A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
dc.bibliographicCitation.seriesTitle | WIAS Preprints | eng |
dc.bibliographicCitation.volume | 2591 | |
dc.contributor.author | Dong, Guozhi | |
dc.contributor.author | Hintermüller, Michael | |
dc.contributor.author | Zhang, Ye | |
dc.date.accessioned | 2022-06-23T09:38:50Z | |
dc.date.available | 2022-06-23T09:38:50Z | |
dc.date.issued | 2019 | |
dc.description.abstract | In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented. | eng |
dc.description.version | publishedVersion | eng |
dc.identifier.uri | https://oa.tib.eu/renate/handle/123456789/9165 | |
dc.identifier.uri | https://doi.org/10.34657/8203 | |
dc.language.iso | eng | |
dc.publisher | Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik | |
dc.relation.doi | https://doi.org/10.20347/WIAS.PREPRINT.2591 | |
dc.relation.hasversion | https://doi.org/10.1137/20M1366277 | |
dc.relation.issn | 2198-5855 | |
dc.rights.license | This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. | eng |
dc.rights.license | Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. | ger |
dc.subject.ddc | 510 | |
dc.subject.other | Quasilinear hyperbolic equation | eng |
dc.subject.other | geometric PDEs | eng |
dc.subject.other | total variation flow | eng |
dc.subject.other | mean curvature flow | eng |
dc.subject.other | level set | eng |
dc.subject.other | second-order dynamics | eng |
dc.subject.other | non-smooth and non-convex variational methods | eng |
dc.subject.other | image denoising | eng |
dc.subject.other | displacement error correction | eng |
dc.title | A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science | eng |
dc.type | Report | eng |
dc.type | Text | eng |
dcterms.extent | 32 S. | |
tib.accessRights | openAccess | |
wgl.contributor | WIAS | |
wgl.subject | Mathematik | |
wgl.type | Report / Forschungsbericht / Arbeitspapier |
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