Exact minimizer for the couple (L∞, BV) and the one-dimensional analogue of the Rudin-Osher-Fatemi model

Abstract

We provide a simple algorithm that constructs an exact minimizer for the E-functional E(t, f ; L∞, BV) = inf ‖g‖L∞≤t ‖ f − g‖BV . Here L∞, BV stand for the space of bounded functions and the space of functions with bounded variation on the interval [a, b], respectively. As a corollary we obtain the following formula for the K-functional K(N, f ; BV, L∞) v sup a≤x0≤···≤xN≤b −N i=1 | f (xi ) − f (xi+1)|. We also discussed the connection between the results and the Rudin–Osher–Fatemi denoising model.