A poset is totally ordered if

YX,pair every for X,Yor YX .4

A totally ordered poset is called a chain.

) , (

y x

• Let be the set of all real numbers, and let have theusual meaning; then is a chain.

• The set d with the relation )y ,...,y ,(y )x...,,x,x(" d21d21 if for all ii y x i" is a poset. It is a chain if and only if d = 1.

• Given a set E, the power set Ρ(E) comprising all subsets of Ebecomes a poset under the inclusion relation, that is, YX "if and only if ."YX The empty set is denoted by .

Examples:

• Let be the set of all nonnegative integers, and put nmif m divides n. Then is a poset.

• Inclusion relation and Hasse diagrams

or

g)(f g)(f .,. ge

and

)( pi is a segment (a subset of a partition) of X containing p

i

j

The connected component labelling

• Sequential algorithm

Initially: lbl (label image) 0; lval (current label) 0; t is the current flat zone

• City-block metric:

• Chessboard metric

• Euclidean distance

Distance functions

= backward 4- or 8-connected neighbors

= forward 4- or 8-connected neighbors