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

Faian Conrad · 2026 · arXiv

DOI: 10.48550/arXiv.2610.29949 · open-access PDF

Indonesian

Model Gaussian Mumford-Shah memecah gambar menjadi wilayah dengan meminimalkan istilah log-probelihood negatif Gaussian multi-variasi berkelanjutan ditambah penalti perimeter. Ini digunakan untuk gambar-gambar bernilai skalar dan vektor, khususnya gambar-gambar hyperspectral, tetapi keberadaan minimizer pada umumnya tidak dijamin. Energi tidak terbatas di bawah setiap kali covariance dari satu wilayah degenerasi, terlepas dari perimeter wilayah. Lantai covariance kuantitatif pada covariance dari semua wilayah, atau kontinuitas Hölder yang setara dengan eksponen paling banyak satu dari semua dimensi

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-dimension

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