<s>
Level-set	B-Algorithm
methods	I-Algorithm
(	O
LSM	O
)	O
are	O
a	O
conceptual	O
framework	O
for	O
using	O
level	B-Algorithm
sets	I-Algorithm
as	O
a	O
tool	O
for	O
numerical	B-General_Concept
analysis	I-General_Concept
of	O
surfaces	O
and	O
shapes	O
.	O
</s>
<s>
The	O
advantage	O
of	O
the	O
level-set	O
model	O
is	O
that	O
one	O
can	O
perform	O
numerical	B-General_Concept
computations	I-General_Concept
involving	O
curves	O
and	O
surfaces	O
on	O
a	O
fixed	O
Cartesian	O
grid	O
without	O
having	O
to	O
parameterize	O
these	O
objects	O
(	O
this	O
is	O
called	O
the	O
Eulerian	O
approach	O
)	O
.	O
</s>
<s>
Also	O
,	O
the	O
level-set	B-Algorithm
method	I-Algorithm
makes	O
it	O
very	O
easy	O
to	O
follow	O
shapes	O
that	O
change	O
topology	B-Architecture
,	O
for	O
example	O
,	O
when	O
a	O
shape	O
splits	O
in	O
two	O
,	O
develops	O
holes	O
,	O
or	O
the	O
reverse	O
of	O
these	O
operations	O
.	O
</s>
<s>
All	O
these	O
make	O
the	O
level-set	B-Algorithm
method	I-Algorithm
a	O
great	O
tool	O
for	O
modeling	O
time-varying	O
objects	O
,	O
like	O
inflation	O
of	O
an	O
airbag	O
,	O
or	O
a	O
drop	O
of	O
oil	O
floating	O
in	O
water	O
.	O
</s>
<s>
The	O
figure	O
on	O
the	O
right	O
illustrates	O
several	O
important	O
ideas	O
about	O
the	O
level-set	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Below	O
it	O
,	O
the	O
red	O
surface	O
is	O
the	O
graph	O
of	O
a	O
level	B-Algorithm
set	I-Algorithm
function	O
determining	O
this	O
shape	O
,	O
and	O
the	O
flat	O
blue	O
region	O
represents	O
the	O
xy	O
plane	O
.	O
</s>
<s>
In	O
the	O
top	O
row	O
we	O
see	O
the	O
shape	O
changing	O
its	O
topology	B-Architecture
by	O
splitting	O
in	O
two	O
.	O
</s>
<s>
It	O
would	O
be	O
quite	O
hard	O
to	O
describe	O
this	O
transformation	O
numerically	B-General_Concept
by	O
parameterizing	O
the	O
boundary	O
of	O
the	O
shape	O
and	O
following	O
its	O
evolution	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
we	O
look	O
at	O
the	O
bottom	O
row	O
,	O
we	O
see	O
that	O
the	O
level	B-Algorithm
set	I-Algorithm
function	O
merely	O
translated	O
downward	O
.	O
</s>
<s>
Thus	O
,	O
in	O
two	O
dimensions	O
,	O
the	O
level-set	B-Algorithm
method	I-Algorithm
amounts	O
to	O
representing	O
a	O
closed	O
curve	O
(	O
such	O
as	O
the	O
shape	O
boundary	O
in	O
our	O
example	O
)	O
using	O
an	O
auxiliary	O
function	O
,	O
called	O
the	O
level-set	O
function	O
.	O
</s>
<s>
and	O
the	O
level-set	B-Algorithm
method	I-Algorithm
manipulates	O
implicitly	O
,	O
through	O
the	O
function	O
.	O
</s>
<s>
This	O
is	O
a	O
partial	O
differential	O
equation	O
,	O
in	O
particular	O
a	O
Hamilton	O
–	O
Jacobi	O
equation	O
,	O
and	O
can	O
be	O
solved	O
numerically	B-General_Concept
,	O
for	O
example	O
,	O
by	O
using	O
finite	B-Algorithm
differences	I-Algorithm
on	O
a	O
Cartesian	O
grid	O
.	O
</s>
<s>
The	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
the	O
level-set	O
