<s>
Box	B-Algorithm
counting	I-Algorithm
is	O
a	O
method	O
of	O
gathering	O
data	O
for	O
analyzing	O
complex	O
patterns	O
by	O
breaking	O
a	O
dataset	B-General_Concept
,	O
object	O
,	O
image	O
,	O
etc	O
.	O
</s>
<s>
In	O
box	B-Algorithm
counting	I-Algorithm
,	O
however	O
,	O
rather	O
than	O
changing	O
the	O
magnification	O
or	O
resolution	O
of	O
a	O
lens	O
,	O
the	O
investigator	O
changes	O
the	O
size	O
of	O
the	O
element	O
used	O
to	O
inspect	O
the	O
object	O
or	O
pattern	O
(	O
see	O
Figure	O
1	O
)	O
.	O
</s>
<s>
Computer	O
based	O
box	B-Algorithm
counting	I-Algorithm
algorithms	O
have	O
been	O
applied	O
to	O
patterns	O
in	O
1-	O
,	O
2-	O
,	O
and	O
3-dimensional	O
spaces	O
.	O
</s>
<s>
The	O
technique	O
arose	O
out	O
of	O
and	O
is	O
used	O
in	O
fractal	B-Application
analysis	O
.	O
</s>
<s>
It	O
also	O
has	O
application	O
in	O
related	O
fields	O
such	O
as	O
lacunarity	O
and	O
multifractal	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Theoretically	O
,	O
the	O
intent	O
of	O
box	B-Algorithm
counting	I-Algorithm
is	O
to	O
quantify	O
fractal	B-Application
scaling	O
,	O
but	O
from	O
a	O
practical	O
perspective	O
this	O
would	O
require	O
that	O
the	O
scaling	O
be	O
known	O
ahead	O
of	O
time	O
.	O
</s>
<s>
In	O
fractal	B-Application
analysis	O
,	O
however	O
,	O
the	O
scaling	O
factor	O
is	O
not	O
always	O
known	O
ahead	O
of	O
time	O
,	O
so	O
box	B-Algorithm
counting	I-Algorithm
algorithms	O
attempt	O
to	O
find	O
an	O
optimized	O
way	O
of	O
cutting	O
a	O
pattern	O
up	O
that	O
will	O
reveal	O
the	O
scaling	O
factor	O
.	O
</s>
<s>
To	O
do	O
this	O
,	O
for	O
each	O
in	O
,	O
a	O
measuring	O
element	O
that	O
is	O
typically	O
a	O
2-dimensional	O
square	O
or	O
3-dimensional	O
box	O
with	O
side	O
length	O
corresponding	O
to	O
is	O
used	O
to	O
scan	O
a	O
pattern	O
or	O
data	B-General_Concept
set	I-General_Concept
(	O
e.g.	O
,	O
an	O
image	O
or	O
object	O
)	O
according	O
to	O
a	O
predetermined	O
scanning	O
plan	O
to	O
cover	O
the	O
relevant	O
part	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
,	O
recording	O
,	O
i.e.	O
,	O
counting	O
,	O
for	O
each	O
step	O
in	O
the	O
scan	O
relevant	O
features	O
captured	O
within	O
the	O
measuring	O
element	O
.	O
</s>
<s>
The	O
relevant	O
features	O
gathered	O
during	O
box	B-Algorithm
counting	I-Algorithm
depend	O
on	O
the	O
subject	O
being	O
investigated	O
and	O
the	O
type	O
of	O
analysis	O
being	O
done	O
.	O
</s>
<s>
Two	O
well-studied	O
subjects	O
of	O
box	B-Algorithm
counting	I-Algorithm
,	O
for	O
instance	O
,	O
are	O
binary	O
(	O
meaning	O
having	O
only	O
two	O
colours	O
,	O
usually	O
black	O
and	O
white	O
)	O
and	O
gray-scale	O
digital	B-Algorithm
images	I-Algorithm
(	O
i.e.	O
,	O
jpegs	O
,	O
tiffs	O
,	O
etc	O
.	O
)	O
.	O
</s>
<s>
Box	B-Algorithm
counting	I-Algorithm
is	O
generally	O
done	O
on	O
patterns	O
extracted	O
from	O
such	O
still	O
images	O
in	O
which	O
case	O
the	O
raw	O
information	O
recorded	O
is	O
typically	O
based	O
on	O
features	O
of	O
pixels	O
such	O
as	O
a	O
predetermined	O
colour	O
value	O
or	O
range	O
of	O
colours	O
or	O
intensities	O
.	O
</s>
<s>
When	O
box	B-Algorithm
counting	I-Algorithm
is	O
done	O
to	O
determine	O
a	O
fractal	B-Application
dimension	O
known	O
as	O
the	O
box	B-Algorithm
counting	I-Algorithm
dimension	O
,	O
the	O
information	O
recorded	O
is	O
usually	O
either	O
yes	O
or	O
no	O
as	O
to	O
whether	O
or	O
not	O
the	O
box	O
contained	O
any	O
pixels	O
of	O
the	O
predetermined	O
colour	O
or	O
range	O
(	O
i.e.	O
,	O
the	O
number	O
of	O
boxes	O
containing	O
relevant	O
pixels	O
at	O
each	O
is	O
counted	O
)	O
.	O
</s>
<s>
Every	O
