloud.
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