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 Morphological Image Operators
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Definition
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.. sidebar:: Maximum vs Supremum, Minimum vs Infimum

   For a finite set of orderable elements (think numbers) we can
   always find the maximal value. For two elements $a$ and $b$ we
   write $a\vee b$. For a finite set of values $a_i$ for
   $i=1,\ldots,n$ we write

   .. math::
      \text{sup} \{a_i|i=1,\ldots,n\} =  \bigvee_{i=1}^n a_i

   For such a finite set of values the supremum is the maximum value
   but for an infinite set that need not be the case. Consider the set
   $a_i = 1 - 1/i$ for $i=1,2,\ldots,\infty$ then for all $i<1$ but we
   cannot define the maximal value as $a_{i+1} > a_i$ for all
   $i$. That is where the **supremum** comes in:

   .. math::
      \text{sup}_i a_i  = \arg\min_b a_i \leq b

   i.e. the supremum is the minimal value $b$ that is greater than or
   equal to any of the values $a_i$.

When considering linear image operators there was only one
**superposition principle** describing how a linear operator
distributes over a weighted sum of images: $\op L(\alpha f + \beta g)
= \alpha\op L f + \beta\op L g$. When we consider morphological
operators we have two, closely related, superposition principles, one
for **erosion** type operators and the second for **dilation** type of
operators.

An erosion $\op E$ is an operator that distributes over the
(additively) weighted pointwise infimum of images

.. math::
   \op E( (\alpha + f) \wedge (\beta + g)) = (\alpha + \op E f) \wedge (\beta + \op E g)