Morphological filters-Part I: Their set-theoretic analysis and relations to linear shift-invariant filters

This 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 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 suprema of erosions. These results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal-shaping strategies.