Shape of structuring elements for morphological gradients I am looking to understand recommended shapes of structuring elements used in calculating morphological gradients. According to Pierre Soille: Morphological Image Analysis:


  Only symmetric structuring elements containing their origin are considered. By doing so, we make sure that the arithmetic difference is always nonnegative.


The arithmetic difference mentioned in the quote is referring to three combinations currently used to calculate the discrete gradient:


arithmetic difference between the dilation and the erosion;
arithmetic difference between the dilation and the original image;
arithmetic difference between the original image and its erosion.


But, I think using a SE containing its origin is enough (it ensures anti-extensivity of dilation and extensivity of erosion). In this case, the following holds and ensures nonnegativity in all three cases:

$\varepsilon_B \leq id \leq \delta_B$ (where $id$ is the identity transform)

I am looking for a reason to enforce the symmetry condition. Intuitively, I understand that using a symmetrical SE is better than using a non-symmetrical one (e.g. examining a symmetrical pixel neighborhood). It was also suggested to me that there might be historical reason for this constraint.

However, I would like specific examples, arguments or references that point to desirable properties of symmetrical SEs (or undesirable properties of non-symmetrical ones).
I looked up in Jaehne, Gonzalez, Soille (the one you've posted as well as Mathematical Morphology and Its Applications to Image and Signal Processing) and some other special morphological papers and haven't found neither any design criteria for the structuring element nor any special hints why it has to be symmetrical.

Personally I think that a symmetrical SE is good for the same symmetrical effects on the object you want to modify.
With my existing experience I would not use a non-symmetrical SE, because I can't take it for any object or scenario and I don't know how it will react for other cases.

Nevertheless it's an interesting question and I am trying to get an answer.