<title>Representation of finite-range increasing filters in the context of computational morphology</title>

Abstract

The classical Matheron representation of gray-scale filters considers images to possess infinite range, in particular so that they are range translation invariant. A key aspect of the theory is that binary morphology algebraically embeds into gray-scale with binary images possessing gray range minus infinity and zero. While this structure causes no algebraic problems, it does create both topological and probabilistic difficulties. In particular, the theory of optimal gray- scale filters does not contain the theory of optimal binary filters as a special case, and the optimal gray-scale filter takes finite-range images, say [0,M], and yields images with range [-M,2M]. These anomalies are mitigated by the theory of computational morphology. Here, morphological filters preserve the gray range and possess very simple Matheron-type representations. Besides range preservation for finite-range images, the key difference in computational morphology is that a filter possesses a vector of bases, not a single basis. The salient feature remains, that of the filter being represented in terms of erosions.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.