  A new proof of the equivalence of the Taut String Algorithm and the
one-dimensional Rudin-Osher-Fatemi model is presented. Based on duality and the
projection theorem in Hilbert space, the proof is strictly elementary.
Existence and uniqueness of solutions to both denoising models follow as
by-products. The standard convergence properties of the denoised signal, as the
regularizing parameter tends to zero, are recalled and efficient proofs
provided. Moreover, a new and fundamental bound on the denoised signal is
derived. This bound implies, among other things, the strong convergence (in the
space of functions of bounded variation) of the denoised signal to the insignal
as the regularization parameter vanishes. The methods developed in the paper
can be modified to cover other interesting applications such as isotonic
regression.
