<s>
Dilation	B-Algorithm
(	O
usually	O
represented	O
by	O
⊕	O
)	O
is	O
one	O
of	O
the	O
basic	O
operations	O
in	O
mathematical	B-Algorithm
morphology	I-Algorithm
.	O
</s>
<s>
Originally	O
developed	O
for	O
binary	B-Algorithm
images	I-Algorithm
,	O
it	O
has	O
been	O
expanded	O
first	O
to	O
grayscale	O
images	O
,	O
and	O
then	O
to	O
complete	O
lattices	O
.	O
</s>
<s>
The	O
dilation	B-Algorithm
operation	O
usually	O
uses	O
a	O
structuring	B-Algorithm
element	I-Algorithm
for	O
probing	O
and	O
expanding	O
the	O
shapes	O
contained	O
in	O
the	O
input	O
image	O
.	O
</s>
<s>
In	O
binary	O
morphology	O
,	O
dilation	B-Algorithm
is	O
a	O
shift-invariant	O
(	O
translation	O
invariant	O
)	O
operator	O
,	O
equivalent	O
to	O
Minkowski	B-Algorithm
addition	I-Algorithm
.	O
</s>
<s>
A	O
binary	B-Algorithm
image	I-Algorithm
is	O
viewed	O
in	O
mathematical	B-Algorithm
morphology	I-Algorithm
as	O
a	O
subset	O
of	O
a	O
Euclidean	O
space	O
Rd	O
or	O
the	O
integer	O
grid	O
Zd	O
,	O
for	O
some	O
dimension	O
d	O
.	O
Let	O
E	O
be	O
a	O
Euclidean	O
space	O
or	O
an	O
integer	O
grid	O
,	O
A	O
a	O
binary	B-Algorithm
image	I-Algorithm
in	O
E	O
,	O
and	O
B	O
a	O
structuring	B-Algorithm
element	I-Algorithm
regarded	O
as	O
a	O
subset	O
of	O
Rd	O
.	O
</s>
<s>
Dilation	B-Algorithm
is	O
commutative	O
,	O
also	O
given	O
by	O
.	O
</s>
<s>
If	O
B	O
has	O
a	O
center	O
on	O
the	O
origin	O
,	O
then	O
the	O
dilation	B-Algorithm
of	O
A	O
by	O
B	O
can	O
be	O
understood	O
as	O
the	O
locus	O
of	O
the	O
points	O
covered	O
by	O
B	O
when	O
the	O
center	O
of	O
B	O
moves	O
inside	O
A	O
.	O
</s>
<s>
The	O
dilation	B-Algorithm
of	O
a	O
square	O
of	O
size	O
10	O
,	O
centered	O
at	O
the	O
origin	O
,	O
by	O
a	O
disk	O
of	O
radius	O
2	O
,	O
also	O
centered	O
at	O
the	O
origin	O
,	O
is	O
a	O
square	O
of	O
side	O
14	O
,	O
with	O
rounded	O
corners	O
,	O
centered	O
at	O
the	O
origin	O
.	O
</s>
<s>
The	O
dilation	B-Algorithm
can	O
also	O
be	O
obtained	O
by	O
,	O
where	O
Bs	O
denotes	O
the	O
symmetric	O
of	O
B	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
Each	O
pixel	O
of	O
every	O
superimposed	O
B	O
is	O
included	O
in	O
the	O
dilation	B-Algorithm
of	O
A	O
by	O
B	O
.	O
</s>
<s>
The	O
dilation	B-Algorithm
of	O
A	O
by	O
B	O
is	O
given	O
by	O
this	O
11	O
x	O
11	O
matrix	O
.	O
</s>
<s>
If	O
the	O
origin	O
of	O
E	O
belongs	O
to	O
the	O
structuring	B-Algorithm
element	I-Algorithm
B	O
,	O
then	O
it	O
is	O
extensive	O
,	O
i.e.	O
,	O
.	O
</s>
<s>
Grayscale	O
structuring	B-Algorithm
elements	I-Algorithm
are	O
also	O
functions	O
of	O
the	O
same	O
format	O
,	O
called	O
"	O
structuring	O
functions	O
"	O
.	O
</s>
<s>
It	O
is	O
common	O
to	O
use	O
flat	O
structuring	B-Algorithm
elements	I-Algorithm
in	O
morphological	O
applications	O
.	O
</s>
<s>
Thus	O
,	O
dilation	B-Algorithm
is	O
a	O
particular	O
case	O
of	O
order	B-General_Concept
statistics	I-General_Concept
filters	O
,	O
returning	O
the	O
maximum	O
value	O
within	O
a	O
moving	O
window	O
(	O
the	O
symmetric	O
of	O
the	O
structuring	O
function	O
support	O
B	O
)	O
.	O
</s>
<s>
A	O
dilation	B-Algorithm
is	O
any	O
operator	O
that	O
distributes	O
over	O
the	O
supremum	O
,	O
and	O
preserves	O
the	O
least	O
element	O
.	O
</s>
