5.3. Morphological Image Operators

5.3.1. Definition

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

\[\text{sup} \{a_i|i=1,\ldots,n\} = \bigvee_{i=1}^n a_i\]

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:

\[\text{sup}_i a_i = \arg\min_b a_i \leq b\]

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

\[\op E( (\alpha + f) \wedge (\beta + g)) = (\alpha + \op E f) \wedge (\beta + \op E g)\]