EDB — 124

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

Having now fixed a set $B$, we define

• the dilation of a set $A\subseteq X$ to be $A\oplus B$;

• the erosion of a set $A\subseteq X$ as

  $$A\ominus B=\{z\in X:(B+z)\subseteq A\};$$

• the closing $A\bullet B=(A\oplus B)\ominus B$;

• the opening $A\circ B=(A\ominus B)\oplus B$.

Where, given $B\subseteq X,z\in X$, we have indicated with $B+z=\{b+z:b\in B\}$ 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.