The Rudin-Osher-Fatemi model (1992) recovers a clean image u from a noisy observation f by minimizing E(u)=21∥u−f∥22+λTV(u),
where TV(u)=∫∣∇u∣ is the total variation. The L1 prior on the gradient is the whole point: unlike a Tikhonov L2 prior, which would penalize edges and smear them, the TV seminorm allows jumps to remain jumps. The minimizer is the closest function to f that does not pay extra for sharp boundaries. Chambolle's dual algorithm (2004) sidesteps the nondifferentiability of TV by working on the dual variable p (a vector field with ∣p∣≤1). One iterates a projected gradient step pi,jn+1=1+τ∣(∇(divpn−f/λ))i,j∣pi,jn+τ(∇(divpn−f/λ))i,j
with step size τ≤1/4, then recovers u=f−λdivp at the end. Watch the right panel: noise dissolves first, while the shape boundaries stay crisp — that staircase-flat look on uniform regions is the signature of TV regularization. Inspired by Gabriel Peyré's image-processing visualizations.