<s>
Geometry	B-Algorithm
processing	I-Algorithm
,	O
or	O
mesh	B-Algorithm
processing	I-Algorithm
,	O
is	O
an	O
area	O
of	O
research	O
that	O
uses	O
concepts	O
from	O
applied	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
and	O
engineering	O
to	O
design	O
efficient	O
algorithms	O
for	O
the	O
acquisition	O
,	O
reconstruction	B-Algorithm
,	O
analysis	O
,	O
manipulation	O
,	O
simulation	O
and	O
transmission	O
of	O
complex	O
3D	O
models	O
.	O
</s>
<s>
As	O
the	O
name	O
implies	O
,	O
many	O
of	O
the	O
concepts	O
,	O
data	O
structures	O
,	O
and	O
algorithms	O
are	O
directly	O
analogous	O
to	O
signal	O
processing	O
and	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
where	O
image	O
smoothing	O
might	O
convolve	O
an	O
intensity	O
signal	O
with	O
a	O
blur	O
kernel	O
formed	O
using	O
the	O
Laplace	O
operator	O
,	O
geometric	B-Algorithm
smoothing	I-Algorithm
might	O
be	O
achieved	O
by	O
convolving	O
a	O
surface	O
geometry	O
with	O
a	O
blur	O
kernel	O
formed	O
using	O
the	O
Laplace-Beltrami	O
operator	O
.	O
</s>
<s>
Applications	O
of	O
geometry	B-Algorithm
processing	I-Algorithm
algorithms	O
already	O
cover	O
a	O
wide	O
range	O
of	O
areas	O
from	O
multimedia	O
,	O
entertainment	O
and	O
classical	O
computer-aided	B-Application
design	I-Application
,	O
to	O
biomedical	O
computing	O
,	O
reverse	O
engineering	O
,	O
and	O
scientific	O
computing	O
.	O
</s>
<s>
Geometry	B-Algorithm
processing	I-Algorithm
is	O
a	O
common	O
research	O
topic	O
at	O
SIGGRAPH	O
,	O
the	O
premier	O
computer	O
graphics	O
academic	O
conference	O
,	O
and	O
the	O
main	O
topic	O
of	O
the	O
annual	O
Symposium	B-General_Concept
on	I-General_Concept
Geometry	I-General_Concept
Processing	I-General_Concept
.	O
</s>
<s>
Geometry	B-Algorithm
processing	I-Algorithm
involves	O
working	O
with	O
a	O
shape	O
,	O
usually	O
in	O
2D	O
or	O
3D	O
,	O
although	O
the	O
shape	O
can	O
live	O
in	O
a	O
space	O
of	O
arbitrary	O
dimensions	O
.	O
</s>
<s>
At	O
its	O
"	O
birth	O
,	O
"	O
a	O
shape	O
can	O
be	O
instantiated	O
through	O
one	O
of	O
three	O
methods	O
:	O
a	O
model	O
,	O
a	O
mathematical	O
representation	O
,	O
or	O
a	O
scan	B-Algorithm
.	O
</s>
<s>
Editing	O
may	O
involve	O
denoising	O
,	O
deforming	O
,	O
or	O
performing	O
rigid	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
Or	O
it	O
can	O
even	O
be	O
fabricated	B-Algorithm
in	O
the	O
real	O
world	O
,	O
through	O
a	O
method	O
such	O
as	O
3D	O
printing	O
or	O
laser	O
cutting	O
.	O
</s>
<s>
Like	O
any	O
other	O
shape	O
,	O
the	O
shapes	O
used	O
in	O
geometry	B-Algorithm
processing	I-Algorithm
have	O
properties	O
pertaining	O
to	O
their	O
geometry	O
and	O
topology	B-Architecture
.	O
</s>
<s>
The	O
topology	B-Architecture
of	O
a	O
shape	O
is	O
a	O
collection	O
of	O
properties	O
that	O
do	O
not	O
change	O
even	O
after	O
smooth	O
transformations	O
have	O
been	O
applied	O
to	O
the	O
shape	O
.	O
</s>
<s>
