MOVE-IIb

Additional information MOVE-IIb

In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set




A
+
B
=
{

a

+

b



|



a

?
A
,


b

?
B
}
.


{\displaystyle A+B=\{\mathbf {a} +\mathbf {b} \,|\,\mathbf {a} \in A,\ \mathbf {b} \in B\}.}
Analogously, the Minkowski difference (or geometric difference) is defined using the complement operation as




A
?
B
=
(

A

c


+
B

)

c




{\displaystyle A-B=(A^{c}+B)^{c}}
In general



A
?
B
?
A
+
(
?
B
)


{\displaystyle A-B\neq A+(-B)}
. For instance, in a one-dimensional case



A
=
[
?
2
,
2
]


{\displaystyle A=[-2,2]}
and



B
=
[
?
1
,
1
]


{\displaystyle B=[-1,1]}
the Minkowski difference



A
?
B
=
[
?
1
,
1
]


{\displaystyle A-B=[-1,1]}
, whereas



A
+
(
?
B
)
=
A
+
B
=
[
?
3
,
3
]
.


{\displaystyle A+(-B)=A+B=[-3,3].}

In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing.

The concept is named for Hermann Minkowski.

Average inclination MOVE-IIb