Deterministic Sampling with Separation of Variables in Spherical Coordinates

Densities separable in spherical coordinates have two advantages: $i$) the normalization constant is easy to compute, as the cumulative distribution can be decomposed into individual scalar integrals, and ii) an orthogonal inverse transform is directly available via a simple, scalar initial value problem and can be used to compute deterministic samples. We propagate uniform low-discrepancy sequences through that orthogonal inverse transform and obtain very homogeneous and even visually appealing deterministic samples. To demonstrate this technique, we exemplarily propose some spherical-coordinate-separable densities in $\mathbb{S}^{2}, \mathbb{R}^{2}$, and $\mathbb{R}^{3}$, including a non-isotropic modification of the von Mises-Fisher distribution. The proposed densities may be used, e.g., to represent uncertain radar measurements and for directional estimation. Furthermore, the framework presented herein allows quite simple design of various more densities tailored to a given scenario.