# Homogenisation of phase-field functionals with linear growth

## TL;DR

This paper establishes a rigorous homogenisation process for elliptic phase-field functionals with linear growth, revealing how fine-scale oscillations influence the limit energy in image segmentation models.

## Contribution

It introduces a novel homogenisation approach for linear growth phase-field functionals, extending classical results to include random integrands and jump-dependent surface energies.

## Key findings

- Homogenisation results for linear growth functionals with explicit dependence on jump amplitude.
- Extension of homogenisation theory to stationary random integrands.
- Identification of the limit energy as a free-discontinuity energy with jump-dependent surface term.

## 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 \emph{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 \emph{stationary random integrands}.