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Published April 4, 2019 | Submitted
Report Open

Reconciling Bayesian and Total Variation Methods for Binary Inversion


A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization. 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 is a natural framework in which to carry this out. The link between Bayesian inversion methods and perimeter regularization, however, is not fully understood. In this paper two links are studied: (i) the MAP objective function of a suitably chosen phase-field Bayesian 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. Furthermore, the level set approach is shown to lead to faster algorithms for uncertainty quantification than the phase field approach.

Additional Information

The research of CME was partially supported by the Royal Society via a Wolfson Research Merit Award; the work of AMS by DARPA contract contract W911NF-15-2-0121; the work of CME and AMS by the EPSRC programme grant EQUIP; the work of MMD and AMS by AFOSR Grant FA9550-17-1-0185 and ONR Grant N00014-17-1-2079; the work of MMD by the EPSRC MASDOC Graduate Training Program; VHH gratefully acknowledges the MOE AcRF Tier 1 grant RG30/16.

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