On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension

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...

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Bibliographic Details
Published in:Journal of mathematical imaging and vision Vol. 61; no. 9; pp. 1276 - 1300
Main Author: Overgaard, Niels Chr
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2019
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Computer Science
Convergence
Hilbert space
Image Processing and Computer Vision
Matematik
Mathematical Methods in Physics
Mathematical Sciences
Natural Sciences
Naturvetenskap
Noise reduction
Parameters
Properties (attributes)
Regression models
Regularization
Restoration
Signal,Image and Speech Processing
Strings
Isotonic regression
Denoising semi-group
Lewy–Stampacchia inequality
Fused Lasso
Higher-order total variation
Total variation minimization
Regression splines
Taut string
ISSN:0924-9907, 1573-7683, 1573-7683
Online Access:Get full text
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Summary: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.
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ISSN:0924-9907
1573-7683
1573-7683
DOI:10.1007/s10851-019-00905-z