Saturation

Axiomatic definition
Axioms:

1. $\forall A \subset X$, $X\setminus$sat$(A)=$cc$(X\setminus A)$ or$\emptyset$;
2. $A\subset B\:\Rightarrow\:$sat$(A) \subset$   sat$(B)$;
3. sat$(X\setminus$sat$(A))=X$ or $\emptyset$;
4. $\operatorname{sat} \circ \operatorname{sat} = \operatorname{sat}$.

• Holes = cc($X\setminus A$) in sat($A$)
• Exterior = $X\setminus$   sat$(A)$
• Shape = saturation of a cc of level set

Summer School on Mathematical Problems in Image Processing, Trieste, Italy, September 4th-18th 2000