{
  "abstract": "In this article, we analyze a recently introduced kinetic model based on Hegselmann-Krause-type interaction dynamics describing consensus-based image segmentation in the mean-field regime, where particle density depends on spatial coordinates and the normalized gray level (feature) value. The model is formulated as an evolutive nonlocal partial differential equation featuring a non-local spatial drift term involving a bounded confidence interaction kernel, depending on spatial and feature proximity, and a spatial diffusion term degenerating in the feature variable representing aleatoric uncertainties. Establishing well-posedness on a bounded domain, essential for practical image processing, presents significant analytical challenges: the lack of uniform control of space derivatives due to kernels with jump discontinuities and degenerate spatial diffusion, complex non-standard boundary terms under homogeneous Neumann conditions, and a lack of compactness arising from the absence of derivatives with respect to the feature variable. To address these difficulties, we introduce a three-level regularization scheme. Specifically, we smooth the interaction kernel, apply structural boundary regularization to the non-local interaction term to simplify boundary conditions, and incorporate artificial diffusion in spatial and feature variables to resolve degeneracy. We prove the well-posedness and regularity of the regularized system via a finite difference scheme and study the asymptotic limit as regularization parameters vanish, proving the existence of weak/distributional solutions of the original model. This approach advances the mathematical theory of kinetic models for image segmentation by extending the framework of distributional solutions to bounded domains and non-smooth interaction kernels with jump discontinuities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39997",
  "author_total": 4,
  "authors": [
    {
      "name": "Abramo Agosti"
    },
    {
      "name": "Pierluigi Colli"
    },
    {
      "name": "Elisabetta Rocca"
    },
    {
      "name": "Mattia Zanella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b2attcaef7bzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39997",
  "fulltext_url": null,
  "id": "b2attcaef7bzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39997",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "1707|Medical Image Segmentation Techniques"
    ]
  ],
  "openalex": "W7215068558",
  "pdf_url": "https://arxiv.org/pdf/2609.39997",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b2attcaef7bzv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39997",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39997"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global existence for Consensus-Based Kinetic Models for Image Segmentation",
  "title_text": "Global existence for Consensus-Based Kinetic Models for Image Segmentation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/b2attcaef7bzv",
  "venue": "arXiv",
  "versions": []
}