# One thing that appeals to me in my study of mathematical morphology is what they call “Structuring element” and that with this structuring element, you can either Erode or Dilate. That’s its basis. It’s very simple. But once you start stacking operations, the order you go into changes outcome. When you consider that most of our imaging ends up on a 2D plane, even if it was processed via layers and masks and mathematical operations affecting binary, grayscale or even color images, having a rigorous field to work within for this is quite smart and has been very fruitful. What draws into it primarily is: set theory, lattice theory, topology, and random functions. As those each can cover a lot of territory on their own, being able to bring them together is practical — and visual.

One thing that appeals to me in my study of mathematical morphology is what they call “Structuring element” and that with this structuring element, you can either Erode or Dilate.

That’s its basis. It’s very simple.

But once you start stacking operations, the order you go into changes outcome.

When you consider that most of our imaging ends up on a 2D plane, even if it was processed via layers and masks and mathematical operations affecting binary, grayscale or even color images, having a rigorous field to work within for this is quite smart and has been very fruitful.

What draws into it primarily is:
set theory, lattice theory, topology, and random functions.

As those each can cover a lot of territory on their own, being able to bring them together is practical — and visual.

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Mathematical morphology’s theoretical foundation has been rigorously shown to be a complete lattice.

https://en.wikipedia.org/wiki/Complete_lattice

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In Category theory, a complete lattice forms a Concrete category of its own:

CompLat

CompLat has limitations. If you can work within these limitations, you’ll find success using any CompLat such as Mathematical Morphology.

But if you need: Predictive Mathematics, MM won’t help you much. You can compare and contrast the two. First:

https://ncatlab.org/nlab/show/CompLat

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“For all practical purposes, CompLat is not available in predicative mathematics.”

https://ncatlab.org/nlab/show/predicative+mathematics

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Also: Complete lattices are rare in https://ncatlab.org/nlab/show/constructive+mathematics – which rejects axiom of choice and excluded middle.

Related is: intuitionistic mathematics and realizability and computability.

Now, my personality is DRAWN towards intuitionistic mathematics and computability. So, I should consequently reject mathematical morphology as a complete lattice is rare in constructive mathematics.

HOWEVER, it’s not impossible to place mathematical morphology (a complete lattice) WITHIN constructive mathematics.

As a reminder:

Truth is replaced by (algorithmic) proof as a primitive notion, and
Existence means constructibility.

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