Inferring Object Boundaries and Their Roughness with Uncertainty Quantification

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries. This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from high-intensity X-ray CT and image inpainting with Gaussian and Laplace additive noise models. We also show that our method can quantify uncertainties for these noise types, various noise levels, and incomplete data scenarios.