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 p in a map ℳ ⊂ ℝd, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a 𝒞∞ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in L2(𝕊d−1), establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm |f|𝒟,n = |Pn f|L2, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity |f|𝒟,n → |f|L2(𝕊d−1) as n → ∞. 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.
Related
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BibTeX
@misc{an2026point,
title = {{Point Feature Descriptor via Directional Partition of Unity on Maps}},
author = {Phan Thanh An and Phiet DauThe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12794},
eprint = {2608.12794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12794},
}
RIS
TY - UNPB
TI - Point Feature Descriptor via Directional Partition of Unity on Maps
AU - Phan Thanh An
AU - Phiet DauThe
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.12794
UR - https://arxiv.org/abs/2608.12794
AB - We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point p in a map ℳ ⊂ ℝ^d, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a 𝒞^∞ 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(𝕊^d−1), establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm |f|_𝒟,n = |P_n f|_L^2, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity |f|_𝒟,n → |f|_L^2(𝕊^d−1) as n → ∞. 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.
ER -