Source-linked AI summary
Algorithms to automatically quantify the geometric similarity of anatomical surfaces
D. Boyer, Y. Lipman, E. St. Clair, J. Puente, T. Funkhouser, B. Patel, J. Jernvall, I. Daubechies
TL;DR
The paper addresses the subjectivity, limited geometric coverage, and labor of landmark-based surface comparison. It develops polynomial algorithms for automatic surface distances and correspondences, yielding correspondences close to true landmarks and cP-based classifications comparable to ODLP and better than cWn across three datasets.
Problem
Landmark-based surface distances depend on subjective landmark choices, use relatively few points, and require time-consuming expert identification.
Method
The paper uses conformal geometry, neighborhood dissimilarity, and optimal-transport-based constructions to compute automatic distances and geometric correspondences between surfaces.
Results
Across three anatomical datasets, cP correspondences typically place propagated landmarks close to true landmarks, while cP classifications match ODLP accuracy and outperform cWn.
Takeaways & Limitations
The resulting correspondence maps capture biologically informative shape variation and support more comprehensive, objective morphological analysis accessible to nonspecialists.
Takeaways & Limitations
The cW distance requires solving a Kantorovich mass-transportation problem for every candidate mapping, making it too computationally heavy for large datasets.
Abstract
from arXiv · showhide
We describe new approaches for distances between pairs of 2-dimensional surfaces (embedded in 3-dimensional space) that use local structures and global information contained in inter-structure geometric relationships. We present algorithms to automatically determine these distances as well as geometric correspondences. This 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 renders these studies inaccessible to non-morphologists, 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.
1. The mathematical components
The mathematical framework combines conformal geometry with optimal mass transportation to represent and compare anatomical surfaces through flattened surface data and transported mass distributions.
- A smooth surface map sends curves and their intersections to corresponding curves and intersections on the target surface.
- Conformal maps preserve the angles between intersecting curves at corresponding points.
- Disk-type surfaces can be conformally flattened to the unit disk, with boundaries mapped to the disk boundary, although area distortion is introduced.
- All conformal maps between two flattened surfaces are represented using disk-preserving Möbius transformations with three parameters.
- Optimal mass transportation compares mass distributions by minimizing the effort needed to transport one distribution into another.
2. New distances between 2-dimensional surfaces
The paper defines surface distances using conformal factors, optimal transport, neighborhood structure, and continuous area-preserving maps, while balancing geometric fidelity against computational cost.
- Conformal Wasserstein distance (cW): Optimal transport compares conformal factors while quotienting over Möbius transformations that represent equivalent flattened conformal factors.
- Conformal Wasserstein distance (cW): The conformal Wasserstein distance is a metric, but evaluating it requires a Kantorovich problem for every candidate Möbius transformation, making large datasets impractical.
- Conformal Wasserstein distance (cW): The neighborhood variant replaces point-to-point hyperbolic distance with dissimilarity between local conformal-factor landscapes and requires only one Kantorovich problem after computing its cost.
- Continuous Procrustes distance between surfaces (cP): cP incorporates extrinsic information and minimizes over rigid motions and area-preserving maps, whereas cW and cWn use intrinsic surface information.
- Continuous Procrustes distance between surfaces (cP): The continuous Procrustes distance replaces subjective landmark subsets with optimized continuous maps constrained to preserve surface area.
3. Application to anatomical datasets
Across three anatomical datasets, the algorithms produced distances and correspondence maps that matched expert landmark-based comparisons and supported taxonomic classification. The correspondence maps also propagated landmarks across specimens and taxa, enabling analysis of controversial anatomical features.
- Datasets: Three datasets comprised 116 mandibular molars, 57 proximal first metatarsals, and 45 distal radii from primates, humans, and close relatives.For every pair of surfaces, the algorithms produced correspondence information and distances.
- Distance comparisons: cP distances matched observer-landmark distances better than cWn distances across all datasets.Mantel significance between ODLP and cP distances exceeded that between ODLP and cWn distances in every case.
- Distance comparisons: The distance-matrix visualization likewise showed cP outperforming cWn, especially for the structure of small distances.The cP-versus-ODLP comparison had better diagonal symmetry than the cWn-versus-ODLP comparison.
- Taxonomic classification: cP-based classifications were as accurate as ODLP-based classifications and outperformed cWn-based classifications for all three anatomical datasets.The classifications used leave-one-out nearest-neighbor assignment to taxonomic groups.
- Correspondence maps: The cP correspondence map propagated observer-placed landmarks between specimens, with propagated landmarks typically close to the corresponding true landmarks.This provided a geometric basis for observer determinations and enabled direct or step-wise propagation between teeth of different taxa.
- Anatomical application: Propagating landmarks from Microcebus to Lepilemur matched the controversial Lepilemur cusp with the Microcebus entoconid, supporting the alternative interpretation reported in the study.The application used direct and intermediate propagation paths across teeth of other taxa.
- Implications: The authors conclude that fast, landmark-free correspondence maps can reproduce detailed feature-point correspondences while preserving taxonomic and observer-landmark distance information.They present the approach as making morphological variation more accessible, objective, and comprehensive to document.