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.