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 $|f|_{\mathcal{D},n} = \|P_n f\|_{L^2}$, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity $|f|_{\mathcal{D},n} \to \|f\|_{L^2(\mathbb{S}^{d-1})}$ as $n \to \infty$. Complementary results include Fr\'echet 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.