{
  "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 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12794",
  "author_total": 2,
  "authors": [
    {
      "name": "Phan Thanh An"
    },
    {
      "name": "Phiet DauThe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8gf9hxrpbe88e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12794",
  "fulltext_url": null,
  "id": "8gf9hxrpbe88e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12794",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2202|Robotics and Sensor-Based Localization"
    ]
  ],
  "openalex": "W7203457552",
  "pdf_url": "https://arxiv.org/pdf/2608.12794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8gf9hxrpbe88e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2608.12794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Point Feature Descriptor via Directional Partition of Unity on Maps",
  "title_text": "Point Feature Descriptor via Directional Partition of Unity on Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/8gf9hxrpbe88e",
  "venue": "arXiv",
  "versions": []
}