Exact Recovery of Discrete Measures from Wigner D-Moments

TL;DR

This paper demonstrates that a sum of Dirac measures on the rotation group SO(3) can be exactly recovered from low-degree Wigner D-moments if the measures are sufficiently separated, using total variation minimization.

Contribution

It establishes the conditions under which exact recovery of discrete measures on SO(3) from moments is possible, including explicit separation bounds and proof of uniqueness.

Findings

Exact recovery from moments up to degree N is possible with sufficient separation.

Total variation minimization yields the unique solution for the measure.

Separation distance bound is explicitly given as 36/(N+1).

Abstract

In this paper, we show the possibility of recovering a sum of Dirac measures on the rotation group $SO(3)$ from its low degree moments with respect to Wigner D-functions only. The main Theorem of the paper states, that exact recovery from moments up to degree $N$ is possible, if the support set of the measure obeys a separation distance of $\frac{36}{N+1}$. In this case, the sought measure is the unique solution of a total variation minimization. The proof of the uniqueness requires localization estimates for interpolation kernels and corresponding derivatives on the rotation group $SO(3)$ with explicit constants.