A Morphological View on Traditional Signal Processing

Abstract

We argue that the fundamentals of mathematical morphology (partial ordered sets, openings, erosions, etc.) could provide a theoretical foundation for signal processing in general. The main observation is that signal processing addresses simpler versions of signals (of a given set S), and this actually determines a partial ordering on S. Another observation, made in the past by Serra, is that ideal filters are in fact algebraic openings. In this paper, these and other ideas are addressed and developed.

In the first part of this paper, we show that several key signal processing tasks (linear filtering, quantization, and decimation) can be seen as particular cases of morphological operators. Specifically, for each of these operators, we show a complete inf-semilattice in which the operator is an erosion. This serves as a background and motivation for investigating the relationship between mathematical morphology and general signal processing.

In the second part, we revisit the foundations of signal processing from the point of view of mathematical morphology. We show that, to every function, one can associate a partial ordering and an ideal filter (algebraic opening in the resulting partial ordered set), which provide a characterization of the “simplification” (information loss) performed by the function. Then, links between classes of signal processing tasks and basic morphological operators are established.