Total Variation Regularization for Image Denoising, I. Geometric Theory

Let Ω be an open subset of R where 2 ≤ n ≤ 7; we assume n ≤ 2 because the case n = 1 has been treated elsewhere (see [Alli]) and is quite different from the case n > 1; we assume n ≤ 7 is that our work will make use of the regularity theory for area minimizing hypersurfaces. Let F(Ω) = L1(Ω) ∩ L∞(Ω)). Suppose s ∈ F(Ω) and Suppose γ : R→ [0,∞) is locally Lipschitzian, positive on R ∼ {0} and zero at zero. Let