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Studying Morphological Variation: Exploring the Shape Space in Evolutionary Anthropology

Shira Faigenbaum-Golovin, Ingrid Daubechies

TL;DR

This paper summarizes the work of many team members other than the authors; in this paper that unites (for the first time) all their results in one joint context, space restrictions prevent a full description of the mathematical details.

Abstract

We present results of a long-term team collaboration of mathematicians and biologists. We focus on building a mathematical framework for the shape space constituted by a collection of homologous bones or teeth from many species. The biological application is to quantitative morphological understanding of the evolutionary history of primates in particular, and mammals more generally. Similar to the practice of biologists, we leverage the power of the whole collection for results that are more robust than can be obtained by only pairwise comparisons, using tools from differential geometry and machine learning. This paper concentrates on the mathematical framework. We review methods for comparing anatomical surfaces, discuss the problem of registration and alignment, and address the computation of different distances. Next, we cover broader questions related to cross-dataset landmark selection, shape segmentation, and shape classification analysis. This paper summarizes the work of many team members other than the authors; in this paper that unites (for the first time) all their results in one joint context, space restrictions prevent a full description of the mathematical details, which are thoroughly covered in the original articles. Although our application is to the study of anatomical surfaces, we believe our approach has much wider applicability.

Studying Morphological Variation: Exploring the Shape Space in Evolutionary Anthropology

TL;DR

This paper summarizes the work of many team members other than the authors; in this paper that unites (for the first time) all their results in one joint context, space restrictions prevent a full description of the mathematical details.

Abstract

We present results of a long-term team collaboration of mathematicians and biologists. We focus on building a mathematical framework for the shape space constituted by a collection of homologous bones or teeth from many species. The biological application is to quantitative morphological understanding of the evolutionary history of primates in particular, and mammals more generally. Similar to the practice of biologists, we leverage the power of the whole collection for results that are more robust than can be obtained by only pairwise comparisons, using tools from differential geometry and machine learning. This paper concentrates on the mathematical framework. We review methods for comparing anatomical surfaces, discuss the problem of registration and alignment, and address the computation of different distances. Next, we cover broader questions related to cross-dataset landmark selection, shape segmentation, and shape classification analysis. This paper summarizes the work of many team members other than the authors; in this paper that unites (for the first time) all their results in one joint context, space restrictions prevent a full description of the mathematical details, which are thoroughly covered in the original articles. Although our application is to the study of anatomical surfaces, we believe our approach has much wider applicability.

Paper Structure

This paper contains 14 sections, 12 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Data Acquisition and Pre-Processing
  3. From Scattered Surfaces to Metric Shape Space
  4. Alignment, Landmarking, and Registration
  5. Distances in the Shape Space
  6. Quantifying Errors of Shape Distances
  7. Manifold Learning
  8. Diffusion Map
  9. Diffusion Distance
  10. Manifolds in which each Sample has Structure: Fiber Bundles
  11. Improved Spatial Correspondence
  12. Other differential geometry tools
  13. Summary and Future Research Directions
  14. Acknowledgements

Figures (12)

  • Figure 1: Comparison of dental morphology based on the diet. (A) Fruit eater; (B) Leaf eater; (C) Predator; (D) Omnivore (source: images wikimedia commons, meshes taken from morphosource (AMNH:Mammals:M-80072, AMNH:Mammals:M-100832, AMNH:Mammals:M-196480, AMNH:Mammals:M-174537) morphosource.
  • Figure 2: Working flow in a shape space. (A) Data acquisition; (B) Alignment, registration, and distance calculation on the manifold of shapes; (C) Automatic Shape analysis (e.g., from top left clockwise, curvature estimation (ariaDNE shan2019ariadne), landmark sampling, consistent segmentation of shapes in the collection ; (D) Phylogeny of different species groups. Images adopted from shan2019ariadnegao2015hypoelliptic.
  • Figure 3: Illustration of the scanning process of a skull: from CT scan to MicroCT cube, to point cloud and surfaces reconstructed and cleaned.
  • Figure 4: Alignment of two different shapes labeled as 1 and 6, by composing a series of incremental different bones (right) found via the MST (left) boyer2015new.
  • Figure 5: (A) Landmark propagation via continuous Procrustes distance. Propagating from Microcebus to Lepilemur is not equivalent if their ancestor Megaladopis is also used boyer2011algorithms; (B) Landmarking via Matching Pursuit algorithm. Points that are in correspondence with each other using an eigenvector (each row is a correspondence to a different eigenvector) gao2015hypoelliptic.
  • ...and 7 more figures

Theorems & Definitions (9)

  • definition 1: Procrustes distance
  • definition 2: Conformal Wasserstein neighborhood distance (cWn)
  • definition 3: Continuous Procrustes distance between surfaces (cP)
  • definition 4: Diffusion Distance (DD)
  • definition 5: Horizontal Base Diffusion Distance (HBDD)
  • definition 6: Diffusion Map
  • definition 7: Semi-group Property
  • definition 8: Semi-group Error
  • definition 9: Fibre Bundle