Celebratio Mathematica

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

D. M. Boyer, J. Puente, J. T. Gladman, C. Glynn, S. Mukherjee, G. S. Yapuncich, and I. Daubechies: “A new fully automated approach for aligning and comparing shapes,” Anat. Rec. 298 : 1 (January 2015), pp. 249–276. article

Three-dimensional geometric morphometric (3DGM) methods for placing landmarks on digitized bones have become increasingly sophisticated in the last 20 years, including greater degrees of automation. One aspect shared by all 3DGM methods is that the researcher must designate initial landmarks. Thus, researcher interpretations of homology and correspondence are required for and influence representations of shape. We present an algorithm allowing fully automatic placement of correspondence points on samples of 3D digital models representing bones of different individuals/species, which can then be input into standard 3DGM software and analyzed with dimension reduction techniques. We test this algorithm against several samples, primarily a dataset of 106 primate calcanei represented by 1,024 correspondence points per bone. Results of our automated analysis of these samples are compared to a published study using a traditional 3DGM approach with 27 landmarks on each bone. Data were analyzed with morphologika2.5 and PAST. Our analyses returned strong correlations between principal component scores, similar variance partitioning among components, and similarities between the shape spaces generated by the automatic and traditional methods. While cluster analyses of both automatically generated and traditional datasets produced broadly similar patterns, there were also differences. Overall these results suggest to us that automatic quantifications can lead to shape spaces that are as meaningful as those based on observer landmarks, thereby presenting potential to save time in data collection, increase completeness of morphological quantification, eliminate observer error, and allow comparisons of shape diversity between different types of bones. We provide an R package for implementing this analysis.

Jesus Puente	Related
Gabriel S. Yapuncich	Related
Douglas Martin Boyer	Related
Sayan Mukherjee	Related
Justin T. Gladman	Related
Chris Glynn	Related

T. Gao, S. Z. Kovalsky, D. M. Boyer, and I. Daubechies: “Gaussian process landmarking for three-dimensional geometric morphometrics,” Siam J. Math. Data Sci. 1 : 1 (2019), pp. 237–267. MR 3949708 ArXiv 1807.11887 article

We demonstrate applications of the Gaussian process-based landmarking algorithm proposed in [T. Gao, S. Z. Kovalsky, and I. Daubechies, SIAM J. Math. Data Sci., 1 (2019), pp.208–236] to geometric morphometrics, a branch of evolutionary biology centered at the analysis and comparisons of anatomical shapes, and compares the automatically sampled landmarks with the “ground truth” landmarks manually placed by evolutionary anthropologists; the results suggest that Gaussian process landmarks perform equally well or better, in terms of both spatial coverage and downstream statistical analysis. We provide a detailed exposition of numerical procedures and feature filtering algorithms for computing high-quality and semantically meaningful diffeomorphisms between disk-type anatomical surfaces.

Tingran Gao	Related
Shahar Z. Kovalsky	Related
Douglas Martin Boyer	Related

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.

Julie M. Winchester	Related
Shan Shan	Related
Shahar Z. Kovalsky	Related
Douglas Martin Boyer	Related

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