In general, morphological filters are not self-dual; that is, they do not process dark and bright parts of an image in the same way. In order to build self-dual filters, the authors introduce another ordering technique called reference ordering. For numbers, x is below y with reference to r if x is between r and y. This can be generalized in any distributive complete lattice, through inequalities involving joins and meets. This leads to a weaker order structure called complete inf-semilattice (CISL), where non-void infima are always defined, but not always suprema. This has some unfortunate consequences, such as dilations that are only partially defined. The authors describe the construction of new types of erosions and dilations for the reference CISL. They also link their approach to that of filtering the positive and negative parts of a function separately.

This framework has been applied by the second author to the processing of video sequences. The authors are aware of my alternate philosophy, which is that the CISL should be completed by adding a greatest element to it (for example, for numerical values with reference ordering, this would be an unsigned infinity), so that one obtains a complete lattice, and everything works as usual.