Coarea formula for BV functions ($|Du|(A)={\int }_{\mathbb{R}}|D{\chi }_{\{u>t\}}|(A)dt$)

For a scalar function in the BV space, derivative variation is the integral of the relative perimeter measures of its superlevel sets. The formula holds on every Borel subset $A$ of the domain, not just for total mass. It converts the continuous ROF denoising problem into linked minimization problems for finite-perimeter sets.