Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value Distribution
The Gaussian Mumford-Shah model segments an image into regions by minimizing a continuous multivariate Gaussian negative log-likelihood term plus a perimeter penalty.
ProofPaper ↗
Key points
- It is used for scalar- and vector-valued images, in particular hyperspectral images, but the existence of a minimizer in general is not guaranteed.
- 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.
- As applications of our theory, we show the convergence of regularized models used in practice and of numerical methods.
Sources (1)
- [1]Existence of Minimizers for the Gaussian Mumford-Shah Model via Covariance Floors on the Value DistributionarXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 21, 04:12 PM
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.
Extractive summary: sentences quoted from the sources.