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

Brassey CA, Gardiner JD - R Soc Open Sci (2015)

Bottom Line: Generating reliable estimates for body mass is therefore a necessary step in many palaeontological studies.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).

View Article: PubMed Central - PubMed

Affiliation: Faculty of Life Sciences , University of Manchester , Manchester M13 9PL, UK.

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.

No MeSH data available.


Related in: MedlinePlus

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.
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RSOS150302F2: 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.

Mentions: When regressing log αvol against log body mass for the densest point cloud dataset (animals represents by 500 000 points), the greatest r2-value of 0.975 occurs at a k-value of 0.427 (figure 2, red line). This refinement coefficient corresponds to an α-shape in which the left and right appendages within the forelimbs and within the hindlimbs are conjoined, but the forelimbs and hindlimbs remain distinct from one another (figure 2c). Additionally, the ribcage is entirely enclosed within the α-shape. The relationship between body mass and α-shape volume for the original dataset when k=0.427 is illustrated in figure 3. The constants of the equation describing this relationship can be found in table 2, as well as the results for the unsegmented convex hull equation and the segmented convex hull from Sellers et al. [26]. A second smaller peak occurs at a k-value of 0.1496 and an r2 of 0.9682, with a corresponding α-shape in which the fit passes within the ribcage and all limbs are distinct (figure 2b). When k exceeds 10, correlation coefficients plateau at values between 0.91 and 0.92 and the α-shapes tend towards a convex hull (figure 2d). Once k-values drop below 0.1, the r2-values of the regression tend to decrease rapidly as the alpha shapes start passing inside the bones and become a discrete set of disconnected volumes rather than one complete ‘shrink wrap’ of the skeleton. Calculated α-shape volumes (αvol) for the modern dataset (500 000 points per skeleton) are given in table 1 for the optimal k-value of 0.427, alongside convex hull volumes for the unsegmented skeletons (Cvol) considered here and the segmented skeletons published elsewhere (Cvol(sub)) [26].Figure 2.


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

Brassey CA, Gardiner JD - R Soc Open Sci (2015)

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.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4555864&req=5

RSOS150302F2: 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.
Mentions: When regressing log αvol against log body mass for the densest point cloud dataset (animals represents by 500 000 points), the greatest r2-value of 0.975 occurs at a k-value of 0.427 (figure 2, red line). This refinement coefficient corresponds to an α-shape in which the left and right appendages within the forelimbs and within the hindlimbs are conjoined, but the forelimbs and hindlimbs remain distinct from one another (figure 2c). Additionally, the ribcage is entirely enclosed within the α-shape. The relationship between body mass and α-shape volume for the original dataset when k=0.427 is illustrated in figure 3. The constants of the equation describing this relationship can be found in table 2, as well as the results for the unsegmented convex hull equation and the segmented convex hull from Sellers et al. [26]. A second smaller peak occurs at a k-value of 0.1496 and an r2 of 0.9682, with a corresponding α-shape in which the fit passes within the ribcage and all limbs are distinct (figure 2b). When k exceeds 10, correlation coefficients plateau at values between 0.91 and 0.92 and the α-shapes tend towards a convex hull (figure 2d). Once k-values drop below 0.1, the r2-values of the regression tend to decrease rapidly as the alpha shapes start passing inside the bones and become a discrete set of disconnected volumes rather than one complete ‘shrink wrap’ of the skeleton. Calculated α-shape volumes (αvol) for the modern dataset (500 000 points per skeleton) are given in table 1 for the optimal k-value of 0.427, alongside convex hull volumes for the unsegmented skeletons (Cvol) considered here and the segmented skeletons published elsewhere (Cvol(sub)) [26].Figure 2.

Bottom Line: Generating reliable estimates for body mass is therefore a necessary step in many palaeontological studies.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).

View Article: PubMed Central - PubMed

Affiliation: Faculty of Life Sciences , University of Manchester , Manchester M13 9PL, UK.

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.

No MeSH data available.


Related in: MedlinePlus