# Aggregated Pairwise Classification of Statistical Shapes

## TL;DR

This paper introduces a novel shape classification method that accounts for the infinite-dimensional, curved nature of shape space by using tangent space projections and pairwise classifiers, improving classification accuracy.

## Contribution

It develops a new approach for shape classification that leverages tangent space projections and pairwise classifiers to handle non-Euclidean, infinite-dimensional shape data.

## Key findings

- Effective shape classification using tangent space projections.
- Improved accuracy with the novel pairwise classifier combination.
- Demonstrated impact of projection point and subspace choice on misclassification.

## Abstract

The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as a vector in a relatively low-dimensional Euclidean space. Statistical shape data have two main properties that suggest a need for a novel approach: (i) shapes are inherently infinite dimensional with strong dependence among the positions of nearby points, and (ii) shape space is not Euclidean, but is fundamentally curved. To accommodate these features of the data, we work with the square-root velocity function of the curves to provide a useful formal description of the shape, pass to tangent spaces of the manifold of shapes at different projection points which effectively separate shapes for pairwise classification in the training data, and use principal components within these tangent spaces to reduce dimensionality. We illustrate the impact of the projection point and choice of subspace on the misclassification rate with a novel method of combining pairwise classifiers.