Functional additive models on manifolds of planar shapes and forms

TL;DR

This paper develops a Riemannian additive modeling framework for planar shapes and forms, enabling analysis of shape data respecting geometric invariances and providing interpretable visualizations.

Contribution

It introduces a novel Riemannian $L_2$-Boosting algorithm for shape regression that accounts for quotient geometry and offers automated model selection.

Findings

Successfully modeled sheep astragalus shapes and cell forms.

Demonstrated effective shape analysis in simulation and real datasets.

Provided interpretable visualizations of covariate effects in shape space.

Abstract

The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian $L_2$-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.