The Mueller matrix cone and its application to filtering

TL;DR

This paper introduces a mathematical framework linking Mueller matrices to Hermitian matrices, enabling optimal filtering and suggesting Riemannian geometry for averaging Mueller matrices.

Contribution

It establishes an isometry between Mueller matrices and Hermitian matrices, leading to new optimal filtering methods and geometric approaches for Mueller matrix analysis.

Findings

Eigenvalue filtering of Mueller matrices is optimal.

The cone of Mueller matrices can be analyzed via Hermitian positive semidefinite matrices.

Riemannian geometry can improve Mueller matrix averaging.

Abstract

We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices which maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality result for the filtering of Mueller matrices, which roughly says that it is always enough to filter the eigenvalues of the corresponding "coherency matrix". Then we further explain how the knowledge of the cone of Hermitian positive semidefinite matrices can be transferred to the cone of Mueller matrices with a special emphasis towards optimisation. In particular, we suggest that means of Mueller matrices should be computed within the corresponding Riemannian geometry.