Celebratio Mathematica — Daubechies

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D. M. Boyer, Y. Lipman, E. S. Clair, J. Puente, B. A. Patel, T. Funkhouser, J. Jernvall, and I. Daubechies: “Algorithms to automatically quantify the geometric similarity of anatomical surfaces,” Proc. Natl. Acad. Sci. USA 108 : 45 (2011), pp. 18221–18226. ArXiv 1110.3649 article

We describe approaches for distances between pairs of two-dimensional surfaces (embedded in three-dimensional space) that use local structures and global information contained in interstructure geometric relationships. We present algorithms to automatically determine these distances as well as geometric correspondences. This approach is motivated by the aspiration of students of natural science to understand the continuity of form that unites the diversity of life. At present, scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points (landmarks). Identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists. This necessity renders these studies inaccessible to nonmorphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns. Unlike other algorithms presented for morphological correspondences, our approach does not require any preliminary marking of special features or landmarks by the user. It also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster, making pairwise comparisons feasible for significantly larger numbers of digitized surfaces. We illustrate our approach using three datasets representing teeth and different bones of primates and humans, and show that it leads to highly accurate results.

Biren A. Patel	Related
Jukka Jernvall	Related
Elizabeth M. St. Clair	Related
Yaron Lipman	Related
Jesus Puente	Related
Thomas Allen Funkhouser	Related
Douglas Martin Boyer	Related

J. M. Bunn, D. M. Boyer, Y. Lipman, E. M. S. Clair, J. Jernvall, and I. Daubechies: “Comparing Dirichlet normal surface energy of tooth crowns, a new technique of molar shape quantification for dietary inference, with previous methods in isolation and in combination,” Am. J. Phys. Anthropol. 145 : 2 (June 2011), pp. 247–261. article

Inferred dietary preference is a major component of paleoecologies of extinct primates. Molar occlusal shape correlates with diet in living mammals, so teeth are a potentially useful structure from which to reconstruct diet in extinct taxa. We assess the efficacy of Dirichlet normal energy (DNE) calculated for molar tooth surfaces for reflecting diet. We evaluate DNE, which uses changes in normal vectors to characterize curvature, by directly comparing this metric to metrics previously used in dietary inference. We also test whether combining methods improves diet reconstructions. The study sample consisted of 146 lower (mandibular) second molars belonging to 24 euarchontan taxa. Five shape quantification metrics were calculated on each molar: DNE, shearing quotient, shearing ratio, relief index, and orientation patch count rotated (OPCR). Statistical analyses were completed for each variable to assess effects of taxon and diet. Discriminant function analysis was used to assess ability of combinations of variables to predict diet. Values differ significantly by diets for all variables, although shearing ratios and OPCR do not distinguish statistically between insectivores and folivores or omnivores and frugivores. Combined analyses were much more effective at predicting diet than any metric alone. Alone, relief index and DNE were most effective at predicting diet. OPCR was the least effective alone but is still valuable as the only quantitative measure of surface complexity. Of all methods considered, DNE was the least methodologically sensitive, and its effectiveness suggests it will be a valuable tool for dietary reconstruction.

Jukka Jernvall	Related
Elizabeth M. St. Clair	Related
Yaron Lipman	Related
Douglas Martin Boyer	Related
Jonathan M. Bunn	Related