# Barnes-Hut Approximation for Point SetGeodesic Shooting

## TL;DR

This paper introduces a Barnes-Hut based approximation method for point set geodesic shooting that significantly speeds up computations while maintaining accuracy, enabling practical applications in clinical research.

## Contribution

It presents a novel Barnes-Hut approximation for point set geodesic shooting, reducing computational complexity from quadratic to near-linear, with comparable accuracy.

## Key findings

- Achieves up to 3-fold speedup in computations.
- Maintains accuracy comparable to exact methods.
- Enables new clinical and practical applications.

## Abstract

Geodesic shooting has been successfully applied to diffeo-morphic registration of point sets. Exact computation of the geodesicshooting between point sets, however, requiresO(N2) calculations each time step on the number of points in the point set. We proposean approximation approach based on the Barnes-Hut algorithm to speedup point set geodesic shooting. This approximation can reduce the al-gorithm complexity toO(N b+N logN). The evaluation of the proposedmethod in both simulated images and the medial temporal lobe thick-ness analysis demonstrates a comparable accuracy to the exact point set geodesic shooting while offering up to 3-fold speed up. This improvementopens up a range of clinical research studies and practical problems towhich the method can be effectively applied.