box	B-Algorithm
counting	I-Algorithm
algorithm	O
has	O
a	O
scanning	O
plan	O
that	O
describes	O
how	O
the	O
data	O
will	O
be	O
gathered	O
,	O
in	O
essence	O
,	O
how	O
the	O
box	O
will	O
be	O
moved	O
over	O
the	O
space	O
containing	O
the	O
pattern	O
.	O
</s>
<s>
A	O
variety	O
of	O
scanning	O
strategies	O
has	O
been	O
used	O
in	O
box	B-Algorithm
counting	I-Algorithm
algorithms	O
,	O
where	O
a	O
few	O
basic	O
approaches	O
have	O
been	O
modified	O
in	O
order	O
to	O
address	O
issues	O
such	O
as	O
sampling	O
,	O
analysis	O
methods	O
,	O
etc	O
.	O
</s>
<s>
To	O
illustrate	O
,	O
Figure	O
2a	O
shows	O
the	O
typical	O
pattern	O
used	O
in	O
software	O
that	O
calculates	O
box	B-Algorithm
counting	I-Algorithm
dimensions	O
from	O
patterns	O
extracted	O
into	O
binary	O
digital	B-Algorithm
images	I-Algorithm
of	O
contours	O
such	O
as	O
the	O
fractal	B-Application
contour	O
illustrated	O
in	O
Figure	O
1	O
or	O
the	O
classic	O
example	O
of	O
the	O
coastline	O
of	O
Britain	O
often	O
used	O
to	O
explain	O
the	O
method	O
of	O
finding	O
a	O
box	B-Algorithm
counting	I-Algorithm
dimension	O
.	O
</s>
<s>
When	O
used	O
to	O
find	O
a	O
box	B-Algorithm
counting	I-Algorithm
dimension	O
,	O
the	O
method	O
is	O
modified	O
to	O
find	O
an	O
optimal	O
covering	O
.	O
</s>
<s>
Sliding	O
box	O
algorithms	O
are	O
often	O
used	O
for	O
analyzing	O
textures	O
in	O
lacunarity	O
analysis	O
and	O
have	O
also	O
been	O
applied	O
to	O
multifractal	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Box	B-Algorithm
counting	I-Algorithm
may	O
also	O
be	O
used	O
to	O
determine	O
local	O
variation	O
as	O
opposed	O
to	O
global	O
measures	O
describing	O
an	O
entire	O
pattern	O
.	O
</s>
<s>
Local	O
variation	O
can	O
be	O
assessed	O
after	O
the	O
data	O
have	O
been	O
gathered	O
and	O
analyzed	O
(	O
e.g.	O
,	O
some	O
software	O
colour	O
codes	O
areas	O
according	O
to	O
the	O
fractal	B-Application
dimension	O
for	O
each	O
subsample	O
)	O
,	O
but	O
a	O
third	O
approach	O
to	O
box	B-Algorithm
counting	I-Algorithm
is	O
to	O
move	O
the	O
box	O
according	O
to	O
some	O
feature	O
related	O
to	O
the	O
pixels	O
of	O
interest	O
.	O
</s>
<s>
In	O
local	O
connected	O
dimension	O
box	B-Algorithm
counting	I-Algorithm
algorithms	O
,	O
for	O
instance	O
,	O
the	O
box	O
for	O
each	O
is	O
centred	O
on	O
each	O
pixel	O
of	O
interest	O
,	O
as	O
illustrated	O
in	O
Figure	O
2c	O
.	O
</s>
<s>
The	O
implementation	O
of	O
any	O
box	B-Algorithm
counting	I-Algorithm
algorithm	O
has	O
to	O
specify	O
certain	O
details	O
such	O
as	O
how	O
to	O
determine	O
the	O
actual	O
values	O
in	O
,	O
including	O
the	O
minimum	O
and	O
maximum	O
sizes	O
to	O
use	O
and	O
the	O
method	O
of	O
incrementing	O
between	O
sizes	O
.	O
</s>
<s>
Many	O
such	O
details	O
reflect	O
practical	O
matters	O
such	O
as	O
the	O
size	O
of	O
a	O
digital	B-Algorithm
image	I-Algorithm
but	O
also	O
technical	O
issues	O
related	O
to	O
the	O
specific	O
analysis	O
that	O
will	O
be	O
performed	O
on	O
the	O
data	O
.	O
</s>
<s>
Another	O
issue	O
that	O
has	O
received	O
considerable	O
attention	O
is	O
how	O
to	O
approximate	O
the	O
so-called	O
"	O
optimal	O
covering	O
"	O
for	O
determining	O
box	B-Algorithm
counting	I-Algorithm
dimensions	O
and	O
assessing	O
multifractal	B-Algorithm
scaling	I-Algorithm
.	O
</s>
<s>
One	O
known	O
issue	O
in	O
this	O
respect	O
is	O
deciding	O
what	O
constitutes	O
the	O
edge	O
of	O
the	O
useful	O
information	O
in	O
a	O
digital	B-Algorithm
image	I-Algorithm
,	O
as	O
the	O
limits	O
employed	O
in	O
the	O
box	B-Algorithm
counting	I-Algorithm
strategy	O
can	O
affect	O
the	O
data	O
gathered	O
.	O
</s>
