<s>
In	O
computer	B-Application
vision	I-Application
,	O
pattern	O
recognition	O
,	O
and	O
robotics	O
,	O
point-set	B-Algorithm
registration	I-Algorithm
,	O
also	O
known	O
as	O
point-cloud	B-Algorithm
registration	O
or	O
scan	O
matching	O
,	O
is	O
the	O
process	O
of	O
finding	O
a	O
spatial	O
transformation	B-Algorithm
(	O
e.g.	O
,	O
scaling	B-Algorithm
,	O
rotation	O
and	O
translation	B-Algorithm
)	O
that	O
aligns	O
two	O
point	B-Algorithm
clouds	I-Algorithm
.	O
</s>
<s>
The	O
purpose	O
of	O
finding	O
such	O
a	O
transformation	B-Algorithm
includes	O
merging	O
multiple	O
data	O
sets	O
into	O
a	O
globally	O
consistent	O
model	O
(	O
or	O
coordinate	O
frame	O
)	O
,	O
and	O
mapping	O
a	O
new	O
measurement	O
to	O
a	O
known	O
data	O
set	O
to	O
identify	O
features	O
or	O
to	O
estimate	B-General_Concept
its	I-General_Concept
pose	I-General_Concept
.	O
</s>
<s>
Raw	O
3D	O
point	B-Algorithm
cloud	I-Algorithm
data	I-Algorithm
are	O
typically	O
obtained	O
from	O
Lidars	O
and	O
RGB-D	B-Algorithm
cameras	I-Algorithm
.	O
</s>
<s>
3D	O
point	B-Algorithm
clouds	I-Algorithm
can	O
also	O
be	O
generated	O
from	O
computer	B-Application
vision	I-Application
algorithms	O
such	O
as	O
triangulation	B-Algorithm
,	O
bundle	B-General_Concept
adjustment	I-General_Concept
,	O
and	O
more	O
recently	O
,	O
monocular	O
image	O
depth	O
estimation	O
using	O
deep	B-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
For	O
2D	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
used	O
in	O
image	O
processing	O
and	O
feature-based	O
image	B-Algorithm
registration	I-Algorithm
,	O
a	O
point	O
set	O
may	O
be	O
2D	O
pixel	O
coordinates	O
obtained	O
by	O
feature	B-Algorithm
extraction	I-Algorithm
from	O
an	O
image	O
,	O
for	O
example	O
corner	B-Algorithm
detection	I-Algorithm
.	O
</s>
<s>
Point	B-Algorithm
cloud	I-Algorithm
registration	I-Algorithm
has	O
extensive	O
applications	O
in	O
autonomous	O
driving	O
,	O
motion	B-Algorithm
estimation	I-Algorithm
and	I-Algorithm
3D	I-Algorithm
reconstruction	I-Algorithm
,	O
object	B-General_Concept
detection	I-General_Concept
and	I-General_Concept
pose	I-General_Concept
estimation	I-General_Concept
,	O
robotic	O
manipulation	O
,	O
simultaneous	B-Application
localization	I-Application
and	I-Application
mapping	I-Application
(	O
SLAM	O
)	O
,	O
panorama	B-Algorithm
stitching	I-Algorithm
,	O
virtual	B-Application
and	I-Application
augmented	I-Application
reality	I-Application
,	O
and	O
medical	B-Application
imaging	I-Application
.	O
</s>
<s>
As	O
a	O
special	O
case	O
,	O
registration	O
of	O
two	O
point	O
sets	O
that	O
only	O
differ	O
by	O
a	O
3D	O
rotation	O
(	O
i.e.	O
,	O
there	O
is	O
no	O
scaling	B-Algorithm
and	O
translation	B-Algorithm
)	O
,	O
is	O
called	O
the	O
Wahba	O
Problem	O
and	O
also	O
related	O
to	O
the	O
orthogonal	O
procrustes	O
problem	O
.	O
</s>
<s>
The	O
problem	O
is	O
to	O
find	O
a	O
transformation	B-Algorithm
to	O
be	O
applied	O
to	O
the	O
moving	O
"	O
model	O
"	O
point	O
set	O
such	O
that	O
the	O
difference	O
(	O
typically	O
defined	O
in	O
the	O
sense	O
of	O
point-wise	O
Euclidean	O
distance	O
)	O
between	O
and	O
the	O
static	O
"	O
scene	O
"	O
set	O
is	O
minimized	O
.	O
</s>
<s>
The	O
transformation	B-Algorithm
model	O
may	O
be	O
written	O
as	O
,	O
using	O
which	O
the	O
transformed	O
,	O
registered	O
model	O
point	O
set	O
is	O
:	O
</s>
<s>
The	O
output	O
of	O
