Stochastic Modelling and Reconstruction of Random Shapes

Finn Lindgren

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2003

ISBN 91-628-5701-0
LUTFMS--1020--2003

Abstract:

This thesis originates from the problem of reconstructing the three-dimensional shape of objects, when the only available data are

two-dimensional images. The solution presented is based on stochastic models for random object shapes and measurements, in combination with

practical surface representation and simulation methods.

As a means to handle general, unknown object types, stochastic models for approximating known smooth surfaces as well as generating random

smooth surfaces is developed. By also constructing statistical models for measured data, shape estimates can be obtained by application of

Bayes' formula. For this purpose, Markov chain Monte Carlo (MCMC) simulation algorithms for the surface models are developed.

Since it is impossible to exactly represent all surfaces in a computer, it is necessary to develop discrete representations, that can be used in estimation algorithms. In this thesis, two spline surface construction methods are developed, one based on triangular Bézier patches, and one based on subdivision techniques. Both methods use control points and normal vectors, so that local control of surface positions and tangent plane orientations is possible.

In addition to surface representations and distributions, an efficient data type and an operator history system are presented, that enable

the practical use of variable dimension MCMC simulation, by taking care of the complicated operations necessary to allow changing the

structure of the spline surface representation during the simulation.

Keywords:

shape reconstruction, deformable templates, stochastic modelling, Bayes' formula, simulation, Markov chain Monte Carlo, variable dimension, triangular splines, geometric continuity, Bézier splines, subdivision splines, Bayes' formula