Hierarchical Discrete Relaxation Richard C. Wilson and Edwin R. Hancock Department of Computer Science, University of York York, Y01 5DD, UK Abstract. Our aim in this paper is to develop a Bayesian framework for matching hierarchical relational models. Such models are widespread in computer vision. The framework that we adopt for this study is provided by iterative discrete relaxation. Here the aim is to assign the discrete matches so as to optimise a global cost function that draws information concerning the consistency of match from different levels of the hierarchy. Our Bayesian development naturally distinguishes between intra-level and inter-level constraints. This allows the impact of reassigning a match to be assessed not only at its own (or peer) level of representation, but also upon its parents and children in the hierarchy. We illustrate the effectiveness of the technique in the matching of line-segment groupings in SAR images of rural scenes. 1 Introduction Marr [9] was one of the first to argue in a principled way for the hierarchical organisation and processing of information in vision systems. In practice the hi- erarchy may either be pyramidal [5], in which case the different levels represent different image resolutions, or it may be a conceptual hierarchy [7] in which case the different levels represent different degrees of abstraction from the available visual information. The key element in the development of techniques for hier- archical information processing is to exploit not only the intra-level constraints applying at the individual levels of representation but also inter-level constraints operating between different levels of the hierarchy. If used effectively these inter- level constraints can be brought to bear on the interpretation of uncertain im- age entities in such a way as to improve the fidelity of interpretation achieved by single level means. Viewed as an additional information source, inter-level constraints can be used to resolve ambiguities that would persist if single-level constraints alone were used. Although hierarchical models clearly provide a powerful source of constraints for intermediate level scene interpretation, the available methodology for effec- tively and objectively utilising them is relatively restricted. In fact, the majority of the algorithms reported in the literature confine themselfs to low or interme- diate level vision [1, 5]. For instance Cohen et. al. have developed hierarchical Markov models for image segmentation [1]. Gidas [5], on the other hand, has used re-normalisation group ideas to improve the efficiency of simulated an- nealing for pyramidal image restoration using Markov-chains. As an example from intermediate-level vision, Henderson [7] has developed a form of syntactic