a	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
algorithm	O
is	O
therefore	O
the	O
optimal	O
transformation	B-Algorithm
such	O
that	O
is	O
best	O
aligned	O
to	O
,	O
according	O
to	O
some	O
defined	O
notion	O
of	O
distance	O
function	O
:	O
</s>
<s>
where	O
denotes	O
the	O
vector	O
2-norm	O
,	O
is	O
the	O
corresponding	B-Algorithm
point	I-Algorithm
in	O
set	O
that	O
attains	O
the	O
shortest	O
distance	O
to	O
a	O
given	O
point	O
in	O
set	O
after	O
transformation	B-Algorithm
.	O
</s>
<s>
Minimizing	O
such	O
a	O
function	O
in	O
rigid	O
registration	O
is	O
equivalent	O
to	O
solving	O
a	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
When	O
the	O
correspondences	O
(	O
i.e.	O
,	O
)	O
are	O
given	O
before	O
the	O
optimization	O
,	O
for	O
example	O
,	O
using	O
feature	O
matching	O
techniques	O
,	O
then	O
the	O
optimization	O
only	O
needs	O
to	O
estimate	O
the	O
transformation	B-Algorithm
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
the	O
correspondences	O
are	O
unknown	O
,	O
then	O
the	O
optimization	O
is	O
required	O
to	O
jointly	O
find	O
out	O
the	O
correspondences	O
and	O
transformation	B-Algorithm
together	O
.	O
</s>
<s>
Given	O
two	O
point	O
sets	O
,	O
rigid	O
registration	O
yields	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
which	O
maps	O
one	O
point	O
set	O
to	O
the	O
other	O
.	O
</s>
<s>
A	O
rigid	B-Algorithm
transformation	I-Algorithm
is	O
defined	O
as	O
a	O
transformation	B-Algorithm
that	O
does	O
not	O
change	O
the	O
distance	O
between	O
any	O
two	O
points	O
.	O
</s>
<s>
Typically	O
such	O
a	O
transformation	B-Algorithm
consists	O
of	O
translation	B-Algorithm
and	O
rotation	O
.	O
</s>
<s>
In	O
robotics	O
and	O
computer	B-Application
vision	I-Application
,	O
rigid	O
registration	O
has	O
the	O
most	O
applications	O
.	O
</s>
<s>
Non-rigid	O
transformations	O
include	O
affine	B-Algorithm
transformations	I-Algorithm
such	O
as	O
scaling	B-Algorithm
and	O
shear	B-Algorithm
mapping	I-Algorithm
.	O
</s>
<s>
However	O
,	O
in	O
the	O
context	O
of	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
,	O
non-rigid	O
registration	O
typically	O
involves	O
nonlinear	O
transformation	B-Algorithm
.	O
</s>
<s>
If	O
the	O
eigenmodes	O
of	O
variation	O
of	O
the	O
point	O
set	O
are	O
known	O
,	O
the	O
nonlinear	O
transformation	B-Algorithm
may	O
be	O
parametrized	O
by	O
the	O
eigenvalues	O
.	O
</s>
<s>
A	O
nonlinear	O
transformation	B-Algorithm
may	O
also	O
be	O
parametrized	O
as	O
a	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
Some	O
approaches	O
to	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
use	O
algorithms	O
that	O
solve	O
the	O
more	O
general	O
graph	B-General_Concept
matching	I-General_Concept
problem	O
.	O
</s>
<s>
In	O
this	O
article	O
,	O
we	O
will	O
only	O
consider	O
algorithms	O
for	O
rigid	O
registration	O
,	O
where	O
the	O
transformation	B-Algorithm
is	O
assumed	O
to	O
contain	O
3D	O
rotations	O
and	O
translations	O
(	O
possibly	O
also	O
including	O
a	O
uniform	B-Algorithm
scaling	I-Algorithm
)	O
.	O
</s>
<s>
The	O
PCL	B-Language
(	O
Point	B-Language
Cloud	I-Language
Library	I-Language
)	O
is	O
an	O
open-source	O
framework	O
for	O
n-dimensional	O
point	B-Algorithm
cloud	I-Algorithm
and	O
3D	O
geometry	O
processing	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
,	O
one	O
can	O
assume	O
that	O
all	O
the	O
correspondences	O
are	O
correct	O
,	O
meaning	O
that	O
