dc.contributor.author Maragos, Petros en_US dc.contributor.author Schafer, R.W. en_US dc.date.accessioned 2007-05-23T09:38:32Z dc.date.available 2007-05-23T09:38:32Z dc.date.issued 1987 en_US dc.identifier.uri http://hdl.handle.net/1903/4642 dc.description.abstract This paper extends the theory of median, order-statistic (OS), and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions, dilations, openings, closings, and open-closings that commute with thresholding. The max-min representation of OS filters is introduced by showing that any median or other OS filter is equal to a maximum of erosions (moving local minima) and also to a minimum of dilations (moving local maxima). Thus, OS filters can be computed by a closed formula that involves a max-min on prespecified sets of numbers and no sorting. Stack filters are established as the class of filters that are composed exactly of a finite number of max-min operations. The kernels of median, OS, and stack filters are collections of input aignals that uniquely represent these filters due to their translation-invariance. The max-min functional definitions of these nonlinear filters is shown to be equivalent to a maximum of erosions by minimal (with respect to a signal ordering) kernel elements, and also to a minimum of dilations by minimal kernel elements of dual filters. The representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum-ofproducts expressions of Boolean functions. It is also shown that median filtering (and its iterations) of any signal by convex 1-D windows is bounded below by openings and above by closings; a signal is a root (fixed point) of the median iff it is a root of both an opening and a closing; the open-closing and close-opening yield median roots in one pass, suppress impulse noise similarly to the median, can discriminate between positive and negative noise impulses, and are computationally less complex than the median. Some similar results are obtained for 2-D median filtering. en_US dc.format.extent 1823312 bytes dc.format.mimetype application/pdf dc.language.iso en_US en_US dc.relation.ispartofseries ISR; TR 1987-134 en_US dc.title Morphological Filters- Part 2: Their Relations to Median, Order- Statistic, and Stack Filters. en_US dc.type Technical Report en_US dc.contributor.department ISR en_US
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