<s>
In	O
the	O
study	O
of	O
image	B-Algorithm
processing	I-Algorithm
,	O
a	O
watershed	B-Algorithm
is	O
a	O
transformation	O
defined	O
on	O
a	O
grayscale	O
image	O
.	O
</s>
<s>
The	O
name	O
refers	O
metaphorically	O
to	O
a	O
geological	O
watershed	B-Algorithm
,	O
or	O
drainage	O
divide	O
,	O
which	O
separates	O
adjacent	O
drainage	O
basins	O
.	O
</s>
<s>
The	O
watershed	B-Algorithm
transformation	O
treats	O
the	O
image	O
it	O
operates	O
upon	O
like	O
a	O
topographic	O
map	O
,	O
with	O
the	O
brightness	O
of	O
each	O
point	O
representing	O
its	O
height	O
,	O
and	O
finds	O
the	O
lines	O
that	O
run	O
along	O
the	O
tops	O
of	O
ridges	O
.	O
</s>
<s>
There	O
are	O
different	O
technical	O
definitions	O
of	O
a	O
watershed	B-Algorithm
.	O
</s>
<s>
In	O
graphs	O
,	O
watershed	B-Algorithm
lines	O
may	O
be	O
defined	O
on	O
the	O
nodes	O
,	O
on	O
the	O
edges	O
,	O
or	O
hybrid	O
lines	O
on	O
both	O
nodes	O
and	O
edges	O
.	O
</s>
<s>
Watersheds	B-Algorithm
may	O
also	O
be	O
defined	O
in	O
the	O
continuous	O
domain	O
.	O
</s>
<s>
There	O
are	O
also	O
many	O
different	O
algorithms	O
to	O
compute	O
watersheds	B-Algorithm
.	O
</s>
<s>
Watershed	B-Algorithm
algorithms	I-Algorithm
are	O
used	O
in	O
image	B-Algorithm
processing	I-Algorithm
primarily	O
for	O
object	O
segmentation	B-Algorithm
purposes	O
,	O
that	O
is	O
,	O
for	O
separating	O
different	O
objects	O
in	O
an	O
image	O
.	O
</s>
<s>
In	O
geology	O
,	O
a	O
watershed	B-Algorithm
is	O
a	O
divide	O
that	O
separates	O
adjacent	O
catchment	O
basins	O
.	O
</s>
<s>
The	O
idea	O
was	O
introduced	O
in	O
1979	O
by	O
S	O
.	O
Beucher	O
and	O
C	O
.	O
Lantuéjoul.	O
<	O
ref>Serge	O
Beucher	O
and	O
Christian	O
Lantuéj	O
workshop	O
on	O
image	B-Algorithm
processing	I-Algorithm
,	O
real-time	O
edge	O
and	O
motion	O
detection	O
(	O
1979	O
)	O
.	O
</s>
<s>
http://cmm.ensmp.fr/~beucher/publi/watershed.pdfxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx1	O
The	O
basic	O
idea	O
consisted	O
of	O
placing	O
a	O
water	O
source	O
in	O
each	O
regional	O
minimum	O
in	O
the	O
relief	O
,	O
to	O
flood	O
the	O
entire	O
relief	O
from	O
sources	O
,	O
and	O
build	O
barriers	O
when	O
different	O
water	O
sources	O
meet	O
.	O
</s>
<s>
The	O
resulting	O
set	O
of	O
barriers	O
constitutes	O
a	O
watershed	B-Algorithm
by	O
flooding	O
.	O
</s>
<s>
Priority-flood	O
:	O
An	O
optimal	O
depression-filling	O
and	O
watershed-labeling	O
algorithm	O
for	O
digital	O
elevation	O
models	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
watershed	B-Algorithm
is	O
a	O
separation	O
of	O
the	O
regional	O
minima	O
from	O
which	O
a	O
drop	O
of	O
water	O
can	O
flow	O
down	O
towards	O
distinct	O
minima	O
.	O
</s>
<s>
Watershed	B-Algorithm
Cuts	O
:	O
Minimum	O
Spanning	O
Forests	O
and	O
the	O
Drop	O
of	O
Water	O
Principle	O
,	O
IEEE	O
Transactions	O
on	O
Pattern	O
Analysis	O
and	O
Machine	O
Intelligence	O
31(8 )	O
pp	O
.	O
</s>
<s>
1362-1374	O
,	O
2009	O
,	O
for	O
defining	O
a	O
watershed	B-Algorithm
of	O
an	O
edge-weighted	O
graph	O
.	O
</s>
<s>
S	O
.	O
Beucher	O
and	O
F	O
.	O
Meyer	O
introduced	O
an	O
algorithmic	O
inter-pixel	O
implementation	O
of	O
the	O
watershed	B-Algorithm
method	O
,	O
Serge	O
Beucher	O
and	O
Fernand	O
Meyer	O
.	O
</s>
<s>
The	O
morphological	O
approach	O
to	O
segmentation	B-Algorithm
:	O
the	O
watershed	B-Algorithm
transformation	O
.	O
</s>
