EDB — 124

[124]For \(A,B⊆ X\) arbitrary subsets, we recall the definition of Minkowski sum \(A ⊕ B=\{ x+y : x∈ A, y∈ B\} \) defined in [11R]↺↻.

Having now fixed a set \(B\), we define

• the dilation of a set \(A⊆ X\) to be \(A ⊕ B\);

• the erosion of a set \(A⊆ X\) as

  \[ A ⊖ B=\{ z∈ X:(B+z)⊆ A\} \quad ; \]

• the closing \(A ∙ B = ( A ⊕ B ) ⊖ B\);

• the opening \( A ∘ B = ( A ⊖ B ) ⊕ B\).

Where, given \(B⊆ X,z∈ X\), we have indicated with \(B+z=\{ b+z:b∈ B\} \) the translation of \(B\) in the direction \(z\). In previous operations \(B\) it is known as ”structural element”, And in applications often \(B\) it’s a puck or a ball.