<s>
Marching	B-Algorithm
cubes	I-Algorithm
is	O
a	O
computer	O
graphics	O
algorithm	O
,	O
published	O
in	O
the	O
1987	O
SIGGRAPH	O
proceedings	O
by	O
Lorensen	O
and	O
Cline	O
,	O
for	O
extracting	O
a	O
polygonal	B-Algorithm
mesh	I-Algorithm
of	O
an	O
isosurface	O
from	O
a	O
three-dimensional	O
discrete	O
scalar	O
field	O
(	O
the	O
elements	O
of	O
which	O
are	O
sometimes	O
called	O
voxels	B-Algorithm
)	O
.	O
</s>
<s>
The	O
applications	O
of	O
this	O
algorithm	O
are	O
mainly	O
concerned	O
with	O
medical	B-Application
visualizations	I-Application
such	O
as	O
CT	O
and	O
MRI	B-Algorithm
scan	O
data	O
images	O
,	O
and	O
special	O
effects	O
or	O
3-D	O
modelling	O
with	O
what	O
is	O
usually	O
called	O
metaballs	O
or	O
other	O
metasurfaces	O
.	O
</s>
<s>
The	O
marching	B-Algorithm
cubes	I-Algorithm
algorithm	I-Algorithm
is	O
meant	O
to	O
be	O
used	O
for	O
3-D	O
;	O
the	O
2-D	O
version	O
of	O
this	O
algorithm	O
is	O
called	O
the	O
marching	B-Algorithm
squares	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
At	O
General	O
Electric	O
they	O
worked	O
on	O
a	O
way	O
to	O
efficiently	O
visualize	O
data	O
from	O
CT	O
and	O
MRI	B-Algorithm
devices	O
.	O
</s>
<s>
For	O
each	O
cube	O
containing	O
a	O
section	O
of	O
the	O
isosurface	O
,	O
a	O
triangular	O
mesh	B-Algorithm
that	O
approximates	O
the	O
behavior	O
of	O
the	O
trilinear	O
interpolant	O
in	O
the	O
interior	O
cube	O
is	O
generated	O
.	O
</s>
<s>
However	O
,	O
due	O
to	O
the	O
existence	O
of	O
ambiguities	O
in	O
the	O
trilinear	O
interpolant	O
behavior	O
in	O
the	O
cube	O
faces	O
and	O
interior	O
,	O
the	O
meshes	B-Algorithm
extracted	O
by	O
the	O
Marching	B-Algorithm
Cubes	I-Algorithm
presented	O
discontinuities	O
and	O
topological	O
issues	O
.	O
</s>
<s>
Similarly	O
,	O
an	O
interior	O
ambiguity	O
occurs	O
when	O
the	O
signs	O
of	O
the	O
cube	O
vertices	O
are	O
insufficient	O
to	O
determine	O
the	O
correct	O
surface	B-Algorithm
triangulation	I-Algorithm
,	O
i.e.	O
,	O
when	O
multiple	O
triangulations	O
are	O
possible	O
for	O
the	O
same	O
cube	O
configuration	O
.	O
</s>
<s>
The	O
popularity	O
of	O
the	O
Marching	B-Algorithm
Cubes	I-Algorithm
and	O
its	O
widespread	O
adoption	O
resulted	O
in	O
several	O
improvements	O
in	O
the	O
algorithm	O
to	O
deal	O
with	O
the	O
ambiguities	O
and	O
to	O
correctly	O
track	O
the	O
behavior	O
of	O
the	O
interpolant	O
.	O
</s>
<s>
Durst	O
in	O
1988	O
was	O
the	O
first	O
to	O
note	O
that	O
the	O
triangulation	O
table	O
proposed	O
by	O
Lorensen	O
and	O
Cline	O
was	O
incomplete	O
,	O
and	O
that	O
certain	O
Marching	B-Algorithm
Cubes	I-Algorithm
cases	O
allow	O
multiple	O
triangulations	O
.	O
</s>
<s>
They	O
proposed	O
a	O
test	O
called	O
Asymptotic	B-Application
Decider	I-Application
to	O
correctly	O
track	O
the	O
interpolant	O
on	O
the	O
faces	O
of	O
the	O
cube	O
.	O
</s>
<s>
In	O
his	O
work	O
,	O
the	O
author	O
proposed	O
a	O
disambiguation	O
test	O
based	O
on	O
the	O
interpolant	O
critical	O
points	O
,	O
and	O
added	O
four	O
new	O
cases	O
to	O
the	O
Marching	B-Algorithm
Cubes	I-Algorithm
triangulation	O
table	O
(	O
subcases	O
of	O
the	O
cases	O
3	O
,	O
4	O
,	O
6	O
and	O
7	O
)	O
.	O
</s>
<s>
At	O
this	O
point	O
,	O
even	O
with	O
all	O
the	O
improvements	O
proposed	O
to	O
the	O
algorithm	O
and	O
its	O
triangulation	O
table	O
,	O
the	O
meshes	B-Algorithm
generated	O
by	O
the	O
Marching	B-Algorithm
Cubes	I-Algorithm
still	O
had	O
topological	O
incoherencies	O
.	O
</s>
<s>
The	O
Marching	B-Algorithm
Cubes	I-Algorithm
33	O
,	O
proposed	O
by	O
Chernyaev	O
in	O
1995	O
,	O
is	O
one	O
of	O
the	O
first	O
isosurface	O
extraction	O
algorithms	O
intended	O
to	O
preserve	O
the	O
topology	O
of	O
the	O
trilinear	O
interpolant	O
.	O
</s>
<s>
He	O
then	O
proposes	O
a	O
different	O
approach	O
to	O
solve	O
the	O
interior	O
ambiguities	O
,	O
which	O
is	O
based	O
on	O
the	O
Asymptotic	B-Application
Decider	I-Application
.	O
</s>
<s>
noted	O
and	O
corrected	O
algorithmic	O
inaccuracies	O
that	O
compromised	O
the	O
topological	O
correctness	O
of	O
the	O
mesh	B-Algorithm
generated	O
by	O
the	O
Marching	B-Algorithm
Cubes	I-Algorithm
33	O
algorithm	O
proposed	O
by	O
Chernyaev	O
.	O
</s>
<s>
Therefore	O
,	O
these	O
normals	O
may	O
be	O
interpolated	O
along	O
the	O
edges	O
of	O
each	O
cube	O
to	O
find	O
the	O
normals	O
of	O
the	O
generated	O
vertices	O
which	O
are	O
essential	O
for	O
shading	O
the	O
resulting	O
mesh	B-Algorithm
with	O
some	O
illumination	O
model	O
.	O
</s>
<s>
An	O
implementation	O
of	O
the	O
marching	B-Algorithm
cubes	I-Algorithm
algorithm	I-Algorithm
was	O
patented	O
as	O
United	O
States	O
Patent	O
4,710,876	O
.	O
</s>
<s>
Another	O
similar	O
algorithm	O
was	O
developed	O
,	O
called	O
marching	B-Algorithm
tetrahedra	I-Algorithm
,	O
in	O
order	O
to	O
circumvent	O
the	O
patent	O
as	O
well	O
as	O
solve	O
a	O
minor	O
ambiguity	O
problem	O
of	O
marching	B-Algorithm
cubes	I-Algorithm
with	O
some	O
cube	O
configurations	O
.	O
</s>
