A note on the umbra transform in gray-scale morphology

Abstract

The umbra transform is the most common tool in mathematical morphology to define gray-scale image operators. Basically, the approach is based on the idea that every increasing set operator which is translation invariant maps an umbra onto an umbra. Although it has been pointed out at several instances in the literature that this statement is false, this has not affected the popularity of the umbra concept. In this note we explain how, by introducing the weaker notion of a pre-umbra, the mistake can be repaired quite easily.