Accepted Paper

Inserted: 27 aug 2025
Last Updated: 6 may 2026

Journal: Calc. Var. Partial Differential Equations
Year: 2025

Abstract:

We propose a first rigorous homogenisation procedure in image-segmentation models by analysing the relative impact of (possibly random) fine-scale oscillations and phase-field regularisations for a family of elliptic functionals of Ambrosio and Tortorelli type, when the regularised volume term grows linearly in the gradient variable. In contrast to the more classical case of superlinear growth, we show that our functionals homogenise to a free-discontinuity energy whose surface term explicitly depends on the jump amplitude of the limit variable. The convergence result as above is obtained under very mild assumptions which allow us to treat, among other, the case of stationary random integrands.

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