Twenty questions, focus of attention, and A^*: A theoretical comparison of optimization strategies

Abstract

Many vision problems involve the detection of the boundary of an object, like a hand, or the tracking of a one-dimensional structure, such as a road in an aerial photograph. These problems can be formulated in terms of Bayesian probability theory and hence expressed as optimization problems on trees or graphs. The twenty questions, or minimum entropy, algorithm has recently been developed by Geman and Jedynak (1994) as a highly effective, and intuitive, tree search algorithm for road tracking. In this paper we analyse this algorithm to understand how it compares to existing algorithms used for vision, and related, optimization problems. First we show that it is a special case of the focus of attention planning strategy used on causal graphs, or Bayes nets, [18]. We then show its relations to standard methods, already successfully applied to vision optimization problems, such as dynamic programming, decision trees, the A^* algorithm used in artificial intelligence [22] and the, closely related, Dijkstra algorithm of computer science [4]. These comparisons show that twenty questions is often equivalent to an algorithm, which we call A+, which tries to explore the most probable paths first. We show that A+ is a greedy, and suboptimal, variant of A^*. This suggests that A+ and twenty questions will be faster than A^* and Dijkstra for certain problems but they may occasionally converge to the wrong answer. However, the fact that A+ and twenty questions maintain a probabilistic estimate of how well they are doing may give warning of faulty convergence and also allow intelligent pruning to speed up the search.