Asymptotic Global Confidence Regions for 3-D
Parametric Shape Estimation in Inverse Problems

by J. C. Ye, P. Moulin and Y. Bresler

This paper derives fundamental performance bounds for statistical
estimation of parametric surfaces embedded in R^3. Unlike 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
our previous work on global confidence region analysis for curves
(2-D shapes), computation of the probability that the entire surface
estimate lies within the confidence region is more challenging,
because a surface estimate is an inhomogeneous random field
continuously indexed by a 2-D variable. We derive an
asymptotic lower bound to this probability by relating it to the
exceedence probability of a higher-dimensional Gaussian random
field, which can in turn be evaluated using the tube formula
due to J.~Sun. Simulation results demonstrate the tightness of the
resulting bound and the usefulness of 3-D global confidence region
approach.