= ROF level-set formulation
{c}
{title2=$\{u>t\}\in\arg\min_E\{\alpha\operatorname{Per}(E;\Omega)+\int_E(t-f)\}$}
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.
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