Asymptotic global confidence regions for 3-D parametric shape estimation in inverse problems

This paper derives fundamental performance bounds for estimating 3-D parametric surfaces in inverse problems. Un-like conventional pixel-based image reconstruction approaches, our problem is reconstruction of the shape of binary or homogeneous objects. The fundamental uncertainty of such estimation problems can be represented by global confidence regions, which facilitate geometric inference and optimization of the imaging system. Compared to two-dimensional global confidence region analysis in our previous work, computation of the probability that the entire 3-D surface estimate lies within the confidence region is, however, more challenging, because a surface estimate is an inhomogeneous random field continuously indexed by a two-dimensional index set. We derive an approximate lower bound to this probability using the so-called tube formula for the tail probability of a Gaussian random field. Simulation results demonstrate the tightness of the resulting bound and the usefulness of 3-D global confidence region approach.