Alpha shapes: Determining 3D shape complexity across morphologically diverse structures

Abstract

Background: Following recent advances in bioimaging, high-resolution 3D models of biological structures are now generated rapidly and at low-cost. To use this data to address evolutionary and ecological questions, an array of tools has been developed to conduct shape analysis and quantify topographic complexity. Here we focus particularly on shape techniques applied to irregular-shaped objects lacking clear homologous landmarks, and propose a new 'alpha-shapes' method for quantifying 3D shape complexity. Methods: We apply alpha-shapes to quantify shape complexity in the mammalian baculum as an example of a morphologically disparate structure. Micro- computed-tomography (μCT) scans of bacula were conducted. Bacula were binarised and converted into point clouds. Following application of a scaling factor to account for absolute size differences, a suite of alpha-shapes was fitted per specimen. An alpha shape is formed from a subcomplex of the Delaunay triangulation of a given set of points, and ranges in refinement from a very coarse mesh (approximating convex hulls) to a very fine fit. 'Optimal' alpha was defined as the refinement necessary in order for alpha-shape volume to equal CT voxel volume, and was taken as a metric of overall 'complexity'. Results: Our results show that alpha-shapes can be used to quantify interspecific variation in shape 'complexity' within biological structures of disparate geometry. The 'stepped' nature of alpha curves is informative with regards to the contribution of specific morphological features to overall 'complexity'. Alpha-shapes agrees with other measures of complexity (dissection index, Dirichlet normal energy) in identifying ursid bacula as having low shape complexity. However, alpha-shapes estimates mustelid bacula as being most complex, contrasting with other shape metrics. 3D fractal dimension is identified as an inappropriate metric of complexity when applied to bacula. Conclusions: Alpha-shapes is used to calculate 'optimal' alpha refinement as a proxy for shape 'complexity' without identifying landmarks. The implementation of alpha-shapes is straightforward, and is automated to process large datasets quickly. We interpret alpha-shapes as being particularly sensitive to concavities in surface topology, potentially distinguishing it from other shape complexity metrics. Beyond genital shape, the alpha-shapes technique holds considerable promise for new applications across evolutionary, ecological and palaeoecological disciplines.