Bayesian assembly of 3D axially symmetric shapes from fragments

Abstract

We present a complete system for the purpose of automatically assembling 3D pots given 3D measurements of their fragments commonly called sherds. A Bayesian approach is formulated which, at present, models the data given a set of sherd geometric parameters. Dense sherd measurement data is obtained by scanning the outside surface of each sherd with a laser scanner. Mathematical models, specified by a set of geometric parameters, represent the sherd outer surface and break curves on the outer surface (where two sherds have broken apart). Optimal alignment of assemblies of sherds, called configurations, is implemented as maximum likelihood estimation (MLE) of the surface and break curve parameters given the measured sherd data for all sherds in a configuration. The assembly process starts with a fast clustering scheme which approximates the MLE solution for all sherd pairs, i.e., configurations of size 2, using a subspace of the geometric parameters, i.e., the sherd break curves. More accurate MLE values based on all parameters, i.e., sherd alignments, are computed when sherd pairs are merged with other sherd configurations. Merges take place in order of constant probability starting at the most probable configuration. This method is robust to missing sherds or groups of sherds which contain sherds from more than one pot. The system represents at least three significant advances over previous 3D puzzle solving approaches : (1) a Bayesian framework which allows for easily combining diverse types of information extracted from each sherd, (2) a search which reduces comparisons on unlikely configurations, and (3) a robust computationally reasonable method for aligning break curves and sherd outer surfaces simultaneously. In addition, a number of insights are given which have not previously been discussed and significantly reduce computation. Methods proposed for (1),(2), and (3) represent important contributions to the field of puzzle assembly, 3D geometry learning, and dataset alignment and are critical to making 3D puzzle solutions tractable to compute. Results are presented which include assembling a 13 sherd pot where only an incomplete set of 10 sherds is available.