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Reconciling Bayesian and perimeter regularization for binary inversion

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Dunbar, Oliver R. A., Dunlop, Matthew M., Elliott, Charles M., Hoang, Viet Ha and Stuart, Andrew M. (2020) Reconciling Bayesian and perimeter regularization for binary inversion. SIAM Journal on Scientific Computing, 42 (4). A1984-A2013. doi:10.1137/18M1179559 ISSN 1064-8275.

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Official URL: https://doi.org/10.1137/18M1179559

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Abstract

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of the total variation. On the other hand, sparse or noisy data often demands a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori (MAP) objective function of a suitably chosen Bayesian phase-field approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess finite perimeter and to have the ability to learn about the true perimeter.

Item Type: Journal Article
Subjects: Q Science > Q Science (General)
Q Science > QA Mathematics
Q Science > QC Physics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Bayesian statistical decision theory , Inverse problems (Differential equations), Level set methods , Convergence, Uncertainty (Information theory)
Journal or Publication Title: SIAM Journal on Scientific Computing
Publisher: Society for Industrial and Applied Mathematics
ISSN: 1064-8275
Official Date: 2020
Dates:
DateEvent
2020Published
6 July 2020Available
9 April 2020Accepted
Volume: 42
Number: 4
Page Range: A1984-A2013
DOI: 10.1137/18M1179559
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): First Published in SIAM Journal on Scientific Computing in 42(4), A1984–A2013. 2020, published by the Society for Industrial and Applied Mathematics (SIAM) Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 20 April 2020
Date of first compliant Open Access: 10 August 2020
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
Wolfson Research Merit AwardRoyal Societyhttp://dx.doi.org/10.13039/501100000288
W911NF-15-2-012Defense Advanced Research Projects Agencyhttp://dx.doi.org/10.13039/100000185
EQUIP[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
AGS-183586National Science Foundationhttp://dx.doi.org/10.13039/501100008982
FA9550-17-1-018Air Force Office of Scientific Researchhttp://dx.doi.org/10.13039/100000181
N00014-17-1-2079Office of Naval Researchhttp://dx.doi.org/10.13039/100000006
MASDOC Graduate Training Program[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
AcRF Tier 1 grant RG30/1Ministry of Educationhttp://dx.doi.org/10.13039/501100002701
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