Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution
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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BibTeX
@misc{conrad2026existence,
title = {{Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution}},
author = {Faian Conrad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29949},
eprint = {2610.29949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29949},
}
RIS
TY - UNPB
TI - Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution
AU - Faian Conrad
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29949
UR - https://arxiv.org/abs/2610.29949
AB - 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.
ER -