the	O
points	O
are	O
generated	O
as	O
follows:where	O
is	O
a	O
uniform	B-Algorithm
scaling	I-Algorithm
factor	O
(	O
in	O
many	O
cases	O
is	O
assumed	O
)	O
,	O
is	O
a	O
proper	O
3D	O
rotation	O
matrix	O
(	O
is	O
the	O
special	O
orthogonal	O
group	O
of	O
degree	O
)	O
,	O
is	O
a	O
3D	O
translation	B-Algorithm
vector	O
and	O
models	O
the	O
unknown	O
additive	O
noise	O
(	O
e.g.	O
,	O
Gaussian	O
noise	O
)	O
.	O
</s>
<s>
Specifically	O
,	O
if	O
the	O
noise	O
is	O
assumed	O
to	O
follow	O
a	O
zero-mean	O
isotropic	O
Gaussian	O
distribution	O
with	O
standard	O
deviation	O
,	O
i.e.	O
,	O
,	O
then	O
the	O
following	O
optimization	O
can	O
be	O
shown	O
to	O
yield	O
the	O
maximum	O
likelihood	O
estimate	O
for	O
the	O
unknown	O
scale	O
,	O
rotation	O
and	O
translation:Note	O
that	O
when	O
the	O
scaling	B-Algorithm
factor	O
is	O
1	O
and	O
the	O
translation	B-Algorithm
vector	O
is	O
zero	O
,	O
then	O
the	O
optimization	O
recovers	O
the	O
formulation	O
of	O
the	O
Wahba	O
problem	O
.	O
</s>
<s>
Horn	O
showed	O
that	O
(	O
)	O
actually	O
admits	O
a	O
closed-form	O
solution	O
,	O
by	O
decoupling	O
the	O
estimation	O
of	O
scale	O
,	O
rotation	O
and	O
translation	B-Algorithm
.	O
</s>
<s>
In	O
addition	O
,	O
in	O
order	O
to	O
find	O
a	O
unique	O
transformation	B-Algorithm
,	O
at	O
least	O
non-collinear	O
points	O
in	O
each	O
point	O
set	O
are	O
required	O
.	O
</s>
<s>
More	O
recently	O
,	O
Briales	O
and	O
Gonzalez-Jimenez	O
have	O
developed	O
a	O
semidefinite	O
relaxation	O
using	O
Lagrangian	B-Algorithm
duality	I-Algorithm
,	O
for	O
the	O
case	O
where	O
the	O
model	O
set	O
contains	O
different	O
3D	O
primitives	O
such	O
as	O
points	O
,	O
lines	O
and	O
planes	O
(	O
which	O
is	O
the	O
case	O
when	O
the	O
model	O
is	O
a	O
3D	O
mesh	O
)	O
.	O
</s>
<s>
The	O
least	B-Algorithm
squares	I-Algorithm
formulation	O
(	O
)	O
is	O
known	O
to	O
perform	O
arbitrarily	O
bad	O
in	O
the	O
presence	O
of	O
outliers	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
one	O
can	O
consider	O
a	O
different	O
generative	O
model	O
as	O
follows:where	O
if	O
the	O
th	O
pair	O
is	O
an	O
inlier	O
,	O
then	O
it	O
obeys	O
the	O
outlier-free	O
model	O
(	O
)	O
,	O
i.e.	O
,	O
is	O
obtained	O
from	O
by	O
a	O
spatial	O
transformation	B-Algorithm
plus	O
some	O
small	O
noise	O
;	O
however	O
,	O
if	O
the	O
th	O
pair	O
is	O
an	O
outlier	O
,	O
then	O
can	O
be	O
any	O
arbitrary	O
vector	O
.	O
</s>
<s>
Since	O
one	O
does	O
not	O
know	O
which	O
correspondences	O
are	O
outliers	O
beforehand	O
,	O
robust	O
registration	O
under	O
the	O
generative	O
model	O
(	O
)	O
is	O
of	O
paramount	O
importance	O
for	O
computer	B-Application
vision	I-Application
and	O
robotics	O
deployed	O
in	O
the	O
real	O
world	O
,	O
because	O
current	O
feature	O
matching	O
techniques	O
tend	O
to	O
output	O
highly	O
corrupted	O
correspondences	O
where	O
over	O
of	O
the	O
correspondences	O
can	O
be	O
outliers	O
.	O
</s>
<s>
Maximum	B-Operating_System
consensus	I-Operating_System
seeks	O
to	O
find	O
the	O
largest	O
set	O
of	O
correspondences	O
that	O
are	O
consistent	O
with	O
the	O
generative	O
model	O
(	O
)	O
for	O
some	O
choice	O
of	O
spatial	O
transformation	B-Algorithm
.	O
</s>
<s>
Formally	O
speaking	O
,	O
maximum	B-Operating_System
consensus	I-Operating_System
solves	O
the	O
following	O
optimization:where	O
denotes	O
the	O
cardinality	B-Application
of	O
the	O
set	O
.	O
</s>
<s>
Unfortunately	O
