Expressing Bayesian Fusion as a Product of Distributions: Application to Randomized Hough Transform

Data fusion is a common issue of mobile robotics, computer assisted medical diagnosis or behavioral control of simulated character for instance. However data sources are often noisy, opinion for experts are not known with absolute precision, and motor commands do not act in the same exact manner on the environment. In these cases, classic logic fails to manage efficiently the fusion process. Confronting different knowledge in an uncertain environment can therefore be adequately formalized in the bayesian framework. Besides, bayesian fusion can be expensive in terms of memory usage and processing time. This paper precisely aims at expressing any bayesian fusion process as a product of probability distributions in order to reduce its complexity. We first study both direct and inverse fusion schemes. We show that contrary to direct models, inverse local models need a specific prior in order to allow the fusion to be computed as a product. We therefore propose to add a consistency variable to each local model and we show that these additional variables allow the use of a product of the local distributions in order to compute the global probability distribution over the fused variable. Finally, we take the example of the Randomized Hough Transform. We rewrite it in the bayesian framework, considering that it is a fusion process to extract lines from couples of dots in a picture. As expected, we can find back the expression of the Randomized Hough Transform from the literature with the appropriate assumptions.