Reconciling Bayesian and Perimeter Regularization for Binary Inversion
Abstract
A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of total variation. On the other hand, sparse or noisy data often demand 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 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 a finite perimeter and to have the ability to learn about the true perimeter.
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Published In
SIAM Journal on Scientific Computing
Pages: A1984 - A2013
DOI: 10.1137/18M1179559
ISSN (online): 1095-7197
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© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 9 April 2018
Accepted: 9 April 2020
Published online: 6 July 2020
Keywords
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Defense Advanced Research Projects Agency https://doi.org/10.13039/100000185 : W911NF-15-2-0121
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Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-17-1-0185
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Office of Naval Research https://doi.org/10.13039/100000006 : N00014-17-1-2079
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Ministry of Education - Singapore https://doi.org/10.13039/501100001459
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National Science Foundation https://doi.org/10.13039/100000001 : AGS-1835860
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Royal Society https://doi.org/10.13039/501100000288
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Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266
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