There is significant interest in the synthesis of discrete-state random fields, particularly those possessing structure over a wide range of scales. However, given a model on some finest, pixellated scale, it is computationally very difficult to synthesize both large- and small-scale structures, motivating research into hierarchical methods. In this paper, we propose a frozen-state approach to hierarchical modeling, in which simulated annealing is performed on each scale, constrained by the state estimates at the parent scale. This approach leads to significant advantages in both modeling flexibility and computational complexity. In particular, a complex structure can be realized with very simple, local, scale-dependent models, and by constraining the domain to be annealed at finer scales to only the uncertain portions of coarser scales; the approach leads to huge improvements in computational complexity. Results are shown for a synthesis problem in porous media.

There is significant interest in the synthesis of discrete-state random fields, particularly those possessing structure over a wide range of scales. However, given a model on some finest, pixellated scale, it is computationally very difficult to synthesize both large and small-scale structures, motivating research into hierarchical methods.This thesis proposes a frozen-state approach to hierarchical modelling, in which simulated annealing is performed on each scale, constrained by the state estimates at the parent scale. The approach leads significant advantages in both modelling flexibility and computational complexity. In particular, a complex structure can be realized with very simple, local, scale-dependent models, and by constraining the domain to be annealed at finer scales to only the uncertain portions of coarser scales, the approach leads to huge improvements in computational complexity. Results are shown for synthesis problems in porous media.v Acknowledgements There are many people without whom I never would have been able to complete this thesis. First and foremost, I would like to thank my supervisor, Prof. Paul Fieguth, for his patience, support, and guidance over the years. Recognition is also due to Simon Alexander, whose Ph.D. research in hierarchical annealing laid the foundation for the research presented here. Prof. Marios Ioannidis provided the porous media images, and was always a source of enthusiasm.