,	O
recent	O
analyses	O
have	O
shown	O
that	O
globally	O
solving	O
problem	O
(	O
cb.4	O
)	O
is	O
NP-Hard	O
,	O
and	O
global	O
algorithms	O
typically	O
have	O
to	O
resort	O
to	O
branch-and-bound	B-Algorithm
(	O
BnB	O
)	O
techniques	O
that	O
take	O
exponential-time	O
complexity	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
popular	O
heuristics	O
is	O
the	O
Random	B-Algorithm
Sample	I-Algorithm
Consensus	I-Algorithm
(	O
RANSAC	B-Algorithm
)	O
scheme	O
.	O
</s>
<s>
RANSAC	B-Algorithm
is	O
an	O
iterative	O
hypothesize-and-verify	O
method	O
.	O
</s>
<s>
RANSAC	B-Algorithm
is	O
highly	O
efficient	O
because	O
the	O
main	O
computation	O
of	O
each	O
iteration	O
is	O
carrying	O
out	O
the	O
closed-form	O
solution	O
in	O
Horn	O
's	O
method	O
.	O
</s>
<s>
However	O
,	O
RANSAC	B-Algorithm
is	O
non-deterministic	O
and	O
only	O
works	O
well	O
in	O
the	O
low-outlier-ratio	O
regime	O
(	O
e.g.	O
,	O
below	O
)	O
,	O
because	O
its	O
runtime	O
grows	O
exponentially	O
with	O
respect	O
to	O
the	O
outlier	O
ratio	O
.	O
</s>
<s>
To	O
fill	O
the	O
gap	O
between	O
the	O
fast	O
but	O
inexact	O
RANSAC	B-Algorithm
scheme	O
and	O
the	O
exact	O
but	O
exhaustive	O
BnB	O
optimization	O
,	O
recent	O
researches	O
have	O
developed	O
deterministic	O
approximate	O
methods	O
to	O
solve	O
consensus	O
maximization	O
.	O
</s>
<s>
Outlier	O
removal	O
methods	O
seek	O
to	O
pre-process	O
the	O
set	O
of	O
highly	O
corrupted	O
correspondences	O
before	O
estimating	O
the	O
spatial	O
transformation	B-Algorithm
.	O
</s>
<s>
The	O
motivation	O
of	O
outlier	O
removal	O
is	O
to	O
significantly	O
reduce	O
the	O
number	O
of	O
outlier	O
correspondences	O
,	O
while	O
maintaining	O
inlier	O
correspondences	O
,	O
so	O
that	O
optimization	O
over	O
the	O
transformation	B-Algorithm
becomes	O
easier	O
and	O
more	O
efficient	O
(	O
e.g.	O
,	O
RANSAC	B-Algorithm
works	O
poorly	O
when	O
the	O
outlier	O
ratio	O
is	O
above	O
but	O
performs	O
quite	O
well	O
when	O
outlier	O
ratio	O
is	O
below	O
)	O
.	O
</s>
<s>
GORE	O
has	O
been	O
shown	O
to	O
be	O
able	O
to	O
drastically	O
reduce	O
the	O
outlier	O
ratio	O
,	O
which	O
can	O
significantly	O
boost	O
the	O
performance	O
of	O
consensus	O
maximization	O
using	O
RANSAC	B-Algorithm
or	O
BnB	O
.	O
</s>
<s>
Yang	O
and	O
Carlone	O
have	O
proposed	O
to	O
build	O
pairwise	O
translation-and-rotation-invariant	O
measurements	O
(	O
TRIMs	O
)	O
from	O
the	O
original	O
set	O
of	O
measurements	O
and	O
embed	O
TRIMs	O
as	O
the	O
edges	O
of	O
a	O
graph	O
whose	O
nodes	O
are	O
the	O
3D	O
points	O
.	O
</s>
<s>
The	O
maximum	O
clique	O
based	O
outlier	O
removal	O
method	O
is	O
also	O
shown	O
to	O
be	O
quite	O
useful	O
in	O
real-world	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
problems	O
.	O
</s>
<s>
M-estimation	O
replaces	O
the	O
least	B-Algorithm
squares	I-Algorithm
objective	O
function	O
in	O
(	O
)	O
with	O
a	O
robust	O
cost	O
function	O
that	O
is	O
less	O
sensitive	O
to	O
outliers	O
.	O
</s>
<s>
Note	O
that	O
choosing	O
recovers	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimation	O
in	O
(	O
)	O
.	O
</s>
<s>
Popular	O
robust	O
cost	O
functions	O
include	O
-norm	O
loss	O
,	O
Huber	O
loss	O
,	O
Geman-McClure	O
loss	O
and	O
truncated	O
least	B-Algorithm
squares	I-Algorithm
loss	O
.	O
</s>
<s>
M-estimation	O
has	O
been	O
one	O
of	O
the	O
most	O
popular	O
paradigms	O
for	O
robust	O
estimation	O
in	O
robotics	O
