An A∗ perspective on deterministic optimization for deformable templates

摘要

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. These optimization problems are difficult because they involve search through a high-dimensional space corresponding to the possible deformations of the object. In this paper, we use the theory of A∗ heuristic algorithms (Pearl, Heuristics, Addison-Wesley, Reading, MA, 1984) to compare three deterministic algorithms – Dijkstra, Dynamic Programming, and Twenty Questions – which have been applied to these problems. We point out that the first two algorithms can be thought of as special cases of A∗ with implicit choices of heuristics. We then analyze the twenty questions, or minimum entropy, algorithm which has recently been developed by Geman and Jedynak (Geman and Jedynak, IEEE Trans. Pattern Anal. Mach. Intell. 18 (1) (1996) 1–14) as a highly effective, and intuitive, tree search algorithm for road tracking. First we discuss its relationship to the focus of attention planning strategy used on causal graphs, or Bayes nets (Pearl, Probalilistic Reasoning in Intelligent Systems, Morgan Kauffman, San Mateo, CA, 1988). Then we prove that, in many cases, twenty questions is equivalent to an algorithm, which we call A+, which is a variant of A∗. Thus all three deterministic algorithms are either exact, or approximate, versions of A∗ with implicit heuristics. We then discuss choices of heuristics for optimization problems and contrast the relative effectiveness of different heuristics. Finally, we briefly summarize some recent work (Yuille and Coughlan, Proceedings NIPS'98, 1998; Yuille and Coughlan, Trans. Pattern Anal. Mach. Intell. (1999), submitted for publication) which show that the Bayesian formulation can lead to a natural choice of heuristics which will be very effective with high probability.