Shapes	O
in	O
geometry	B-Algorithm
processing	I-Algorithm
are	O
usually	O
represented	O
as	O
triangle	B-Algorithm
meshes	I-Algorithm
,	O
which	O
can	O
be	O
seen	O
as	O
a	O
graph	O
.	O
</s>
<s>
Each	O
triangle	O
forms	O
a	O
face	O
of	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
These	O
are	O
combinatoric	O
in	O
nature	O
and	O
encode	O
the	O
topology	B-Architecture
of	O
the	O
shape	O
.	O
</s>
<s>
In	O
addition	O
to	O
triangles	O
,	O
a	O
more	O
general	O
class	O
of	O
polygon	B-Algorithm
meshes	I-Algorithm
can	O
also	O
be	O
used	O
to	O
represent	O
a	O
shape	O
.	O
</s>
<s>
More	O
advanced	O
representations	O
like	O
progressive	O
meshes	B-Algorithm
encode	O
a	O
coarse	O
representation	O
along	O
with	O
a	O
sequence	O
of	O
transformations	O
,	O
which	O
produce	O
a	O
fine	O
or	O
high	O
resolution	O
representation	O
of	O
the	O
shape	O
once	O
applied	O
.	O
</s>
<s>
These	O
meshes	B-Algorithm
are	O
useful	O
in	O
a	O
variety	O
of	O
applications	O
,	O
including	O
geomorphs	O
,	O
progressive	O
transmission	O
,	O
mesh	B-Algorithm
compression	O
,	O
and	O
selective	O
refinement	O
.	O
</s>
<s>
A	O
concrete	O
example	O
of	O
this	O
is	O
a	O
mesh	B-Algorithm
of	O
a	O
pair	O
of	O
pants	O
.	O
</s>
<s>
To	O
bring	O
this	O
into	O
the	O
discrete	O
world	O
,	O
the	O
Euler	O
characteristic	O
of	O
a	O
mesh	B-Algorithm
is	O
computed	O
in	O
terms	O
of	O
its	O
vertices	O
,	O
edges	O
,	O
and	O
faces	O
.	O
</s>
<s>
To	O
transform	O
the	O
surface	O
points	O
into	O
a	O
mesh	B-Algorithm
,	O
the	O
Poisson	O
reconstruction	B-Algorithm
strategy	O
can	O
be	O
employed	O
.	O
</s>
<s>
The	O
task	O
of	O
reconstruction	B-Algorithm
then	O
becomes	O
a	O
variational	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
As	O
a	O
variational	B-Algorithm
problem	I-Algorithm
,	O
one	O
can	O
view	O
the	O
minimizer	O
as	O
a	O
solution	O
of	O
Poisson	O
's	O
equation	O
.	O
</s>
<s>
After	O
obtaining	O
a	O
good	O
approximation	O
for	O
and	O
a	O
value	O
for	O
which	O
the	O
points	O
with	O
lie	O
on	O
the	O
surface	O
to	O
be	O
reconstructed	O
,	O
the	O
marching	B-Algorithm
cubes	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
to	O
construct	O
a	O
triangle	B-Algorithm
mesh	I-Algorithm
from	O
the	O
function	O
,	O
which	O
can	O
then	O
be	O
applied	O
in	O
subsequent	O
computer	O
graphics	O
applications	O
.	O
</s>
<s>
One	O
common	O
problem	O
encountered	O
in	O
geometry	B-Algorithm
processing	I-Algorithm
is	O
how	O
to	O
merge	O
multiple	O
views	O
of	O
a	O
single	O
object	O
captured	O
from	O
different	O
angles	O
or	O
positions	O
.	O
</s>
<s>
This	O
problem	O
is	O
known	O
as	O
registration	B-Algorithm
.	O
</s>
<s>
In	O
registration	B-Algorithm
,	O
we	O
wish	O
to	O
find	O
an	O
optimal	O
rigid	B-Algorithm
transformation	I-Algorithm
that	O
will	O
align	O
surface	O
with	O
surface	O
.	O
</s>
<s>
An	O
iterative	O
solution	O
such	O
as	O
Iterative	B-General_Concept