and	O
computer	B-Application
vision	I-Application
.	O
</s>
<s>
Because	O
robust	O
objective	O
functions	O
are	O
typically	O
non-convex	O
(	O
e.g.	O
,	O
the	O
truncated	O
least	B-Algorithm
squares	I-Algorithm
loss	O
v.s.	O
</s>
<s>
the	O
least	B-Algorithm
squares	I-Algorithm
loss	O
)	O
,	O
algorithms	O
for	O
solving	O
the	O
non-convex	O
M-estimation	O
are	O
typically	O
based	O
on	O
local	O
optimization	O
,	O
where	O
first	O
an	O
initial	O
guess	O
is	O
provided	O
,	O
following	O
by	O
iterative	O
refinements	O
of	O
the	O
transformation	B-Algorithm
to	O
keep	O
decreasing	O
the	O
objective	O
function	O
.	O
</s>
<s>
Therefore	O
,	O
at	O
each	O
level	O
of	O
the	O
hyper-parameter	O
,	O
the	O
following	O
optimization	O
is	O
solved:Black	O
and	O
Rangarajan	O
proved	O
that	O
the	O
objective	O
function	O
of	O
each	O
optimization	O
(	O
)	O
can	O
be	O
dualized	O
into	O
a	O
sum	O
of	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
and	O
a	O
so-called	O
outlier	O
process	O
function	O
on	O
the	O
weights	O
that	O
determine	O
the	O
confidence	O
of	O
the	O
optimization	O
in	O
each	O
pair	O
of	O
measurements	O
.	O
</s>
<s>
Using	O
Black-Rangarajan	O
duality	B-Algorithm
and	O
GNC	O
tailored	O
for	O
the	O
Geman-McClure	O
function	O
,	O
Zhou	O
et	O
al	O
.	O
</s>
<s>
showed	O
that	O
the	O
joint	O
use	O
of	O
GNC	O
(	O
tailored	O
to	O
the	O
Geman-McClure	O
function	O
and	O
the	O
truncated	O
least	B-Algorithm
squares	I-Algorithm
function	O
)	O
and	O
Black-Rangarajan	O
duality	B-Algorithm
can	O
lead	O
to	O
a	O
general-purpose	O
solver	O
for	O
robust	O
registration	O
problems	O
,	O
including	O
point	B-Algorithm
clouds	I-Algorithm
and	O
mesh	O
registration	O
.	O
</s>
<s>
has	O
developed	O
the	O
first	O
certifiably	O
robust	O
registration	O
algorithm	O
,	O
named	O
Truncated	O
least	B-Algorithm
squares	I-Algorithm
Estimation	O
And	O
SEmidefinite	O
Relaxation	O
(	O
TEASER	O
)	O
.	O
</s>
<s>
For	O
point	B-Algorithm
cloud	I-Algorithm
registration	I-Algorithm
,	O
TEASER	O
not	O
only	O
outputs	O
an	O
estimate	O
of	O
the	O
transformation	B-Algorithm
,	O
but	O
also	O
quantifies	O
the	O
optimality	O
of	O
the	O
given	O
estimate	O
.	O
</s>
<s>
TEASER	O
adopts	O
the	O
following	O
truncated	O
least	B-Algorithm
squares	I-Algorithm
(	O
TLS	O
)	O
estimator:which	O
is	O
obtained	O
by	O
choosing	O
the	O
TLS	O
robust	O
cost	O
function	O
,	O
where	O
is	O
a	O
pre-defined	O
constant	O
that	O
determines	O
the	O
maximum	O
allowed	O
residuals	O
to	O
be	O
considered	O
inliers	O
.	O
</s>
<s>
TEASER	O
solves	O
(	O
)	O
as	O
follows	O
:	O
(	O
i	O
)	O
It	O
builds	O
invariant	O
measurements	O
such	O
that	O
the	O
estimation	O
of	O
scale	O
,	O
rotation	O
and	O
translation	B-Algorithm
can	O
be	O
decoupled	O
and	O
solved	O
separately	O
,	O
a	O
strategy	O
that	O
is	O
inspired	O
by	O
the	O
original	O
Horn	O
's	O
method	O
;	O
(	O
ii	O
)	O
The	O
same	O
TLS	O
estimation	O
is	O
applied	O
for	O
each	O
of	O
the	O
three	O
sub-problems	O
,	O
where	O
the	O
scale	O
TLS	O
problem	O
can	O
be	O
solved	O
exactly	O
using	O
an	O
algorithm	O
called	O
adaptive	O
voting	O
,	O
the	O
rotation	O
TLS	O
problem	O
can	O
relaxed	O
to	O
a	O
semidefinite	O
program	O
(	O
SDP	O
)	O
where	O
the	O
relaxation	O
is	O
exact	O
in	O
practice	O
,	O
even	O
with	O
large	O
amount	O
of	O
outliers	O
;	O
the	O
translation	B-Algorithm
TLS	O
problem	O
can	O
solved	O
using	O
component-wise	O
adaptive	O
