On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension
(2019) In Journal of Mathematical Imaging and Vision 61(9). p.1276-1300- Abstract
We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string... (More)
We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.
(Less)
- author
- Overgaard, Niels Chr LU
- organization
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Denoising semi-group, Fused Lasso, Higher-order total variation, Isotonic regression, Lewy–Stampacchia inequality, Regression splines, Taut string, Total variation minimization
- in
- Journal of Mathematical Imaging and Vision
- volume
- 61
- issue
- 9
- pages
- 25 pages
- publisher
- Springer
- external identifiers
-
- scopus:85074206778
- ISSN
- 0924-9907
- DOI
- 10.1007/s10851-019-00905-z
- language
- English
- LU publication?
- yes
- id
- 9f3b6483-c425-4612-a349-5b6b2c345892
- date added to LUP
- 2019-11-05 11:03:19
- date last changed
- 2022-05-04 02:15:15
@article{9f3b6483-c425-4612-a349-5b6b2c345892,
abstract = {{<p>We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.</p>}},
author = {{Overgaard, Niels Chr}},
issn = {{0924-9907}},
keywords = {{Denoising semi-group; Fused Lasso; Higher-order total variation; Isotonic regression; Lewy–Stampacchia inequality; Regression splines; Taut string; Total variation minimization}},
language = {{eng}},
number = {{9}},
pages = {{1276--1300}},
publisher = {{Springer}},
series = {{Journal of Mathematical Imaging and Vision}},
title = {{On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension}},
url = {{http://dx.doi.org/10.1007/s10851-019-00905-z}},
doi = {{10.1007/s10851-019-00905-z}},
volume = {{61}},
year = {{2019}},
}