{
  "abstract": "The Gaussian Mumford-Shah model segments an image into regions by minimizing a continuous multivariate Gaussian negative log-likelihood term plus a perimeter penalty. It is used for scalar- and vector-valued images, in particular hyperspectral images, but the existence of a minimizer in general is not guaranteed. The energy is unbounded below whenever the covariance of a single region degenerates, regardless of the perimeter of the regions. A quantitative covariance floor on the covariances of all regions, or equivalently uniform Hölder continuity with exponent at most one of all one-dimensional marginals of the histogram, guarantees a lower bound on the energy and the existence of a minimizer. Our approach therefore shows that existence follows from a property of the histogram of the image, and we furthermore argue that scale-independent covariance floors cannot exist. Admitting a quantitative covariance floor is generic: it holds almost surely for all images with suitable noise. As applications of our theory, we show the convergence of regularized models used in practice and of numerical methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29949",
  "author_total": 1,
  "authors": [
    {
      "name": "Faian Conrad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6287kvpjstw6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29949",
  "fulltext_url": null,
  "id": "6287kvpjstw6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29949",
  "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",
      "cs.CV"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1707|Medical Image Segmentation Techniques"
    ]
  ],
  "openalex": "W7223209019",
  "pdf_url": "https://arxiv.org/pdf/2610.29949",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6287kvpjstw6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29949",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29949"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution",
  "title_text": "Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/6287kvpjstw6b",
  "venue": "arXiv",
  "versions": []
}