# On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

### Niels Chr Overgaard

#### 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 (in the continuous case) 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 estimate on the denoised signal is derived. It implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the in-signal as the regularization parameter vanishes.

Download#### Paper Citation

#### in Harvard Style

Overgaard N. (2018). **On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model**.In *Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-758-276-9, pages 233-244. DOI: 10.5220/0006720402330244

#### in Bibtex Style

@conference{icpram18,

author={Niels Chr Overgaard},

title={On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model},

booktitle={Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2018},

pages={233-244},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006720402330244},

isbn={978-989-758-276-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

SN - 978-989-758-276-9

AU - Overgaard N.

PY - 2018

SP - 233

EP - 244

DO - 10.5220/0006720402330244