<s>
In	O
mathematical	B-Algorithm
morphology	I-Algorithm
,	O
opening	O
is	O
the	O
dilation	B-Algorithm
of	O
the	O
erosion	B-Algorithm
of	O
a	O
set	O
A	O
by	O
a	O
structuring	B-Algorithm
element	I-Algorithm
B	O
:	O
</s>
<s>
where	O
and	O
denote	O
erosion	B-Algorithm
and	O
dilation	B-Algorithm
,	O
respectively	O
.	O
</s>
<s>
Together	O
with	O
closing	B-Algorithm
,	O
the	O
opening	O
serves	O
in	O
computer	B-Application
vision	I-Application
and	O
image	B-Algorithm
processing	I-Algorithm
as	O
a	O
basic	O
workhorse	O
of	O
morphological	O
noise	O
removal	O
.	O
</s>
<s>
Opening	O
removes	O
small	O
objects	O
from	O
the	O
foreground	O
(	O
usually	O
taken	O
as	O
the	O
bright	O
pixels	O
)	O
of	O
an	O
image	O
,	O
placing	O
them	O
in	O
the	O
background	O
,	O
while	O
closing	B-Algorithm
removes	O
small	O
holes	O
in	O
the	O
foreground	O
,	O
changing	O
small	O
islands	O
of	O
background	O
into	O
foreground	O
.	O
</s>
<s>
Opening	O
can	O
be	O
used	O
to	O
find	O
things	O
into	O
which	O
a	O
specific	O
structuring	B-Algorithm
element	I-Algorithm
can	O
fit	O
(	O
edges	O
,	O
corners	O
,	O
...	O
)	O
.	O
</s>
<s>
Opening	O
and	O
closing	B-Algorithm
satisfy	O
the	O
duality	O
,	O
where	O
denotes	O
closing	B-Algorithm
.	O
</s>
<s>
In	O
morphological	O
opening	O
,	O
the	O
erosion	B-Algorithm
operation	O
removes	O
objects	O
that	O
are	O
smaller	O
than	O
structuring	B-Algorithm
element	I-Algorithm
B	O
and	O
the	O
dilation	B-Algorithm
operation	O
(	O
approximately	O
)	O
restores	O
the	O
size	O
and	O
shape	O
of	O
the	O
remaining	O
objects	O
.	O
</s>
<s>
However	O
,	O
restoration	O
accuracy	O
in	O
the	O
dilation	B-Algorithm
operation	O
depends	O
highly	O
on	O
the	O
type	O
of	O
structuring	B-Algorithm
element	I-Algorithm
and	O
the	O
shape	O
of	O
the	O
restoring	O
objects	O
.	O
</s>
<s>
The	O
opening	O
by	O
reconstruction	O
method	O
is	O
able	O
to	O
restore	O
the	O
objects	O
more	O
completely	O
after	O
erosion	B-Algorithm
has	O
been	O
applied	O
.	O
</s>
<s>
It	O
is	O
defined	O
as	O
the	O
reconstruction	O
by	O
geodesic	O
dilation	B-Algorithm
of	O
erosions	B-Algorithm
of	O
by	O
with	O
respect	O
to	O
:	O
</s>
<s>
where	O
denotes	O
a	O
marker	O
image	O
and	O
is	O
a	O
mask	O
image	O
in	O
morphological	O
reconstruction	O
by	O
dilation	B-Algorithm
.	O
</s>
<s>
denotes	O
geodesic	O
dilation	B-Algorithm
with	O
iterations	O
until	O
stability	O
,	O
i.e.	O
,	O
such	O
that	O
Since	O
,	O
the	O
marker	O
image	O
is	O
limited	O
in	O
the	O
growth	O
region	O
by	O
the	O
mask	O
image	O
,	O
so	O
the	O
dilation	B-Algorithm
operation	O
on	O
the	O
marker	O
image	O
will	O
not	O
expand	O
beyond	O
the	O
mask	O
image	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
structuring	B-Algorithm
element	I-Algorithm
is	O
an	O
8-pixel	O
vertical	O
line	O
which	O
is	O
applied	O
in	O
the	O
erosion	B-Algorithm
operation	O
in	O
order	O
to	O
find	O
objects	O
of	O
interest	O
.	O
</s>
<s>
Moreover	O
,	O
morphological	O
reconstruction	O
by	O
dilation	B-Algorithm
,	O
iterates	O
times	O
until	O
the	O
resulting	O
image	O
converges	O
.	O
</s>
