On the relation of order-statistics filters and template matching: optimal morphological pattern recognition

In this paper, we investigate methods for optimal morphological pattern recognition. The task of optimal pattern recognition is posed as a solution to a hypothesis testing problem. A minimum probability of error decision rule—maximum a posteriori filter—is sought. The classical solution to the minimum probability of error hypothesis testing problem, in the presence of independent and identically distributed noise degradation, is provided by template matching (TM). A modification of this task, seeking a solution to the minimum probability of error hypothesis testing problem, in the presence of composite (mixed) independent and identically distributed noise degradation, is demonstrated to be given by weighted composite template matching (WCTM). As a consequence of our investigation, the relationship of the order-statistics filter (OSF) and TM—in both the standard as well as the weighted and composite implementations—is established. This relationship is based on the thresholded cross-correlation representation of the OSF. The optimal order and weights of the OSF for pattern recognition are subsequently derived. An additional outcome of this representation is a fast method for the implementation of the OSF.