<s>
Erosion	B-Algorithm
(	O
usually	O
represented	O
by	O
⊖	O
)	O
is	O
one	O
of	O
two	O
fundamental	O
operations	O
(	O
the	O
other	O
being	O
dilation	B-Algorithm
)	O
in	O
morphological	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
from	O
which	O
all	O
other	O
morphological	B-Algorithm
operations	I-Algorithm
are	O
based	O
.	O
</s>
<s>
It	O
was	O
originally	O
defined	O
for	O
binary	B-Algorithm
images	I-Algorithm
,	O
later	O
being	O
extended	O
to	O
grayscale	O
images	O
,	O
and	O
subsequently	O
to	O
complete	O
lattices	O
.	O
</s>
<s>
The	O
erosion	B-Algorithm
operation	O
usually	O
uses	O
a	O
structuring	B-Algorithm
element	I-Algorithm
for	O
probing	O
and	O
reducing	O
the	O
shapes	O
contained	O
in	O
the	O
input	O
image	O
.	O
</s>
<s>
This	O
simple	O
"	O
probe	O
"	O
is	O
called	O
structuring	B-Algorithm
element	I-Algorithm
,	O
and	O
is	O
itself	O
a	O
binary	B-Algorithm
image	I-Algorithm
(	O
i.e.	O
,	O
a	O
subset	O
of	O
the	O
space	O
or	O
grid	O
)	O
.	O
</s>
<s>
Let	O
E	O
be	O
a	O
Euclidean	O
space	O
or	O
an	O
integer	O
grid	O
,	O
and	O
A	O
a	O
binary	B-Algorithm
image	I-Algorithm
in	O
E	O
.	O
</s>
<s>
The	O
erosion	B-Algorithm
of	O
the	O
binary	B-Algorithm
image	I-Algorithm
A	O
by	O
the	O
structuring	B-Algorithm
element	I-Algorithm
B	O
is	O
defined	O
by	O
:	O
</s>
<s>
When	O
the	O
structuring	B-Algorithm
element	I-Algorithm
B	O
has	O
a	O
center	O
(	O
e.g.	O
,	O
a	O
disk	O
or	O
a	O
square	O
)	O
,	O
and	O
this	O
center	O
is	O
located	O
on	O
the	O
origin	O
of	O
E	O
,	O
then	O
the	O
erosion	B-Algorithm
of	O
A	O
by	O
B	O
can	O
be	O
understood	O
as	O
the	O
locus	O
of	O
points	O
reached	O
by	O
the	O
center	O
of	O
B	O
when	O
B	O
moves	O
inside	O
A	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
erosion	B-Algorithm
of	O
a	O
square	O
of	O
side	O
10	O
,	O
centered	O
at	O
the	O
origin	O
,	O
by	O
a	O
disc	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
6	O
centered	O
at	O
the	O
origin	O
.	O
</s>
<s>
The	O
erosion	B-Algorithm
of	O
A	O
by	O
B	O
is	O
also	O
given	O
by	O
the	O
expression	O
:	O
,	O
where	O
A−b	O
denotes	O
the	O
translation	O
of	O
A	O
by	O
-b	O
.	O
</s>
<s>
This	O
is	O
more	O
generally	O
also	O
known	O
as	O
a	O
Minkowski	B-Algorithm
difference	I-Algorithm
.	O
</s>
<s>
Therefore	O
the	O
Erosion	B-Algorithm
of	O
A	O
by	O
B	O
is	O
given	O
by	O
this	O
13	O
x	O
13	O
matrix	O
.	O
</s>
<s>
The	O
erosion	B-Algorithm
is	O
translation	O
invariant	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
the	O
erosion	B-Algorithm
is	O
anti-extensive	O
,	O
i.e.	O
,	O
.	O
</s>
<s>
The	O
erosion	B-Algorithm
satisfies	O
,	O
where	O
denotes	O
the	O
morphological	B-Algorithm
dilation	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
the	O
erosion	B-Algorithm
of	O
a	O
point	O
is	O
the	O
minimum	O
of	O
the	O
points	O
in	O
its	O
neighborhood	O
,	O
with	O
that	O
neighborhood	O
defined	O
by	O
the	O
structuring	B-Algorithm
element	I-Algorithm
.	O
</s>
<s>
In	O
this	O
way	O
it	O
is	O
similar	O
to	O
many	O
other	O
kinds	O
of	O
image	O
filters	O
like	O
the	O
median	B-Algorithm
filter	I-Algorithm
and	O
the	O
gaussian	O
filter	O
.	O
</s>
<s>
An	O
erosion	B-Algorithm
in	O
is	O
any	O
operator	O
that	O
distributes	O
over	O
the	O
infimum	O
,	O
and	O
preserves	O
the	O
universe	O
.	O
</s>
