ROF level-set formulation ($\{u>t\}\in \arg {\min }_E\{\alpha \mathrm{Per}(E;\Omega )+{\int }_E(t-f)\}$)

The scalar ROF denoising minimizer has perimeter-plus-forcing minimizing superlevel sets at almost every height. Conversely any finite-energy function with that property minimizes the full energy. The signed layer-cake identity for quadratic fidelity and coarea formula for BV functions prove sufficiency. To prove necessity, construct nested minimizing sets using comparison of perimeter minimizers with ordered forcing, reconstruct a function from rational levels, clip to establish energy bounds, and use uniqueness from strict convexity of the squared fidelity.

For bounded BV space data and $s<t$, nested ROF level-set formulation minimizing sets cannot share an interface of positive surface measure away from data jumps. On regular graph patches, upward and downward first variations use the BV traces on a hypersurface. At contacts outside $J_f$ the forcing BV trace on a hypersurface agrees, whereas the two levels differ by $t-s$. Graph regularity and contact differentiation give the contradiction. Bounded forcing is a hypothesis of this argument; arbitrary unbounded $L^2\cap BV$ data are not covered by this statement.