<s>
In	O
computer	B-Application
vision	I-Application
and	O
image	B-Algorithm
processing	I-Algorithm
,	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
named	O
after	O
,	O
is	O
used	O
to	O
perform	O
automatic	O
image	O
thresholding	B-Algorithm
.	O
</s>
<s>
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
one-dimensional	O
discrete	O
analogue	O
of	O
Fisher	O
's	O
Discriminant	O
Analysis	O
,	O
is	O
related	O
to	O
Jenks	B-Algorithm
optimization	I-Algorithm
method	I-Algorithm
,	O
and	O
is	O
equivalent	O
to	O
a	O
globally	O
optimal	O
k-means	B-Algorithm
performed	O
on	O
the	O
intensity	O
histogram	O
.	O
</s>
<s>
The	O
extension	O
to	O
multi-level	O
thresholding	B-Algorithm
was	O
described	O
in	O
the	O
original	O
paper	O
,	O
and	O
computationally	O
efficient	O
implementations	O
have	O
since	O
been	O
proposed	O
.	O
</s>
<s>
Matlab	O
has	O
built-in	O
functions	O
graythresh( )	O
and	O
multithresh( )	O
in	O
the	O
Image	B-Algorithm
Processing	I-Algorithm
Toolbox	O
which	O
are	O
implemented	O
with	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
and	O
Multi	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
respectively	O
.	O
</s>
<s>
This	O
implementation	O
requires	O
the	O
NumPy	B-Application
library.Python	O
libraries	O
dedicated	O
to	O
image	B-Algorithm
processing	I-Algorithm
such	O
as	O
OpenCV	B-Language
and	O
Scikit-image	B-Library
propose	O
built-in	O
implementations	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
performs	O
well	O
when	O
the	O
histogram	O
has	O
a	O
bimodal	O
distribution	O
with	O
a	O
deep	O
and	O
sharp	O
valley	O
between	O
the	O
two	O
peaks	O
.	O
</s>
<s>
Like	O
all	O
other	O
global	O
thresholding	B-Algorithm
methods	O
,	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
performs	O
badly	O
in	O
case	O
of	O
heavy	O
noise	O
,	O
small	O
objects	O
size	O
,	O
inhomogeneous	O
lighting	O
and	O
larger	O
intra-class	O
than	O
inter-class	O
variance	O
.	O
</s>
<s>
In	O
those	O
cases	O
,	O
local	O
adaptations	O
of	O
the	O
Otsu	B-Algorithm
method	I-Algorithm
have	O
been	O
developed	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
mathematical	O
grounding	O
of	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
models	O
the	O
histogram	O
of	O
the	O
image	O
as	O
a	O
mixture	O
of	O
two	O
Normal	O
distributions	O
with	O
equal	O
variance	O
and	O
equal	O
size	O
.	O
</s>
<s>
Otsu	O
's	O
thresholding	B-Algorithm
may	O
however	O
yield	O
satisfying	O
results	O
even	O
when	O
these	O
assumptions	O
are	O
not	O
met	O
,	O
in	O
the	O
same	O
way	O
statistical	O
tests	O
(	O
to	O
which	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
heavily	O
connected	O
)	O
can	O
perform	O
correctly	O
even	O
when	O
the	O
working	O
assumptions	O
are	O
not	O
fully	O
satisfied	O
.	O
</s>
<s>
However	O
,	O
several	O
variations	O
of	O
Otsu	B-Algorithm
's	I-Algorithm
methods	I-Algorithm
have	O
been	O
proposed	O
to	O
account	O
for	O
more	O
severe	O
deviations	O
from	O
these	O
assumptions	O
,	O
such	O
as	O
the	O
Kittler-Illingworth	O
method	O
.	O
</s>
<s>
A	O
popular	O
local	O
adaptation	O
is	O
the	O
two-dimensional	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
which	O
performs	O
better	O
for	O
the	O
object	O
segmentation	O
task	O
in	O
noisy	O
images	O
.	O
</s>
<s>
And	O
the	O
2-dimensional	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
developed	O
based	O
on	O
the	O
2-dimensional	O
histogram	O
as	O
follows	O
.	O
</s>
<s>
Similar	O
to	O
one-dimensional	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
the	O
optimal	O
threshold	O
is	O
obtained	O
by	O
maximizing	O
.	O
</s>
<s>
The	O
and	O
is	O
obtained	O
iteratively	O
which	O
is	O
similar	O
with	O
one-dimensional	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
However	O
,	O
even	O
with	O
the	O
dynamic	O
programming	O
approach	O
,	O
2d	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
still	O
has	O
large	O
time	O
complexity	O
.	O
</s>
<s>
The	O
Kittler-Illingworth	O
algorithm	O
(	O
also	O
known	O
as	O
Minimum	O
Error	O
thresholding	B-Algorithm
)	O
is	O
a	O
variation	O
of	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
to	O
handle	O
such	O
cases	O
.	O
</s>
<s>
While	O
this	O
algorithm	O
could	O
seem	O
superior	O
to	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
it	O
introduces	O
new	O
parameters	O
to	O
be	O
estimated	O
,	O
and	O
this	O
can	O
result	O
in	O
the	O
algorithm	O
being	O
over-parametrized	O
and	O
thus	O
unstable	O
.	O
</s>
<s>
In	O
many	O
cases	O
where	O
the	O
assumptions	O
from	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
seem	O
at	O
least	O
partially	O
valid	O
,	O
it	O
may	O
be	O
preferable	O
to	O
favor	O
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
over	O
the	O
Kittler-Illingworth	O
algorithm	O
,	O
following	O
Occam	O
's	O
razor	O
.	O
</s>
<s>
Iterative	O
triclass	O
thresholding	B-Algorithm
algorithm	O
is	O
a	O
variation	O
of	O
the	O
Otsu	O
’s	O
method	O
to	O
circumvent	O
this	O
limitation	O
.	O
</s>
<s>
Given	O
an	O
image	O
,	O
at	O
the	O
first	O
iteration	O
,	O
the	O
triclass	O
thresholding	B-Algorithm
algorithm	O
calculates	O
a	O
threshold	O
using	O
the	O
Otsu	O
’s	O
method	O
.	O
</s>
