Estimating a-priori Unknown 3D Axially Symmetric Surfaces from Noisy Measurements of Their Fragments

In this paper, we present a computationally efficient technique for solving the difficult problem of estimating the global shape of a ceramic pot from measurements of its fragments. Each unknown pot is modeled as a surface of revolution, i.e., a 3D line- the central axis of the pot- and a 2D profile curve with respect to that axis. For each fragment, a probabilistic distribution is estimated which models both the geometric shape of the fragment and the variability of the estimated fragment shape. Estimation of the global pot shape is then a Maximum Likelihood Estimation (MLE) problem where we seek the values of the Euclidean transformation parameters that maximize the joint probability of the matched fragments´ axis/profile-curve models (which includes the additional constraint that the matched fragments must share a common central axis). This is a new type of curve-analysis problem and our solution is a new and effective approach applicable for generic constrained 2D curve alignment and for modeling of 3D axially-symmetric surfaces and for comparing geometric models which may correspond over a subset of the complete model.