Stochastic 3D geometric models for classification, deformation, and estimation

In the thesis two new 3D stochastic shape models are proposed: (1) a model for the estimation and classification of 3D axially symmetric shapes, and (2) a model for the deformation of 3D free-form shapes.; In (1), a system is described for the purpose of automatically estimating mathematical models for archaeological vessels given 3D measurements of their fragments commonly called sherds where it is assumed that the vessel is symmetric about a central axis. The unique approach integrates solutions to 4 different problems: (i) an algorithm for accurately estimating the surface geometry of individual sherds, (ii) an algorithm for accurately aligning assemblies of sherds, called configurations , (iii) a Bayesian performance measure for sherd configurations, and (iv) a performance-driven search algorithm. Estimation of the unknown geometry of the object is implemented as maximum likelihood estimation (MLE) where we seek to find the axially symmetric geometry which best explains the measured fragment data. For a configuration consisting of N sherds, this requires simultaneous estimation of N - 1 3D sherd alignment transformations, the matched sherd boundary curves, and the global shape of the axially symmetric surface. Configurations are constructed using a Bayesian performance measure which is the log of the probability of the computed configuration. The system merges sherds from a set of pairwise matches to generate larger sherd configurations and merges are done along contours of constant probability starting at the most probable merge.; In (2), a new stochastic surface model for deformable 3D surfaces is proposed and its application to totally unconstrained 3D free-form surface sculpting is presented. A 3D surface is a sample of a Markov Random Field (MRF) defined on the vertices of a 3D mesh where MRF sites coincide with mesh vertices and MRF cliques consist of subsets of sites. Each site has 3D coordinates (x,y,z) as random variables and is a random vector in one or more clique potentials which are functions defined on the sites within a clique and describe stochastic dependencies among sites. Data, which is used to deform the surface, can consist of, but is not limited to, an unorganized set of 3D points and is modeled by a conditional probability distribution given the 3D surface. A deformed surface is a MAP (Maximum A-posteriori Probability) estimate and is the deformed surface for which the joint distribution of the MRF surface model and the data is maximum. Included in the development is the introduction of a virtual sculpting system which integrates new data models, new anisotropic clique potentials, and cliques which involve sites that are spatially far apart.