In a binary image, the area opening (with a size parameter s) removes all connected components of the figure whose size is less than s (the size s being the Lebesgue measure in the Euclidean case, and the number of pixels in the digital case). Similarly, the area closing fills all holes (connected components of the background) whose size is less than s. In gray-level images, the area opening darkens small bright connected components, while the area closing brightens small dark connected components. The authors start with a theoretical analysis of these gray-level image filters in the continuous framework, where images are upper-semicontinuous functions from a Euclidean domain to the extended real line.

The area opening and closing do not commute; composing them does not therefore lead to a self-dual filter (one that acts in the same way on bright and dark regions). The authors propose another filter, called the grain filter. Given a figure, the saturation is the operation that fills all the figure’s internal holes (connected components of the background that do not contain a preselected “infinity” point). The grain filter (with a size parameter s) keeps in a figure all connected components whose saturation has size at least s, and fills their internal holes of size less than s. This filter acts self-dually on binary images, except when the saturation of a connected component or of a hole has size exactly s. The authors analyze the gray-level extension of this filter in the same continuous framework, and show that for continuous functions it is self-dual.

This study in the continuous framework is mainly of theoretical interest. However, the authors have also included a section on the implementation of the filter in the digital framework.