# Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms

## TL;DR

This paper introduces a symmetric space approach to shape analysis with diffeomorphisms, reducing numerical errors in registration by leveraging symmetries and involutions, and demonstrates improved consistency on cardiac shape data.

## Contribution

It proposes a novel symmetric space framework for diffeomorphic shape analysis, enhancing numerical stability and accuracy in registration tasks.

## Key findings

- Improved numerical consistency in shape registration.
- Effective use of symmetries to reduce registration errors.
- Successful application to cardiac shape data.

## Abstract

In computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to important numerical errors. In many cases, it is well known that the error can be drastically reduced in the presence of a symmetry. In this work, the leading idea is to approximate the space of deformations and images with a possibly non-metric symmetric space structure using an involution, with the aim to perform parallel transport. Through basic properties of symmetries, we investigate how the implementations of a midpoint and the involution compare with the ones of the Riemannian exponential and logarithm on diffeomorphisms and propose a modification of these maps using registration errors. This leads us to identify transvections, the composition of two symmetries, as a mean to measure how far from symmetric the underlying structure is. We test our method on a set of 138 cardiac shapes and demonstrate improved numerical consistency in the Pole Ladder scheme.