Celebratio Mathematica — Daubechies

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S. Shan, S. Z. Kovalsky, J. M. Winchester, D. M. Boyer, and I. Daubechies: “ariaDNE: A robustly implemented algorithm for Dirichlet energy of the normal,” Methods. Ecol. Evol. 10 : 4 (April 2019), pp. 541–552. ArXiv 1901.06318 article

Shape characterizers are metrics that quantify aspects of the overall geometry of a 3D digital surface. When computed for biological objects, the values of a shape characterizer are largely independent of homology interpretations and often contain a strong ecological and functional signal. Thus shape characterizers are useful for understanding evolutionary processes. Dirichlet Normal Energy (DNE) is a widely used shape characterizer in morphological studies.
Recent studies found that DNE is sensitive to various procedures for preparing 3D mesh from raw scan data, raising concerns regarding comparability and objectivity when utilizing DNE in morphological research. We provide a robustly implemented algorithm for computing the Dirichlet energy of the normal (ariaDNE) on 3D meshes.
We show through simulation that the effects of preparation-related mesh surface attributes such as triangle count, mesh representation, noise, smoothing and boundary triangles are much more limited on ariaDNE than DNE. Furthermore, ariaDNE retains the potential of DNE for biological studies, illustrated by its effectiveness in differentiating species by dietary preferences.
Use of ariaDNE can dramatically enhance assessment of ecological aspects of morphological variation by its stability under different 3D model acquisition methods and preparation procedure. Towards this goal, we provide scripts for computing ariaDNE and ariaDNE values for specimens used in previously published DNE analyses.