<s>
In	O
Mathematical	O
Morphology	O
in	O
Image	B-Algorithm
Processing	I-Algorithm
(	O
Ed	O
.	O
</s>
<s>
The	O
topological	O
watershed	B-Algorithm
was	O
introduced	O
by	O
M	O
.	O
Couprie	O
and	O
G	O
.	O
Bertrand	O
in	O
1997	O
,	O
M	O
.	O
</s>
<s>
Topological	O
gray-scale	O
watershed	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
function	O
W	O
is	O
a	O
watershed	B-Algorithm
of	O
a	O
function	O
F	O
if	O
and	O
only	O
if	O
W	O
≤	O
F	O
and	O
W	O
preserves	O
the	O
contrast	O
between	O
the	O
regional	O
minima	O
of	O
F	O
;	O
where	O
the	O
contrast	O
between	O
two	O
regional	O
minima	O
M1	O
and	O
M2	O
is	O
defined	O
as	O
the	O
minimal	O
altitude	O
to	O
which	O
one	O
must	O
climb	O
in	O
order	O
to	O
go	O
from	O
M1	O
to	O
M2.G	O
.	O
</s>
<s>
On	O
topological	O
watersheds	B-Algorithm
.	O
</s>
<s>
Quasi-linear	O
algorithms	O
for	O
the	O
topological	O
watershed	B-Algorithm
.	O
</s>
<s>
Different	O
approaches	O
may	O
be	O
employed	O
to	O
use	O
the	O
watershed	B-Algorithm
principle	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
.	O
</s>
<s>
Local	O
minima	O
of	O
the	O
gradient	O
of	O
the	O
image	O
may	O
be	O
chosen	O
as	O
markers	O
,	O
in	O
this	O
case	O
an	O
over-segmentation	O
is	O
produced	O
and	O
a	O
second	O
step	O
involves	O
region	O
merging	O
.	O
</s>
<s>
Marker	O
based	O
watershed	B-Algorithm
transformation	O
make	O
use	O
of	O
specific	O
marker	O
positions	O
which	O
have	O
been	O
either	O
explicitly	O
defined	O
by	O
the	O
user	O
or	O
determined	O
automatically	O
with	O
morphological	O
operators	O
or	O
other	O
ways	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
common	O
watershed	B-Algorithm
algorithms	I-Algorithm
was	O
introduced	O
by	O
F	O
.	O
Meyer	O
in	O
the	O
early	O
1990s	O
,	O
though	O
a	O
number	O
of	O
improvements	O
,	O
collectively	O
called	O
Priority-Flood	O
,	O
have	O
since	O
been	O
made	O
to	O
this	O
algorithm	O
,	O
Barnes	O
,	O
R.	O
,	O
Lehman	O
,	O
C.	O
,	O
Mulla	O
,	O
D.	O
,	O
2014	O
.	O
</s>
<s>
Priority-flood	O
:	O
An	O
optimal	O
depression-filling	O
and	O
watershed-labeling	O
algorithm	O
for	O
digital	O
elevation	O
models	O
.	O
</s>
<s>
During	O
the	O
successive	O
flooding	O
of	O
the	O
grey	O
value	O
relief	O
,	O
watersheds	B-Algorithm
with	O
adjacent	O
catchment	O
basins	O
are	O
constructed	O
.	O
</s>
<s>
Normally	O
this	O
will	O
lead	O
to	O
an	O
over-segmentation	O
of	O
the	O
image	O
,	O
especially	O
for	O
noisy	O
image	O
material	O
,	O
e.g.	O
</s>
<s>
The	O
non-labeled	O
pixels	O
are	O
the	O
watershed	B-Algorithm
lines	O
.	O
</s>
<s>
thumb|right|Example	O
of	O
a	O
marker-supported	O
watershed	B-Algorithm
transformation	O
for	O
a	O
population	O
of	O
pharmaceutical	O
pellets	O
.	O
</s>
<s>
Watershed	B-Algorithm
lines	O
are	O
superimposed	O
in	O
black	O
on	O
the	O
CT	O
image	O
stack.Doerr	O
,	O
F	O
.	O
J	O
.	O
S.	O
,	O
&	O
Florence	O
,	O
A	O
.	O
J	O
.	O
</s>
<s>
Watersheds	B-Algorithm
as	O
optimal	O
spanning	O
forest	O
have	O
been	O
introduced	O
by	O
Jean	O
Cousty	O
et	O
al.Jean	O
Cousty	O
,	O
Gilles	O
Bertrand	O
,	O
Laurent	O
Najman	O
,	O
and	O
Michel	O
Couprie	O
.	O
</s>
<s>
Watershed	B-Algorithm
Cuts	O
:	O
Minimum	O
Spanning	O
Forests	O
and	O
the	O
Drop	O
of	O
Water	O
Principle	O
.	O
</s>
<s>
They	O
establish	O
the	O
consistency	O
of	O
these	O
watersheds	B-Algorithm
:	O
they	O
can	O
be	O
equivalently	O
defined	O
