Set of finite perimeter ($\mathrm{Per}(E;\Omega )=|D{\chi }_E|(\Omega )$)

A measurable set $E$ has finite perimeter in $\Omega$ when ${\chi }_E$ belongs to the BV space. Its relative perimeter is $\mathrm{Per}(E;\Omega )=|D{\chi }_E|(\Omega )$. Relative perimeter excludes the exterior image signal boundary and measures the interior interface in Caccioppoli partitions.

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.