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

Y. Lipman, J. Puente, and I. Daubechies: “Conformal Wasserstein distance, II: Computational aspects and extensions,” Math. Comp. 82 : 281 (2013), pp. 331–381. Part I was published in Adv. Math 227:3 (2011). MR 2983027 Zbl 1281.65034 ArXiv 1103.4681 article

This paper is a companion paper to [Yaron Lipman and Ingrid Daubechies, “Conformal Wasserstein distances: Comparing surfaces in polynomial time”, Adv. in Math. (ELS), 227 (2011), no. 3, 1047–1077, (2011)]. We provide numerical procedures and algorithms for computing the alignment of and distance between two disk-type surfaces. We provide a convergence analysis of the discrete approximation to the arising mass-transportation problems. We furthermore generalize the framework to support sphere-type surfaces, and prove a result connecting this distance to local geodesic distortion. Finally, we perform numerical experiments on several surface datasets and compare them to state-of-the-art methods.

Yaron Lipman	Related
Jesus Puente	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

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