<s>
In	O
mathematical	B-Algorithm
morphology	I-Algorithm
,	O
granulometry	B-Algorithm
is	O
an	O
approach	O
to	O
compute	O
a	O
size	O
distribution	O
of	O
grains	O
in	O
binary	B-Algorithm
images	I-Algorithm
,	O
using	O
a	O
series	O
of	O
morphological	B-Algorithm
opening	I-Algorithm
operations	O
.	O
</s>
<s>
It	O
was	O
introduced	O
by	O
Georges	O
Matheron	O
in	O
the	O
1960s	O
,	O
and	O
is	O
the	O
basis	O
for	O
the	O
characterization	O
of	O
the	O
concept	O
of	O
size	O
in	O
mathematical	B-Algorithm
morphology	I-Algorithm
.	O
</s>
<s>
Let	O
B	O
be	O
a	O
structuring	B-Algorithm
element	I-Algorithm
in	O
a	O
Euclidean	O
space	O
or	O
grid	O
E	O
,	O
and	O
consider	O
the	O
family	O
,	O
,	O
given	O
by	O
:	O
</s>
<s>
where	O
denotes	O
morphological	B-Algorithm
dilation	I-Algorithm
.	O
</s>
<s>
Let	O
X	O
be	O
a	O
set	O
(	O
i.e.	O
,	O
a	O
binary	B-Algorithm
image	I-Algorithm
in	O
mathematical	B-Algorithm
morphology	I-Algorithm
)	O
,	O
and	O
consider	O
the	O
series	O
of	O
sets	O
,	O
,	O
given	O
by	O
:	O
</s>
<s>
where	O
denotes	O
the	O
morphological	B-Algorithm
opening	I-Algorithm
.	O
</s>
<s>
The	O
granulometry	B-Algorithm
function	O
is	O
the	O
cardinality	B-Application
(	O
i.e.	O
,	O
area	O
or	O
volume	O
,	O
in	O
continuous	O
Euclidean	O
space	O
,	O
or	O
number	O
of	O
elements	O
,	O
in	O
grids	O
)	O
of	O
the	O
image	O
:	O
</s>
<s>
The	O
pattern	B-Algorithm
spectrum	I-Algorithm
or	O
size	O
distribution	O
of	O
X	O
is	O
the	O
collection	O
of	O
sets	O
,	O
,	O
given	O
by	O
:	O
</s>
<s>
The	O
parameter	O
k	O
is	O
referred	O
to	O
as	O
size	O
,	O
and	O
the	O
component	O
k	O
of	O
the	O
pattern	B-Algorithm
spectrum	I-Algorithm
provides	O
a	O
rough	O
estimate	O
for	O
the	O
amount	O
of	O
grains	O
of	O
size	O
k	O
in	O
the	O
image	O
X	O
.	O
</s>
<s>
A	O
granulometry-generating	O
family	O
of	O
operators	O
should	O
satisfy	O
the	O
above	O
three	O
axioms	O
.	O
</s>
<s>
In	O
the	O
above	O
case	O
(	O
granulometry	B-Algorithm
generated	O
by	O
a	O
structuring	B-Algorithm
element	I-Algorithm
)	O
,	O
.	O
</s>
<s>
Another	O
example	O
of	O
granulometry-generating	O
family	O
is	O
when	O
,	O
where	O
is	O
a	O
set	O
of	O
linear	O
structuring	B-Algorithm
elements	I-Algorithm
with	O
different	O
directions	O
.	O
</s>
