Efficiency, approximation guarantees, and parameter selection for the conformal Wasserstein (cW/cWn) and continuous Procrustes (cP) surface distances in automated anatomical morphometrics

The 2011 PNAS paper by Boyer, Lipman, St. Clair, Puente, Patel, Funkhouser, Jernvall, and Daubechies introduced three landmark-free distances between genus-zero anatomical surfaces embedded in 3D — the conformal Wasserstein distance (cW), its neighborhood variant (cWn), and the continuous Procrustes distance (cP) — together with algorithms that produce surface correspondences and dissimilarities matching observer-determined landmarks on three primate datasets. The framework was explicitly acknowledged to rest on unproven practical foundations: cW requires solving a Kantorovich optimal-transport problem for every candidate Möbius transformation and is too slow for large datasets; cWn depends on a neighborhood scale parameter $R$ whose choice is unprincipled; the cP algorithm replaces an infinite-dimensional search over area-preserving maps by small deformations of conformal maps with no error bound relative to the true infimum; and the propagation of correspondences across taxa requires criteria that were not formulated. Later literature has addressed these gaps only partially: the authors' own group documented that the cP approximation fails precisely for geometrically dissimilar surfaces and remedied it heuristically with globally transitive (synchronization) methods, fast approximate optimal transport and alternative landmark-free metrics appeared, but no work computes cW/cWn at scale with certified error, no principled selector for $R$ exists, no approximation theorems for cP are proved, and no formal criteria for choosing correspondence-propagation channels in evolutionary inference have been established. The problem — turning this conformal-geometry comparison framework into a certified, scalable, and biologically reliable instrument — remains open.

Background

Students of evolutionary biology quantify anatomical shape through geometric morphometrics: trained observers place homologous correspondence points ("landmarks") on each digitized specimen, and shape differences are summarized, classically by the Procrustes distance — the residual after optimally superimposing the two landmark configurations by a rigid motion. This is slow, expert-dependent, and discards all geometry between the landmarks, so phenomics — the systematic, large-scale study of organismal form — lags behind genomics. The cited paper proposes to replace landmarks by whole-surface comparisons. Its ingredients are standard in differential geometry and optimal transport but need introducing. By the uniformization theorem, any genus-zero surface (topologically a sphere, or a disk if one point is removed) can be mapped conformally — angle-preservingly, though not distance-preservingly — onto a common flat domain, the unit disk $D$; the local stretch of such a map is encoded in a positive density, the conformal factor. The conformal maps from a disk-type surface onto $D$ are the disk-preserving Möbius transformations, a three-parameter family. Kantorovich optimal mass transport (the mathematics behind the earth mover's distance) computes the cheapest way to move a "mass" distribution onto another when the cost of moving a unit of mass is a prescribed ground cost. Lipman and Daubechies (Advances in Mathematics, 2011) combined these: identify each surface with its conformal factor on $D$, and define the conformal Wasserstein distance $\mathscr{D}_{\mathrm{cW}}$ as the minimum, over all Möbius transformations $m$ acting on $D$, of the transport cost between the two conformal-factor masses measured with the hyperbolic ground distance of $D$. This is a true metric, computable in polynomial time, and it is invariant under isometries of the surfaces; but a separate transport problem must be solved for every Möbius candidate, which is why the paper calls it too heavy for large datasets. The neighborhood variant cWn instead compares the two conformal-factor "landscapes" around corresponding points using a disk of radius $R$ transported by Möbius maps, solving only a single transport problem; the definition, and hence the resulting dissimilarity matrix, depends on the choice of $R$, and the paper reports runtimes of roughly 20 seconds per pair for cP versus 5 minutes for cWn. The continuous Procrustes distance cP is extrinsic: it minimizes the usual Procrustes functional over all area-preserving diffeomorphisms (a map whose Jacobian transports surface area exactly — maps of this kind exist by a classical theorem of Moser) between the two surfaces, modulo rigid motions. That minimizing family is an infinite-dimensional manifold, so the algorithm restricts the search to maps of the form $\chi\circ\varrho\circ m$, where $m$ is a Möbius transformation, $\varrho$ heuristically aligns peaks of the two conformal factors, and $\chi$ is a diffusion-based (Moser-type) correction that restores area preservation up to approximation error; the paper relies on the theoretical observation, proved in the companion work of Al-Aifari, Daubechies, and Lipman (Communications on Pure and Applied Mathematics, 2013), that for "reasonable" surfaces near-optimal maps are close to conformal — a qualitative statement without a quantitative error bound. The computational aspects and discretization-convergence analysis of the transport steps appear in Lipman, Puente, and Daubechies (Mathematics of Computation, 2012). The authors validated the pipeline on 116 prosimian lower molars, 57 first metatarsals, and 45 distal radii, showing high Mantel correlation — the standard statistic for the agreement between two distance matrices — with observer-determined landmark distances, and demonstrated that propagating observer landmarks through the computed correspondence maps can adjudicate a classical homology controversy (homology: whether structures in different species derive from the same ancestral structure) on lemur molars (the Lepilemur entoconid), while noting that different propagation paths support different hypotheses — which path is appropriate is itself unsettled. The authors' own later work developed auto3dgm, a fully automated alignment-and-correspondence pipeline (Boyer et al., The Anatomical Record, 2015), and a reassessment by Gao, Yapuncich, Daubechies, Mukherjee, and Boyer (The Anatomical Record, 2017) found that a substantial percentage of the 2011 cP alignments and pairwise maps were inaccurate for geometrically dissimilar specimens, remedying this only heuristically with collection-level, globally transitive (synchronization) methods — jointly adjusting all pairwise maps so that correspondences compose consistently across the whole collection — that in turn degrade similar-object correspondences through accumulated numerical error. Fast approximate optimal transport on surfaces (Solomon et al., ACM Transactions on Graphics, 2015), fast Möbius search (Le, Chin, and Suter, CVPR 2016), and alternative landmark-free metrics (Koehl and Hass, Journal of the Royal Society Interface, 2015; Hass and Koehl, 2017; Shi and Wang, IEEE TPAMI, 2020; the recent computations by Koga (2024–2026) of a 'weak distance' between surface area measures) address neighboring computational problems, while applied morphometrics has largely shifted to functional-map frameworks (which compare shapes through maps between spaces of functions defined on the shapes) and learned-descriptor frameworks (e.g., Thomas et al., PLOS Computational Biology, 2023; Klatzow et al., 2022).