Closest	I-General_Concept
Point	I-General_Concept
(	O
ICP	O
)	O
is	O
therefore	O
employed	O
to	O
solve	O
for	O
small	O
transformations	O
iteratively	O
,	O
instead	O
of	O
solving	O
for	O
the	O
potentially	O
large	O
transformation	O
in	O
one	O
go	O
.	O
</s>
<s>
In	O
order	O
to	O
sample	O
points	O
uniformly	O
at	O
random	O
across	O
the	O
surface	O
of	O
the	O
triangle	B-Algorithm
mesh	I-Algorithm
,	O
the	O
random	O
sampling	O
is	O
broken	O
into	O
two	O
stages	O
:	O
uniformly	O
sampling	O
points	O
within	O
a	O
triangle	O
;	O
and	O
non-uniformly	O
sampling	O
triangles	O
,	O
such	O
that	O
each	O
triangle	O
's	O
associated	O
probability	O
is	O
proportional	O
to	O
its	O
surface	O
area	O
.	O
</s>
<s>
Reducing	O
noise	O
on	O
the	O
former	O
is	O
known	O
as	O
data	O
denoising	O
,	O
while	O
noise	O
reduction	O
on	O
the	O
latter	O
is	O
known	O
as	O
surface	B-Algorithm
fairing	I-Algorithm
.	O
</s>
<s>
The	O
task	O
of	O
geometric	B-Algorithm
smoothing	I-Algorithm
is	O
analogous	O
to	O
signal	O
noise	O
reduction	O
,	O
and	O
consequently	O
employs	O
similar	O
approaches	O
.	O
</s>
<s>
Because	O
the	O
variation	O
is	O
free	O
,	O
this	O
results	O
in	O
a	O
self-adjoint	O
linear	O
problem	O
to	O
solve	O
with	O
a	O
parameter	O
:	O
When	O
working	O
with	O
triangle	B-Algorithm
meshes	I-Algorithm
one	O
way	O
to	O
determine	O
the	O
values	O
of	O
the	O
Laplacian	O
matrix	O
is	O
through	O
analyzing	O
the	O
geometry	O
of	O
connected	O
triangles	O
on	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
The	O
mass	B-General_Concept
matrix	I-General_Concept
M	O
as	O
an	O
operator	O
computes	O
the	O
local	O
integral	O
of	O
a	O
function	O
's	O
value	O
and	O
is	O
often	O
set	O
for	O
a	O
mesh	B-Algorithm
with	O
m	O
triangles	O
as	O
follows	O
:	O
</s>
<s>
This	O
process	O
is	O
known	O
as	O
parameterization	B-General_Concept
.	O
</s>
<s>
In	O
this	O
manner	O
,	O
parameterization	B-General_Concept
can	O
be	O
seen	O
as	O
an	O
optimization	O
problem	O
.	O
</s>
<s>
One	O
of	O
the	O
major	O
applications	O
of	O
mesh	B-General_Concept
parameterization	I-General_Concept
is	O
texture	O
mapping	O
.	O
</s>
<s>
Where	O
is	O
the	O
set	O
of	O
mesh	B-Algorithm
edges	O
and	O
is	O
the	O
set	O
of	O
vertices	O
.	O
</s>
<s>
Borrowing	O
an	O
idea	O
from	O
graph	O
theory	O
,	O
we	O
apply	O
the	O
Tutte	O
Mapping	O
and	O
restrict	O
the	O
boundary	O
vertices	O
of	O
the	O
mesh	B-Algorithm
onto	O
a	O
unit	O
circle	O
or	O
other	O
convex	O
polygon	O
.	O
</s>
<s>
The	O
Tutte	O
Mapping	O
,	O
however	O
,	O
still	O
suffers	O
from	O
severe	O
distortions	O
as	O
it	O
attempts	O
to	O
make	O
the	O
edge	O
lengths	O
equal	O
,	O
and	O
hence	O
does	O
not	O
correctly	O
account	O
for	O
the	O
triangle	O
sizes	O
on	O
the	O
actual	O
surface	B-Algorithm
mesh	I-Algorithm
.	O
</s>
<s>
Typically	O
,	O
these	O
transformations	O
are	O
continuous	O
and	O
do	O
not	O
alter	O
the	O
topology	B-Architecture
of	O
