Bayesian Inference for Inverse Scattering Problems with Topological Priors
We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. The forward models are transmission Helmholtz problems which we solve by boundary elements. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution by either Markov Chain Monte Carlo sampling or by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate, at a lower computational cost. When the number of objects is unknown, we devise a stochastic model selection framework.