voting	O
.	O
</s>
<s>
also	O
prove	O
that	O
,	O
under	O
some	O
mild	O
conditions	O
on	O
the	O
point	B-Algorithm
cloud	I-Algorithm
data	I-Algorithm
,	O
TEASER	O
's	O
estimated	O
transformation	B-Algorithm
has	O
bounded	O
errors	O
from	O
the	O
ground-truth	O
transformation	B-Algorithm
.	O
</s>
<s>
The	O
iterative	B-General_Concept
closest	I-General_Concept
point	I-General_Concept
(	O
ICP	O
)	O
algorithm	O
was	O
introduced	O
by	O
Besl	O
and	O
McKay	O
.	O
</s>
<s>
The	O
algorithm	O
performs	O
rigid	O
registration	O
in	O
an	O
iterative	O
fashion	O
by	O
alternating	O
in	O
(	O
i	O
)	O
given	O
the	O
transformation	B-Algorithm
,	O
finding	O
the	B-Algorithm
closest	I-Algorithm
point	I-Algorithm
in	O
for	O
every	O
point	O
in	O
;	O
and	O
(	O
ii	O
)	O
given	O
the	O
correspondences	O
,	O
finding	O
the	O
best	O
rigid	B-Algorithm
transformation	I-Algorithm
by	O
solving	O
the	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
(	O
)	O
.	O
</s>
<s>
In	O
pseudocode	B-Language
,	O
the	O
basic	O
algorithm	O
is	O
implemented	O
as	O
follows	O
:	O
</s>
<s>
Here	O
,	O
the	O
function	O
least_squares	O
performs	O
least	B-Algorithm
squares	I-Algorithm
optimization	O
to	O
minimize	O
the	O
distance	O
in	O
each	O
of	O
the	O
pairs	O
,	O
using	O
the	O
closed-form	O
solutions	O
by	O
Horn	O
and	O
Arun	O
.	O
</s>
<s>
Because	O
the	O
cost	O
function	O
of	O
registration	O
depends	O
on	O
finding	O
the	B-Algorithm
closest	I-Algorithm
point	I-Algorithm
in	O
to	O
every	O
point	O
in	O
,	O
it	O
can	O
change	O
as	O
the	O
algorithm	O
is	O
running	O
.	O
</s>
<s>
Nonetheless	O
,	O
because	O
ICP	O
is	O
intuitive	O
to	O
understand	O
and	O
straightforward	O
to	O
implement	O
,	O
it	O
remains	O
the	O
most	O
commonly	O
used	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
algorithm	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
expectation	B-Algorithm
maximization	I-Algorithm
algorithm	I-Algorithm
is	O
applied	O
to	O
the	O
ICP	O
algorithm	O
to	O
form	O
the	O
EM-ICP	O
method	O
,	O
and	O
the	O
Levenberg-Marquardt	B-Algorithm
algorithm	I-Algorithm
is	O
applied	O
to	O
the	O
ICP	O
algorithm	O
to	O
form	O
the	O
LM-ICP	O
method	O
.	O
</s>
<s>
The	O
method	O
performs	O
registration	O
using	O
deterministic	B-Algorithm
annealing	I-Algorithm
and	O
soft	O
assignment	O
of	O
correspondences	O
between	O
point	O
sets	O
.	O
</s>
<s>
The	O
problem	O
is	O
then	O
defined	O
as	O
:	O
Given	O
two	O
point	O
sets	O
and	O
find	O
the	O
Affine	B-Algorithm
transformation	I-Algorithm
and	O
the	O
match	O
matrix	O
that	O
best	O
relates	O
them	O
.	O
</s>
<s>
Knowing	O
the	O
optimal	O
transformation	B-Algorithm
makes	O
it	O
easy	O
to	O
determine	O
the	O
match	O
matrix	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
The	O
transformation	B-Algorithm
may	O
be	O
decomposed	O
into	O
a	O
translation	B-Algorithm
vector	O
and	O
a	O
transformation	B-Algorithm
matrix	O
:	O
</s>
<s>
The	O
matrix	O
in	O
2D	O
is	O
composed	O
of	O
four	O
separate	O
parameters	O
,	O
which	O
are	O
scale	O
,	O
rotation	O
,	O
and	O
the	O
vertical	O
and	O
horizontal	O
shear	B-Algorithm
components	O
respectively	O
.	O
</s>
<s>
The	O
function	O
serves	O
to	O
regularize	O
the	O
Affine	B-Algorithm
transformation	I-Algorithm
by	O
penalizing	O
large	O
values	O
of	O
the	O
scale	O
and	O
shear	B-Algorithm
components	O
:	O
</s>
<s>
In	O
the	O
deterministic	B-Algorithm
annealing	I-Algorithm
method	O
