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.

The perimeter of a measurable set relative to an open domain is the total variation seminorm of its indicator function there. For a smooth region it equals the Hausdorff measure of the boundary lying inside the domain. Global perimeter uses the entire space as the domain; an internal interface shared by two regions appears in both relative perimeters and must be counted only once in a segmentation energy.

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.

The total variation seminorm of the indicator function within an open set counts only its interior interface. The perimeter of a zero extension may add a boundary contribution.

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

Suppose $E_g$ minimizes $\mathrm{Per}(E)-{\int }_{E}g$ and $E_h$ minimizes $\mathrm{Per}(E)-{\int }_{E}h$. Compare them with their intersection and union. Adding their optimality inequalities and using submodularity of relative perimeter gives ${\int }_{E_g\setminus E_h}(h-g)\le 0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested ROF denoising superlevel sets.

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.