**Using Markov Random Fields and Algebraic Geometry to Extract 3D Symmetry Properties**

Presented at 3DPVT 2008 June 18-20, 2008 Atlanta, Georgia

Fourth International Symposium on

3D Data Processing, Visualization, and Transmission

** Yunfeng Sui and Andrew Willis (UNC-Charlotte, USA) **

In this paper, we present a new technique for solving the difficult problem of estimating the axis of symmetry for axially-symmetric surfaces. Accurate solutions to this problem are important in archaeology for systems that seek to reconstruct pottery vessels from measurements of their fragments. Our approach estimates quadratic surfaces at each measured surface point and uses a Markov Random Field superimposed on the measured surface mesh to estimate a collection of surface patches, each of which lies close to a single 3D quadratic surface. For each surface patch we estimate an quadratic implicit polynomial whose coefficients directly provide an estimate of the unknown axis location and orientation. Competing estimates of the global axis are combined using a Maximum Likelihood Estimation (MLE) framework that reflects the uncertainty present in the estimates computed from each surface patch. Our approach differs from past approaches by combining estimates derived from large surface regions that include many measurements instead of combining many local (often pointwise) estimates of the surface to determine the global estimate. Estimates from these large regions are more robust to noise and have sufficient data to generate statistics that accurately reflect the uncertainty in the computed estimates. As such, each estimate of the central axis is less susceptible to outliers and the overall axis estimate is significantly improved.