Problem Statement

Make the landmark-free conformal-geometry surface-comparison framework of Boyer, Lipman, St. Clair, Puente, Patel, Funkhouser, Jernvall, and Daubechies (2011) — the conformal Wasserstein distance $\mathscr{D}_{\mathrm{cW}}$, its neighborhood variant cWn with scale parameter $R$, and the continuous Procrustes distance computed via maps $\chi\circ\varrho\circ m$ — a certified, scalable, and biologically reliable instrument for comparing anatomical surfaces. The common objective is that each of the four gaps acknowledged in or immediately derivable from the source paper, and left open by the subsequent literature (reconstructed in the background above), be closed for the framework itself (not merely replaced by a different metric): (1) algorithms, or provably reliable approximations, that compute $\mathscr{D}_{\mathrm{cW}}$ and cWn for large collections of digitized surfaces within practical runtimes, with the approximation error to the defining infimum controlled; (2) a principled treatment of the cWn neighborhood radius $R$ — how the resulting dissimilarity and downstream biological performance (correlation with observer-determined landmark distances, taxonomic signal) depend on $R$, and criteria or a selection rule for choosing it; (3) a priori approximation bounds for the computed continuous Procrustes value relative to the true infimum over all area-preserving maps, together with a characterization of the classes of surface pairs for which restricting the search to small deformations of conformal maps (peak alignment $\varrho$ plus diffusion-based Moser-type correction $\chi$) is provably near-optimal, in light of the empirical finding of Gao et al. (2017) that the current algorithm is inaccurate precisely for geometrically dissimilar specimens; and (4) formal criteria or guidelines for selecting correspondence-propagation channels (direct versus stepwise through intermediate taxa) so that evolutionary inferences drawn from propagated correspondences — such as the resolution of transformational-homology controversies (judging the sameness of two structures by the hypothesized transformation linking one to the other across taxa) — are methodologically justified rather than path-dependent. These four parts are independent sub-problems of the objective: each may be pursued and answered on its own. The objective is to establish, for the cW/cWn/cP family itself, which computed distances and correspondences can be trusted, at what computational cost, with which parameter choices, and under which propagation conventions.

References

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