Browsing by Author "Schafer, R.W."
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Item Morphological Filters- Part 1: Their Set-Theoretic Analysis and Relations to Linear Shift-Invariant Filters.(1987) Maragos, Petros; Schafer, R.W.; ISRThis paper examines the set-theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations, which involve supremum/infimum operations and additions. Binary signals are classified as sets and multilevel signals as functions. Two set theoretic representations of signals are reviewed. Filters are classified as set-processing (SP) or function-processing (FP). Conditions are provided for certain FP filters that pass binary signals as binary to commute with signal thresholding, because then they can be analyzed and implemented as SP filters. The basic morphological operations of set erosion, dilation, opening, and closing are related to Minkowski set operations and are used to construct FP morphological filters. Emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions; the latter case allows the definition of morphological convolutions and correlations. Toward this goal various properties of FP morphological filters are also examined. Linear shift-invariant filters (due to their translation- invariance) are uniquely characterized by their kernel, which is a special collection of input signals. Increasing linear filters are represented as the supremum of erosions by their kernel functions. If the filters are also discrete and have a finite- extent impulse response, they can be represented as the supremum of erosions only by their minimal (with respect to a signal ordering) kernel functions. Stable linear filters can be represented as the sum of (at most) two weighted supreme of erosions. These results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal-shaping strategies.Item Morphological Filters- Part 2: Their Relations to Median, Order- Statistic, and Stack Filters.(1987) Maragos, Petros; Schafer, R.W.; ISRThis 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.