Factor Graphs for Computer Vision and Image Processing

Factor graphs have been used extensively in the decoding of error correcting codes such as turbo codes, and in signal processing. However, while computer vision and pattern recognition are awash with graphical model usage, it is some-what surprising that factor graphs are still somewhat under-researched in these communities. This is surprising because factor graphs naturally generalise both Markov random fields and Bayesian networks. Moreover, they are useful in modelling relationships between variables that are not necessarily probabilistic and allow for efficient marginalisation via a sum-product of probabilities. In this thesis, we present and illustrate the utility of factor graphs in the vision community through some of the field’s popular problems. The thesis does so with a particular focus on maximum a posteriori (MAP) inference in graphical structures with layers. To this end, we are able to break-down complex problems into factored representations and more computationally realisable constructions. Firstly, we present a sum-product framework that uses the explicit factorisation in local subgraphs from the partitioned factor graph of a layered structure to perform inference. This provides an efficient method to perform inference since exact inference is attainable in the resulting local subtrees. Secondly, we extend this framework to the entire graphical structure without partitioning, and discuss preliminary ways to combine outputs from a multilevel construction. Lastly, we further our endeavour to combine evidence from different methods through a simplicial spanning tree reparameterisation of the factor graph in a way that ensures consistency, to produce an ensembled and improved result. Throughout the thesis, the underlying feature we make use of is to enforce adjacency constraints using Delaunay triangulations computed by adding points dynamically, or using a convex hull algorithm. The adjacency relationships from Delaunay triangulations aid the factor graph approaches in this thesis to be both efficient and competitive for computer vision tasks. This is because of the low treewidth they provide in local subgraphs, as well as the reparameterised interpretation of the graph they form through the spanning tree of simplexes. While exact inference is known to be intractable for junction trees obtained from the loopy graphs in computer vision, in this thesis we are able to effect exact inference on our spanning tree of simplexes. More importantly, the approaches presented here are not restricted to the computer vision and image processing fields, but are extendable to more general applications that involve distributed computations