,	O
the	O
control	O
parameter	O
is	O
slowly	O
increased	O
as	O
the	O
algorithm	O
runs	O
.	O
</s>
<s>
this	O
is	O
known	O
as	O
the	O
softmax	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
This	O
is	O
straightforward	O
,	O
except	O
that	O
now	O
the	O
constraints	O
on	O
are	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
constraints	O
:	O
and	O
.	O
</s>
<s>
To	O
satisfy	O
the	O
constraints	O
,	O
it	O
is	O
possible	O
to	O
use	O
a	O
result	O
due	O
to	O
Sinkhorn	O
,	O
which	O
states	O
that	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
obtained	O
from	O
any	O
square	O
matrix	O
with	O
all	O
positive	O
entries	O
by	O
the	O
iterative	O
process	O
of	O
alternating	O
row	O
and	O
column	O
normalizations	O
.	O
</s>
<s>
where	O
the	O
deterministic	B-Algorithm
annealing	I-Algorithm
control	O
parameter	O
is	O
initially	O
set	O
to	O
and	O
increases	O
by	O
factor	O
until	O
it	O
reaches	O
the	O
maximum	O
value	O
.	O
</s>
<s>
As	O
such	O
the	O
th	O
and	O
th	O
elements	O
are	O
slack	B-Algorithm
variables	I-Algorithm
.	O
</s>
<s>
Hence	O
the	O
structure	O
of	O
the	O
algorithm	O
remains	O
unchanged	O
,	O
with	O
the	O
main	O
difference	O
being	O
how	O
the	O
rotation	O
and	O
translation	B-Algorithm
matrices	O
are	O
solved	O
.	O
</s>
<s>
The	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
robust	O
point	O
matching	O
(	O
TPS-RPM	O
)	O
algorithm	O
by	O
Chui	O
and	O
Rangarajan	O
augments	O
the	O
RPM	O
method	O
to	O
perform	O
non-rigid	O
registration	O
by	O
parametrizing	O
the	O
transformation	B-Algorithm
as	O
a	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
However	O
,	O
because	O
the	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
parametrization	O
only	O
exists	O
in	O
three	O
dimensions	O
,	O
the	O
method	O
cannot	O
be	O
extended	O
to	O
problems	O
involving	O
four	O
or	O
more	O
dimensions	O
.	O
</s>
<s>
The	O
kernel	O
correlation	O
(	O
KC	O
)	O
approach	O
of	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
was	O
introduced	O
by	O
Tsin	O
and	O
Kanade	O
.	O
</s>
<s>
The	O
kernel	O
function	O
chosen	O
for	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
is	O
typically	O
symmetric	O
and	O
non-negative	O
kernel	O
,	O
similar	O
to	O
the	O
ones	O
used	O
in	O
the	O
Parzen	B-General_Concept
window	I-General_Concept
density	O
estimation	O
.	O
</s>
<s>
The	O
cost	O
function	O
of	O
the	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
algorithm	O
for	O
some	O
transformation	B-Algorithm
parameter	O
is	O
defined	O
thus	O
:	O
</s>
<s>
Furthermore	O
,	O
assuming	O
rigid	O
registration	O
,	O
is	O
invariant	O
when	O
is	O
changed	O
because	O
the	O
Euclidean	O
distance	O
between	O
every	O
pair	O
of	O
points	O
stays	O
the	O
same	O
under	O
rigid	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
are	O
defined	O
as	O
:	O
</s>
<s>
The	O
cost	O
function	O
can	O
then	O
be	O
shown	O
to	O
be	O
the	O
correlation	O
of	O
the	O
two	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
:	O
</s>
<s>
Having	O
established	O
the	O
cost	O
function	O
,	O
the	O
algorithm	O
simply	O
uses	O
gradient	B-Algorithm
descent	I-Algorithm
to	O
find	O
the	O
optimal	O
transformation	B-Algorithm
.	O
</s>
<s>
The	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
can	O
be	O
evaluated	O
at	O
grid	O
points	O
and	O
stored	O
in	O
a	O
lookup	B-Data_Structure
table	I-Data_Structure
.	O
</s>
<s>
