# Small sphere distributions for directional data with application to medical imaging

## TL;DR

This paper introduces new small-sphere distribution models for multivariate directional data, particularly useful in medical imaging for analyzing shape and shape changes of 3D objects with directions concentrated on small circles.

## Contribution

The paper develops novel multivariate small-sphere distributions that model association among directions and facilitate inference, with applications in medical imaging and shape analysis.

## Key findings

- Effective modeling of shape changes in 3D objects.
- Application to human knee gait data.
- Advantages over existing small-circle data analysis methods.

## Abstract

We propose new small-sphere distributional families for modeling multivariate directional data on $(\mathbb{S}^{p-1})^K$ for $p \ge 3$ and $K \ge 1$. In a special case of univariate directions in $\Re^3$, the new densities model random directions on $\mathbb{S}^2$ with a tendency to vary along a small circle on the sphere, and with a unique mode on the small circle. The proposed multivariate densities enable us to model association among multivariate directions, and are useful in medical imaging, where multivariate directions are used to represent shape and shape changes of 3-dimensional objects. When the underlying objects are rotationally deformed under noise, for instance, twisted and/or bend, corresponding directions tend to follow the proposed small-sphere distributions. The proposed models have several advantages over other methods analyzing small-circle-concentrated data, including inference procedures on the association and small-circle fitting. We demonstrate the use of the proposed multivariate small-sphere distributions in analyses of skeletally-represented object shapes and human knee gait data.