Mathematical Morphology on Irregularly Sampled Data in One Dimension

Abstract Mathematical morphology (MM) on grayscale images is commonly performed in the discrete domain on regularly sampled data. However, if the intention is to characterize or quantify continuous-domain objects, then the discrete-domain morphology is affected by discretization errors that may be alleviated by considering the underlying continuous signal. Given a band-limited image, for example, a real image projected through a lens system, which has been correctly sampled, the continuous signal may be reconstructed. Using information from the continuous signal when applying morphology to the discrete samples can then aid in approximating the continuous morphology. Additionally, there are a number of applications where MM would be useful and the data is irregularly sampled. A common way to deal with this is to resample the data onto a regular grid. Often this creates problems where data is interpolated in areas with too few samples. In this paper, an alternative way of thinking about the morphological operators is presented. This leads to a new type of discrete operators that work on irregularly sampled data. These operators are shown to be morphological operators that are consistent with the regular, morphological operators under the same conditions, and yield accurate results under certain conditions where traditional morphology performs poorly.