Caccioppoli partition (${\sum }_i\mathrm{Per}(E_i;\Omega )<\infty$)

A Caccioppoli partition is a countable measurable partition, up to null sets, into sets of finite perimeter with finite sum of relative perimeters. Its interior interfaces are counted twice by that sum. It represents regions of a piecewise constant image signal in the piecewise-constant Mumford–Shah problem.