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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

Chinese

加西亚的Mumford-Shah模型将图像分为区域,通过减少连续多变的加西亚的负日志概率术语和周边处罚来减少。它用于规模和矢量值的图像,特别是超光谱图像,但总体上没有保证一个最小化器的存在。 能量在一个单一区域的可变性退化时不受限制。 因此,我们的方法表明,在所有区域的可变性上存在的定量可变性地板,或者相当均匀的Hölder连续性,在 histogram的所有所有维度边界中最多有一处呈现,保证了能量较低的边界和最低化器的存在。 因此,我们的方法表明,存在是由图像的历史图像的常规转换属性所产生的,我们进一步认为

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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