Point Feature Descriptor via Directional Partition of Unity on Maps

Abstract

We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point $\mathbf{p}$ in a map $\mathcal{M}\subset {\mathbb{R}}^d$, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a ${\mathcal{C}}^{\infty }$ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in $L^2({\mathbb{S}}^{d-1})$, establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm $|f{|}_{\mathcal{D},n}=\Vert P_{n}f{\Vert }_{L^2}$, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity $|f{|}_{\mathcal{D},n}\to \Vert f{\Vert }_{L^2({\mathbb{S}}^{d-1})}$ as $n\to \infty$. Complementary results include Fréchet differentiability, lower semicontinuity under occlusion, and explicit Lipschitz stability bounds with constants depending on the kernel and temperature. These properties underpin a localization theory in which the descriptor grid enables nearest-neighbor position recovery with a certifiable static error bound, while robot motion generates an observability Gramian whose smallest eigenvalue controls a dynamic error bound and generically resolves symmetry-induced ambiguities that persist under single-observation matching.

24 pages, 1 figure