Bayesian Network and Probabilistic Image Processing —Statistical Aspect of Belief Propagation Method—

Though the Bayesian network is one of methods for probabilistic inferences in the artificial intelligence, also probabilistic models in the image processing based on the Bayesian statistics are regarded as Bayesian networks[1, 2, 3]. As one of approximate algorithms for probabilistic inferences by using Bayesian networks, belief propagation has been investigated[4, 5, 6, 7]. Recently, the belief propagation has been applied to the probabilistic image processing[8, 9, 10]. In this talk, the statistical aspect and the practical schemes of Bayesian network to probabilistic image processing are reviewed. We consider an image on a square lattice Ω≡{1, 2, · · ·, L}. The states at pixel i(∈Ω) in the original image and the observed image are regarded as random variables denoted by Ai and Di, respectively. Then the random fields of states in the original image and the observed image are represented by A = (A1, A2, · · ·, AL) and D = (D1, D2, · · ·, DL). The actual original image and the observed image are denoted by a = (a1, a2, · · ·, aL) and d = (d1, d2, · · ·, dL), respectively. The probability that the original image is a, Pr{ A = a}, is called the a priori probability of the image. In the Bayes formula, the a posteriori probability Pr{ A = a| D = d}, that the original image is a when the given observed image is d, is expressed as