Density estimation on the rotation group using diffusive wavelets

TL;DR

This paper introduces two novel methods for estimating probability density functions on the rotation group SO(3), demonstrating that the heat-kernel wavelet approach achieves superior convergence and guarantees positivity.

Contribution

It proposes two new density estimation techniques on SO(3), including a wavelet-based method using the heat kernel, with theoretical error analysis and numerical comparison.

Findings

Heat-kernel wavelet estimator shows faster convergence.

Proposed methods guarantee positive density estimates.

Numerical results outperform existing kernel estimator.

Abstract

This paper considers the problem of estimating probability density functions on the rotation group $SO(3)$. Two distinct approaches are proposed, one based on characteristic functions and the other on wavelets using the heat kernel. Expressions are derived for their Mean Integrated Squared Errors. The performance of the estimators is studied numerically and compared with the performance of an existing technique using the De La Vall\'ee Poussin kernel estimator. The heat-kernel wavelet approach appears to offer the best convergence, with faster convergence to the optimal bound and guaranteed positivity of the estimated probability density function.