On the Taut String Interpretation of the One-dimensional Rudin-Osher-Fatemi Model: A New Proof, a Fundamental Estimate and Some Applications

TL;DR

This paper offers a new elementary proof of the equivalence between the Taut String Algorithm and the 1D Rudin-Osher-Fatemi model, establishing fundamental estimates and exploring applications like isotonic regression.

Contribution

It provides a novel proof based on duality and projection in Hilbert space, along with new bounds and convergence results for the denoising model.

Findings

Elementary proof of equivalence between Taut String and ROF models

New fundamental bound on the denoised signal

Strong convergence of denoised signal as regularization vanishes

Abstract

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.