Celebratio Mathematica

Y. Lipman and I. Daubechies: Surface comparison with mass transportation. Preprint, December 2009. ArXiv 0912.3488 techreport

We use mass-transportation as a tool to compare surfaces (2-manifolds). In particular, we determine the “similarity” of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Möbius transformations. Our approach provides a constructive way of defining a metric in the abstract space of simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type); this metric can also be used to define meaningful intrinsic distances between pairs of “patches” in the two surfaces, which allows automatic alignment of the surfaces. We provide numerical experiments on “real-life” surfaces to demonstrate possible applications in natural sciences.

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

Y. Lipman and I. Daubechies: “Conformal Wasserstein distances: Comparing surfaces in polynomial time,” Adv. Math. 227 : 3 (2011), pp. 1047–1077. Part II was published in Math. Comp. 82:281 (2013). MR 2799600 Zbl 1217.53026 ArXiv 1103.4408 article

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

Y. Lipman, R. Al-Aifari, and I. Daubechies: “Continuous Procrustes distance between two surfaces,” Comm. Pure Appl. Math. 66 : 6 (2013), pp. 934–964. Full version of work presented first at Third Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment (in conjunction with CVPR’10), June 2010. MR 3043386 Zbl 1276.68157 ArXiv 1106.4588 article

The Procrustes distance is used to quantify the similarity or dissimilarity of (three-dimensional) shapes and extensively used in biological morphometrics. Typically each (normalized) shape is represented by \( N \) landmark points, chosen to be homologous, as far as possible, and the Procrustes distance is then computed as \[ \inf_R\sum_{j=1}^N \| Rx_j - x^{\prime}_j\|^2 ,\] where the minimization is over all euclidean transformations, and the correspondences \( x_j \leftrightarrow x^{\prime}_j \) are picked in an optimal way.

The discrete Procrustes distance has the drawback that each shape is represented by only a finite number of points, which may not capture all the geometric aspects of interest; a need has been expressed for alternatives that are still easy to compute. We propose in this paper the concept of continuous Procrustes distance and prove that it provides a true metric for two-dimensional surfaces embedded in three dimensions. We also propose an efficient algorithm to calculate approximations to this new distance.

Rima Alaifari	Related
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