*2.2.1 Morphological filter*

In signal processing, the term "filter" is not very precise, it depends on the context in which it is used. It sometimes implies convolution, sometimes also any operation that produces a new function. On the contrary, mathematical morphology defines it very precisely [34–36]:

Any increasing and idempotent transformation on a trellis defines a morphological filter.

• Increase:

This hypothesis is the most fundamental. It ensures that the basic structure of the trellis, i.e. the order relation, is preserved during a morphological filtering.

This property causes the filter to generally lose information.

• Idempotence:

By definition, an idempotent transformation transforms the signal into an invariant.

This property often appears, but implicitly, in the descriptions. We say of an optical filter that it is red, or of an amp that its bandwidth is 50 kHz. Here, we will pose it as an axiom.

Idempotence is reached either after a single pass or as a limit by iteration. More generally, a sequence of operations, taken as a whole, can be idempotent.

Finally, note that when linear filters are idempotent, then they do not admit an inverse: they lose information, which brings them near to morphological filters.
