Stochastic Modelling and Reconstruction of Random Shapes
Finn Lindgren
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2003
ISBN 9162857010
LUTFMS10202003

Abstract:

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

twodimensional 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