the	O
shape	O
.	O
</s>
<s>
Modern	O
mesh-based	O
shape	O
deformation	O
methods	O
satisfy	O
user	O
deformation	O
constraints	O
at	O
handles	O
(	O
selected	O
vertices	O
or	O
regions	O
on	O
the	O
mesh	B-Algorithm
)	O
and	O
propagate	O
these	O
handle	O
deformations	O
to	O
the	O
rest	O
of	O
shape	O
smoothly	O
and	O
without	O
removing	O
or	O
distorting	O
details	O
.	O
</s>
<s>
Some	O
common	O
forms	O
of	O
interactive	O
deformations	O
are	O
point-based	O
,	O
skeleton-based	O
,	O
and	O
cage-based	O
.	O
</s>
<s>
Skeleton-based	O
deformation	O
defines	O
a	O
skeleton	B-Application
for	O
the	O
shape	O
,	O
which	O
allows	O
a	O
user	O
to	O
move	O
the	O
bones	O
and	O
rotate	O
the	O
joints	O
.	O
</s>
<s>
The	O
As-Rigid-As-Possible	O
deformation	O
scheme	O
applies	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
to	O
each	O
handle	O
i	O
,	O
where	O
is	O
a	O
rotation	O
matrix	O
and	O
is	O
a	O
translation	O
vector	O
.	O
</s>
<s>
While	O
seemingly	O
trivial	O
,	O
in	O
many	O
cases	O
,	O
determining	O
the	O
inside	O
from	O
the	O
outside	O
of	O
a	O
triangle	B-Algorithm
mesh	I-Algorithm
is	O
not	O
an	O
easy	O
problem	O
.	O
</s>
<s>
This	O
mesh	B-Algorithm
clearly	O
has	O
a	O
semantic	O
inside-and-outside	O
,	O
despite	O
there	O
being	O
holes	O
at	O
the	O
waist	O
and	O
the	O
legs	O
.	O
</s>
<s>
In	O
the	O
limit	O
of	O
shooting	O
many	O
,	O
many	O
rays	O
,	O
this	O
method	O
handles	O
open	O
meshes	B-Algorithm
,	O
however	O
it	O
in	O
order	O
to	O
become	O
accurate	O
,	O
far	O
too	O
many	O
rays	O
are	O
required	O
for	O
this	O
method	O
to	O
be	O
computationally	O
ideal	O
.	O
</s>
<s>
Inspired	O
by	O
the	O
2D	O
winding	O
number	O
,	O
this	O
approach	O
uses	O
the	O
solid	O
angle	O
at	O
of	O
each	O
triangle	O
in	O
the	O
mesh	B-Algorithm
to	O
determine	O
if	O
is	O
inside	O
or	O
outside	O
.	O
</s>
<s>
The	O
value	O
of	O
the	O
Generalized	O
Winding	O
Number	O
at	O
,	O
is	O
proportional	O
to	O
the	O
sum	O
of	O
the	O
solid	O
angle	O
contribution	O
from	O
each	O
triangle	O
in	O
the	O
mesh	B-Algorithm
:	O
</s>
<s>
For	O
a	O
closed	O
mesh	B-Algorithm
,	O
is	O
equivalent	O
to	O
the	O
characteristic	O
function	O
for	O
the	O
volume	O
represented	O
by	O
.	O
</s>
<s>
Because	O
is	O
a	O
harmonic	O
function	O
,	O
it	O
degrades	O
gracefully	O
,	O
meaning	O
the	O
inside-outside	O
segmentation	O
would	O
not	O
change	O
much	O
if	O
we	O
poked	O
holes	O
in	O
a	O
closed	O
mesh	B-Algorithm
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
Generalized	O
Winding	O
Number	O
handles	O
open	O
meshes	B-Algorithm
robustly	O
.	O
</s>
<s>
The	O
boundary	O
between	O
inside	O
and	O
outside	O
smoothly	O
passes	O
over	O
holes	O
in	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
3D	O
Surface	O
Reconstruction	B-Algorithm
,	O
e.g.	O
</s>
<s>
Image-to-world	O
Registration	B-Algorithm
,	O
e.g.	O
</s>
<s>
Visualization	B-Application
(	O
graphics	O
)	O
e.g.	O
</s>
