Optimal stack filtering and the estimation and structural approaches to image processing

Two approaches have been used in the past to design rank-order based nonlinear filters for enhancing or restoring images: the structural approach and the estimation approach. The first approach requires structural descriptions of the image and the process which has altered it, whereas the second requires statistical descriptions of the image and the process which has altered it. The many different classes of rank-order-based filters that have been developed over the last few decades are reviewed in the context of these two approaches. One of the filter classes, stack filters, is then emphasized. These filters, which are defined by a weak superposition property and an ordering property, cover all 2-D rank-order operations. The theory of minimum-mean-absolute-error (MMAE) stack filtering is reviewed and extended to two dimensions. A theory of optimal stack filtering under structural constraints and goals is then developed. These two optimal stack filtering theories are combined into a single design theory for rank-order based filters