by	O
their	O
“	O
catchment	O
basins	O
”	O
(	O
through	O
a	O
steepest	O
descent	O
property	O
)	O
or	O
by	O
the	O
“	O
dividing	O
lines	O
”	O
separating	O
these	O
catchment	O
basins	O
(	O
through	O
the	O
drop	O
of	O
water	O
principle	O
)	O
.	O
</s>
<s>
"	O
Some	O
links	O
between	O
min-cuts	O
,	O
optimal	O
spanning	O
forests	O
and	O
watersheds	B-Algorithm
"	O
,	O
Image	O
and	O
Vision	O
Computing	O
,	O
2009	O
.	O
established	O
links	O
relating	O
Graph	B-Algorithm
Cuts	I-Algorithm
to	O
optimal	O
spanning	O
forests	O
.	O
</s>
<s>
More	O
precisely	O
,	O
they	O
show	O
that	O
when	O
the	O
power	O
of	O
the	O
weights	O
of	O
the	O
graph	O
is	O
above	O
a	O
certain	O
number	O
,	O
the	O
cut	O
minimizing	O
the	O
graph	B-Algorithm
cuts	I-Algorithm
energy	O
is	O
a	O
cut	O
by	O
maximum	O
spanning	O
forest	O
.	O
</s>
<s>
The	O
image	B-Algorithm
foresting	I-Algorithm
transform	I-Algorithm
(	O
IFT	O
)	O
of	O
Falcao	O
et	O
al.Falcao	O
,	O
A.X.	O
</s>
<s>
Stolfi	O
,	O
J	O
.	O
de	O
Alencar	O
Lotufo	O
,	O
R	O
.	O
:	O
"	O
The	O
image	B-Algorithm
foresting	I-Algorithm
transform	I-Algorithm
:	O
theory	O
,	O
algorithms	O
,	O
and	O
applications	O
"	O
,	O
In	O
PAMI	O
,	O
2004	O
is	O
a	O
procedure	O
for	O
computing	O
shortest	O
path	O
forests	O
.	O
</s>
<s>
Watershed	B-Algorithm
cuts	O
:	O
thinnings	O
,	O
shortest-path	O
forests	O
and	O
topological	O
watersheds	B-Algorithm
.	O
</s>
<s>
that	O
when	O
the	O
markers	O
of	O
the	O
IFT	O
corresponds	O
to	O
extrema	O
of	O
the	O
weight	O
function	O
,	O
the	O
cut	O
induced	O
by	O
the	O
forest	O
is	O
a	O
watershed	B-Algorithm
cut	O
.	O
</s>
<s>
The	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
segmentation	B-Algorithm
algorithm	O
solving	O
the	O
combinatorial	O
Dirichlet	O
problem	O
,	O
adapted	O
to	O
image	B-Algorithm
segmentation	I-Algorithm
by	O
L	O
.	O
Grady	O
in	O
2006.Grady	O
,	O
L.	O
:	O
"	O
Random	O
walks	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
"	O
.	O
</s>
<s>
proved	O
that	O
when	O
the	O
power	O
of	O
the	O
weights	O
of	O
the	O
graph	O
converge	O
toward	O
infinity	O
,	O
the	O
cut	O
minimizing	O
the	O
random	B-Algorithm
walker	I-Algorithm
energy	O
is	O
a	O
cut	O
by	O
maximum	O
spanning	O
forest.Camille	O
Couprie	O
,	O
Leo	O
Grady	O
,	O
Laurent	O
Najman	O
and	O
Hugues	O
Talbot	O
,	O
"	O
Power	O
Watersheds	B-Algorithm
:	O
A	O
Unifying	O
Graph-Based	O
Optimization	O
Framework	O
”	O
,	O
IEEE	O
Transactions	O
on	O
Pattern	O
Analysis	O
and	O
Machine	O
Intelligence	O
,	O
Vol	O
.	O
</s>
<s>
A	O
hierarchical	O
watershed	B-Algorithm
transformation	O
converts	O
the	O
result	O
into	O
a	O
graph	O
display	O
(	O
i.e.	O
</s>
<s>
the	O
neighbor	O
relationships	O
of	O
the	O
segmented	O
regions	O
are	O
determined	O
)	O
and	O
applies	O
further	O
watershed	B-Algorithm
transformations	O
recursively	O
.	O
</s>
<s>
Geodesic	O
Saliency	O
of	O
Watershed	B-Algorithm
Contours	O
and	O
Hierarchical	O
Segmentation	B-Algorithm
.	O
</s>
<s>
A	O
theory	O
linking	O
watershed	B-Algorithm
to	O
hierarchical	O
segmentations	B-Algorithm
has	O
been	O
developed	O
inLaurent	O
Najman	O
.	O
</s>
<s>
On	O
the	O
equivalence	O
between	O
hierarchical	O
segmentations	B-Algorithm
and	O
ultrametric	O
watersheds	B-Algorithm
.	O
</s>
<s>
Watersheds	B-Algorithm
in	O
digital	O
spaces	O
:	O
an	O
efficient	O
algorithm	O
based	O
on	O
immersion	O
simulations	O
.	O
</s>
