Order parameters for detecting target curves in images:: When does high level knowledge help?

Many problems in vision can be formulated as Bayesian inference. It is important to determine the accuracy of these inferences and how they depend on the problem domain. In this paper, we provide a theoretical framework based on Bayesian decision theory which involves evaluating performance based on an ensemble of problem instances. We pay special attention to the task of detecting a target in the presence of background clutter. This framework is then used to analyze the detectability of curves in images. We restrict ourselves to the case where the probability models are ergodic (both for the geometry of the curve and for the imaging). These restrictions enable us to use techniques from large deviation theory to simplify the analysis. We show that the detectability of curves depend on a parameter K which is a function of the probability distributions characterizing the problem. At critical values of K the target becomes impossible to detect on average. Our framework also enables us to determine whether a simpler approximate model is sufficient to detect the target curve and hence clarify how much information is required to perform specific tasks. These results generalize our previous work (Yuille, A.L. and Coughlan, J.M. 2000. Pattern Analysis and Machine Intelligence PAMI, 22(2):160-173) by placing it in a Bayesian decision theory framework, by extending the class of probability models which can be analyzed, and by analysing the case where approximate models are used for inference.