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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).

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How can I download the article that you mention? –  Andrey Oct 29 '12 at 16:23
    
@Andrey It's a book, not an article. And, sorry, can't help you there, I have a paper copy. –  penelope Oct 29 '12 at 16:37
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3 Answers

up vote 1 down vote accepted

For planar elements (implied by the wording "structuring element") the containment of origin is enough to maintain the properties of anti-extensivity for erosion, and extensivity for dilation as can be found in many texts and you also pointed that out. So, yes, this is enough for the non-negativity for the arithmetic difference (this is directly shown by contradiction). The reason this piece of text is present in Pierre's book might be a simple one: a mistake. This statement is supported by other papers (like "Morphological Gradients" by Rivest, Soille, Beucher; or "An Overview of Morphological Filtering" by Serra, Vincent) on the morphological gradient defined by Beucher on his thesis. Now, I expect the most common situation to be the application of a gradient in an isotropic way, implying a symmetrical structuring element.

Now, to the second part of the question (supposing isotropy is not enough to conclude the answer). The first reason I can give for using symmetrical elements is to eliminate the burden of dealing with the multiple definitions for erosion and dilation present in the literature. It turns out that when you consider symmetrical elements, the distinct definitions become the same, guaranteeing the same behavior among different implementations. Using anisotropic elements will also translate your objects, which might be only useful for some certain applications. Also, some structuring elements are trivially decomposed when they are symmetric, enabling faster applications of morphological operations.

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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.

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Could you please expand on "symmetrical SE is good for the same symmetrical effects on the object you want to modify"? And, just to emphasize (not sure if you got this from your answer): I'm not asking about non-symmetrical SEs for all the morph. operations (e.g. erosion, dilation, opening): the question is about SE shape for morphological gradients. Nevertheless, thank you for your interest. –  penelope Oct 29 '12 at 15:19
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The asymmetric structuring elements produce a translation dilation on the original set or image. The size of the translation is determined by the offset in the center of the structuring element. For example you could try this using matlab for the dilation operator:

I = imread('circles.png');
se = strel('disk',10); %you could see it with se = strel('line',5,180) 
%too but have to make sure that the origin still lies in the se.
se2 = translate(se,[-5,-5]) %offset the center by 5 pixels
figure, imshow(imdilate(I,se))
figure, imshow(imdilate(I,se2))

One avoids this since it introduces anisotropy using the asymmetric structuring element. But one can use this for applications in edge detection i guess.

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