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Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution

Faian Conrad · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.29949 · cited by 0 works (OpenAlex) · open-access PDF

Korean

사우스시안 Mumford-Shah 모델은 지속적인 다변화 된 Gaussian 부정적인 log-probabilities 용어와 범위 처벌을 최소화함으로써 영역으로 이미지를 분할합니다. 그것은 스칼라 및 벡터 가치가있는 이미지, 특히 하이퍼 스펙트럼 이미지를 위해 사용되지만 일반적으로 최소화기의 존재는 보장되지 않습니다. 에너지는 단일 영역의 covariance가 지역의 범위와 상관없이 퇴행 할 때마다 무제한입니다. 따라서 우리의 접근 방식은 모든 영역의 covariance에 양적 covariance 바닥이 있음을 보여줍니다, 또는 동등하게 동일한 Hölder 연속성은 histogram의 모든 차원 마진 중 하나 이상에 나타납니다. 에너지의 낮은 한계와 최소화의

Machine translation (Meta M2M100); the English original follows.

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.

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Topics: Covariance, Mathematics, Gaussian, Multivariate normal distribution, Applied mathematics, Covariance function

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