<s>
In	O
computer	B-General_Concept
science	I-General_Concept
a	O
level	B-Algorithm
set	I-Algorithm
data	B-General_Concept
structure	I-General_Concept
is	O
designed	O
to	O
represent	O
discretely	O
sampled	O
dynamic	O
level	B-Algorithm
sets	I-Algorithm
functions	O
.	O
</s>
<s>
A	O
common	O
use	O
of	O
this	O
form	O
of	O
data	B-General_Concept
structure	I-General_Concept
is	O
in	O
efficient	O
image	O
rendering	O
.	O
</s>
<s>
The	O
underlying	O
method	O
constructs	O
a	O
signed	B-Algorithm
distance	I-Algorithm
field	I-Algorithm
that	O
extends	O
from	O
the	O
boundary	O
,	O
and	O
can	O
be	O
used	O
to	O
solve	O
the	O
motion	O
of	O
the	O
boundary	O
in	O
this	O
field	O
.	O
</s>
<s>
The	O
powerful	O
level-set	B-Algorithm
method	I-Algorithm
is	O
due	O
to	O
Osher	O
and	O
Sethian	O
1988	O
.	O
</s>
<s>
However	O
,	O
the	O
straightforward	O
implementation	O
via	O
a	O
dense	O
d-dimensional	O
array	B-Data_Structure
of	O
values	O
,	O
results	O
in	O
both	O
time	O
and	O
storage	O
complexity	O
of	O
,	O
where	O
is	O
the	O
cross	O
sectional	O
resolution	O
of	O
the	O
spatial	O
extents	O
of	O
the	O
domain	O
and	O
is	O
the	O
number	O
of	O
spatial	O
dimensions	O
of	O
the	O
domain	O
.	O
</s>
<s>
The	O
narrow	O
band	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
,	O
introduced	O
in	O
1995	O
by	O
Adalsteinsson	O
and	O
Sethian	O
,	O
restricted	O
most	O
computations	O
to	O
a	O
thin	O
band	O
of	O
active	O
voxels	B-Algorithm
immediately	O
surrounding	O
the	O
interface	O
,	O
thus	O
reducing	O
the	O
time	O
complexity	O
in	O
three	O
dimensions	O
to	O
for	O
most	O
operations	O
.	O
</s>
<s>
Periodic	O
updates	O
of	O
the	O
narrowband	O
structure	O
,	O
to	O
rebuild	O
the	O
list	O
of	O
active	O
voxels	B-Algorithm
,	O
were	O
required	O
which	O
entailed	O
an	O
operation	O
in	O
which	O
voxels	B-Algorithm
over	O
the	O
entire	O
volume	O
were	O
accessed	O
.	O
</s>
<s>
This	O
time	O
complexity	O
was	O
eliminated	O
in	O
the	O
approximate	O
"	O
sparse	O
field	O
"	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
introduced	O
by	O
Whitaker	O
in	O
1998	O
.	O
</s>
<s>
The	O
sparse	O
field	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
employs	O
a	O
set	O
of	O
linked	O
lists	O
to	O
track	O
the	O
active	O
voxels	B-Algorithm
around	O
the	O
interface	O
.	O
</s>
<s>
While	O
consistently	O
efficient	O
in	O
time	O
,	O
storage	O
space	O
is	O
still	O
required	O
by	O
the	O
sparse	O
field	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
sparse	O
block	O
grid	O
method	O
,	O
introduced	O
by	O
Bridson	O
in	O
2003	O
,	O
divides	O
the	O
entire	O
bounding	B-Algorithm
volume	I-Algorithm
of	O
size	O
into	O
small	O
cubic	O
blocks	O
of	O
voxels	B-Algorithm
each	O
.	O
</s>
<s>
A	O
coarse	O
grid	O
of	O
size	O
then	O
stores	O
pointers	O
only	O
to	O
those	O
blocks	O
that	O
intersect	O
the	O
narrow	O
band	O
of	O
the	O
level	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
octree	B-Data_Structure
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
,	O
introduced	O
by	O
Strain	O
in	O
1999	O
and	O
refined	O
by	O
Losasso	O
,	O
Gibou	O
and	O
Fedkiw	O
,	O
and	O
more	O
recently	O
by	O
Min	O
and	O
Gibou	O
uses	O
a	O
tree	O
of	O
nested	O
cubes	O
of	O
which	O
the	O
leaf	O
nodes	O
contain	O
signed	O
distance	O
values	O
.	O
</s>
<s>
Octree	B-Data_Structure
level	B-Algorithm
sets	I-Algorithm
currently	O
require	O
uniform	O
refinement	O
along	O
the	O
interface	O
(	O
i.e.	O
</s>
<s>
This	O
representation	O
is	O
efficient	O
in	O
terms	O
of	O
storage	O
,	O
and	O
relatively	O
efficient	O
in	O
terms	O
of	O
access	O
queries	O
,	O
An	O
advantage	O
of	O
the	O
level	O
method	O
on	O
octree	B-Data_Structure
data	B-General_Concept
structures	I-General_Concept
is	O
that	O
one	O
can	O
solve	O
the	O
partial	O
differential	O
equations	O
associated	O
with	O
typical	O
free	O
boundary	O
problems	O
that	O
use	O
the	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
run-length	B-Algorithm
encoding	I-Algorithm
(	O
RLE	O
)	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
,	O
introduced	O
in	O
2004	O
,	O
applies	O
the	O
RLE	O
scheme	O
to	O
compress	O
regions	O
away	O
from	O
the	O
narrow	O
band	O
to	O
just	O
their	O
sign	O
representation	O
while	O
storing	O
with	O
full	O
precision	O
the	O
narrow	O
band	O
.	O
</s>
<s>
The	O
sequential	O
traversal	O
of	O
the	O
narrow	O
band	O
is	O
optimal	O
and	O
storage	O
efficiency	O
is	O
further	O
improved	O
over	O
the	O
octree	B-Data_Structure
level	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
Hash	O
Table	O
Local	O
Level	B-Algorithm
Set	I-Algorithm
method	I-Algorithm
,	O
introduced	O
in	O
2011	O
by	O
Eyiyurekli	O
and	O
Breen	O
and	O
extended	O
in	O
2012	O
by	O
Brun	O
,	O
Guittet	O
and	O
Gibou	O
,	O
only	O
computes	O
the	O
level	B-Algorithm
set	I-Algorithm
data	O
in	O
a	O
band	O
around	O
the	O
interface	O
,	O
as	O
in	O
the	O
Narrow	O
Band	O
Level-Set	B-Algorithm
Method	I-Algorithm
,	O
but	O
also	O
only	O
stores	O
the	O
data	O
in	O
that	O
same	O
band	O
.	O
</s>
<s>
A	O
hash	O
table	O
data	B-General_Concept
structure	I-General_Concept
is	O
used	O
,	O
which	O
provides	O
an	O
access	O
to	O
the	O
data	O
.	O
</s>
<s>
Corbett	O
in	O
2005	O
introduced	O
the	O
point-based	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Instead	O
of	O
using	O
a	O
uniform	O
sampling	O
of	O
the	O
level	B-Algorithm
set	I-Algorithm
,	O
the	O
continuous	O
level	B-Algorithm
set	I-Algorithm
function	O
is	O
reconstructed	O
from	O
a	O
set	O
of	O
unorganized	O
point	O
samples	O
via	O
moving	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
