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 impo rtant 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 ou rselves 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 sh ow that the detectability of curves depend on a parameter K which is a func tion of the probability distributions characterizing the problem. At critic al values of K the target becomes impossible to detect on average. Our fram ework also enables us to determine whether a simpler approximate model is s ufficient to detect the target curve and hence clarify how much information is required to perform specific tasks. These results generalize our previo us 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 the ory framework, by extending the class of probability models which can be an alyzed, and by analysing the case where approximate models are used for inf erence.