equation	O
,	O
however	O
,	O
requires	O
sophisticated	O
techniques	O
.	O
</s>
<s>
Simple	O
finite-difference	B-Algorithm
methods	O
fail	O
quickly	O
.	O
</s>
<s>
Upwinding	B-Algorithm
methods	O
,	O
such	O
as	O
the	O
Godunov	O
method	O
,	O
fare	O
better	O
;	O
however	O
,	O
the	O
level-set	B-Algorithm
method	I-Algorithm
does	O
not	O
guarantee	O
the	O
conservation	O
of	O
the	O
volume	O
and	O
the	O
shape	O
of	O
the	O
level	B-Algorithm
set	I-Algorithm
in	O
an	O
advection	O
field	O
that	O
does	O
conserve	O
the	O
shape	O
and	O
size	O
,	O
for	O
example	O
,	O
uniform	O
or	O
rotational	O
velocity	O
field	O
.	O
</s>
<s>
Instead	O
,	O
the	O
shape	O
of	O
the	O
level	B-Algorithm
set	I-Algorithm
may	O
get	O
severely	O
distorted	O
,	O
and	O
the	O
level	B-Algorithm
set	I-Algorithm
may	O
vanish	O
over	O
several	O
time	O
steps	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
high-order	O
finite-difference	B-Algorithm
schemes	O
are	O
generally	O
required	O
,	O
such	O
as	O
high-order	O
essentially	B-Algorithm
non-oscillatory	I-Algorithm
(	O
ENO	O
)	O
schemes	O
,	O
and	O
even	O
then	O
the	O
feasibility	O
of	O
long-time	O
simulations	O
is	O
questionable	O
.	O
</s>
<s>
Further	O
sophisticated	O
methods	O
to	O
deal	O
with	O
this	O
difficulty	O
have	O
been	O
developed	O
,	O
e.g.	O
,	O
combinations	O
of	O
the	O
level-set	B-Algorithm
method	I-Algorithm
with	O
tracing	O
marker	O
particles	O
advected	O
by	O
the	O
velocity	O
field	O
.	O
</s>
<s>
In	O
combustion	B-Application
,	O
this	O
method	O
is	O
used	O
to	O
describe	O
the	O
instantaneous	O
flame	O
surface	O
,	O
known	O
as	O
the	O
G	O
equation	O
.	O
</s>
<s>
The	O
level-set	B-Algorithm
method	I-Algorithm
was	O
developed	O
in	O
1979	O
by	O
Alain	O
Dervieux	O
,	O
and	O
subsequently	O
popularized	O
by	O
Stanley	O
Osher	O
and	O
James	O
Sethian	O
.	O
</s>
<s>
It	O
has	O
become	O
popular	O
in	O
many	O
disciplines	O
,	O
such	O
as	O
image	B-Algorithm
processing	I-Algorithm
,	O
computer	O
graphics	O
,	O
computational	O
geometry	O
,	O
optimization	O
,	O
computational	O
fluid	O
dynamics	O
,	O
and	O
computational	O
biology	O
.	O
</s>
<s>
A	O
number	O
of	O
level-set	B-Algorithm
data	I-Algorithm
structures	I-Algorithm
have	O
been	O
developed	O
to	O
facilitate	O
the	O
use	O
of	O
the	O
level-set	B-Algorithm
method	I-Algorithm
in	O
computer	O
applications	O
.	O
</s>
<s>
Therefore	O
we	O
need	O
to	O
apply	O
a	O
specific	O
function	O
:	O
Compact	O
Level	B-Algorithm
Set	I-Algorithm
Method	I-Algorithm
.	O
</s>
<s>
But	O
,	O
how	O
is	O
the	O
Heaviside	O
function	O
inserted	O
in	O
the	O
Compact	O
Level	B-Algorithm
Set	I-Algorithm
Method	I-Algorithm
?	O
</s>
<s>
To	O
minimize	O
these	O
problems	O
,	O
the	O
Level	B-Algorithm
Set	I-Algorithm
method	I-Algorithm
uses	O
a	O
smooth	O
,	O
cell-related	O
Heaviside	O
function	O
that	O
explicitly	O
defines	O
the	O
interface	O
position	O
( ∅	O
=	O
0	O
)	O
.	O
</s>
