J un 2 01 7 BAYESIAN FORMULATIONS OF MULTIDIMENSIONAL BARCODE INVERSION

Abstract. A pair of Bayesian approaches to the reconstruction of binary functions in R, d = 2, 3, is adopted; one is based on a Ginzburg-Landau penalized Gaussian prior, the other on a Bayesian level set formulation. For the Ginzburg-Landau approach a link is made to classical methods based on least squares constrained to piecewise constant functions, with a total variation regularization term which penalizes interfaces. This link rests on a careful choice, and scaling with respect to noise amplitude, of the prior and of the Ginzburg-Landau penalty. The key technical tool used to establish links between the Bayesian and classical approaches is the use of Γ−limits to study the MAP estimator. Furthermore, the parameter choices and scalings are shown, by means of numerical experiments, to lead to posterior concentration around a point which adequately encapsulates the truth. However, the Bayesian level set method is also shown to produce reconstructions of similar quality, at considerably lower computational cost, suggesting that the Bayesian level set method may be a viable alternative to total variation regularization for certain problems. The numerical studies are conducted using function-space MCMC.