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

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Názov: On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension
Autori: Overgaard, Niels Chr
Prispievatelia: Lund University, Faculty of Science, Centre for Mathematical Sciences, Research groups at the Centre for Mathematical Sciences, Mathematical Imaging Group, Lunds universitet, Naturvetenskapliga fakulteten, Matematikcentrum, Forskargrupper vid Matematikcentrum, Mathematical Imaging Group, Originator, Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematics (Faculty of Engineering), Lunds universitet, Naturvetenskapliga fakulteten, Matematikcentrum, Matematik LTH, Originator, Lund University, Profile areas and other strong research environments, Strategic research areas (SRA), ELLIIT: the Linköping-Lund initiative on IT and mobile communication, Lunds universitet, Profilområden och andra starka forskningsmiljöer, Strategiska forskningsområden (SFO), ELLIIT: the Linköping-Lund initiative on IT and mobile communication, Originator, Lund University, Profile areas and other strong research environments, Strategic research areas (SRA), eSSENCE: The e-Science Collaboration, Lunds universitet, Profilområden och andra starka forskningsmiljöer, Strategiska forskningsområden (SFO), eSSENCE: The e-Science Collaboration, Originator
Zdroj: Journal of Mathematical Imaging and Vision. 61(9):1276-1300
Predmety: Natural Sciences, Mathematical Sciences, Naturvetenskap, Matematik
Popis: 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.
Prístupová URL adresa: https://doi.org/10.1007/s10851-019-00905-z
Databáza: SwePub
Popis
Abstrakt: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.
ISSN:09249907
15737683
DOI:10.1007/s10851-019-00905-z