The	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
are	O
sums	O
of	O
Gaussians	O
and	O
may	O
therefore	O
be	O
represented	O
as	O
Gaussian	O
mixture	O
models	O
(	O
GMM	O
)	O
.	O
</s>
<s>
Jian	O
and	O
Vemuri	O
use	O
the	O
GMM	O
version	O
of	O
the	O
KC	O
registration	O
algorithm	O
to	O
perform	O
non-rigid	O
registration	O
parametrized	O
by	O
thin	B-Algorithm
plate	I-Algorithm
splines	I-Algorithm
.	O
</s>
<s>
Unlike	O
earlier	O
approaches	O
to	O
non-rigid	O
registration	O
which	O
assume	O
a	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
transformation	B-Algorithm
model	O
,	O
CPD	O
is	O
agnostic	O
with	O
regard	O
to	O
the	O
transformation	B-Algorithm
model	O
used	O
.	O
</s>
<s>
The	O
expectation	B-Algorithm
maximization	I-Algorithm
algorithm	I-Algorithm
is	O
used	O
to	O
optimize	O
the	O
cost	O
function	O
.	O
</s>
<s>
The	O
expectation	B-Algorithm
maximization	I-Algorithm
(	O
EM	O
)	O
algorithm	O
is	O
used	O
to	O
find	O
and	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
consists	O
of	O
two	O
steps	O
.	O
</s>
<s>
Thus	O
,	O
the	O
algorithm	O
can	O
be	O
expressed	O
using	O
the	O
following	O
pseudocode	B-Language
,	O
where	O
the	O
point	O
sets	O
and	O
are	O
represented	O
as	O
and	O
matrices	O
and	O
respectively	O
:	O
</s>
<s>
For	O
example	O
,	O
in	O
rigid	O
registration	O
,	O
the	O
output	O
is	O
a	O
scale	O
,	O
a	O
rotation	O
matrix	O
,	O
and	O
a	O
translation	B-Algorithm
vector	O
.	O
</s>
<s>
which	O
is	O
initialized	O
to	O
one	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
a	O
column	O
vector	O
of	O
zeroes	O
:	O
</s>
<s>
For	O
affine	O
registration	O
,	O
where	O
the	O
goal	O
is	O
to	O
find	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
instead	O
of	O
a	O
rigid	O
one	O
,	O
the	O
output	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
matrix	I-Algorithm
and	O
a	O
translation	B-Algorithm
such	O
that	O
the	O
aligned	O
point	O
set	O
is	O
:	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
use	O
CPD	O
with	O
non-rigid	O
registration	O
using	O
a	O
parametrization	O
derived	O
using	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
.	O
</s>
<s>
A	O
variant	O
of	O
coherent	O
point	O
drift	O
,	O
called	O
Bayesian	O
coherent	O
point	O
drift	O
(	O
BCPD	O
)	O
,	O
was	O
derived	O
through	O
a	O
Bayesian	O
formulation	O
of	O
point	B-Algorithm
set	I-Algorithm
registration	I-Algorithm
.	O
</s>
<s>
An	O
variant	O
of	O
coherent	O
point	O
drift	O
called	O
CPD	O
with	O
Local	O
Surface	O
Geometry	O
(	O
LSG-CPD	O
)	O
for	O
rigid	O
point	B-Algorithm
cloud	I-Algorithm
registration	I-Algorithm
.	O
</s>
<s>
is	O
the	O
anisotropic	O
covariance	O
matrix	O
of	O
the	O
m-th	O
point	O
in	O
the	O
target	O
set	O
;	O
is	O
the	O
normal	O
vector	O
corresponding	O
to	O
the	O
same	O
point	O
;	O
is	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
serving	O
as	O
a	O
regularizer	O
,	O
pulling	O
the	O
problem	O
away	O
from	O
ill-posedness	O
.	O
</s>
<s>
The	O
point	B-Algorithm
cloud	I-Algorithm
registration	I-Algorithm
is	O
formulated	O
as	O
a	O
maximum	O
likelihood	O
estimation	O
(	O
MLE	O
)	O
problem	O
and	O
solve	O
it	O
with	O
the	O
Expectation-Maximization	B-Algorithm
(	O
EM	O
)	O
algorithm	O
.	O
</s>
<s>
In	O
the	O
M	O
step	O
,	O
an	O
unconstrained	O
optimization	O
on	O
a	O
matrix	O
Lie	O
group	O
is	O
designed	O
to	O
efficiently	O
update	O
the	O
rigid	B-Algorithm
transformation	I-Algorithm
of	O
the	O
registration	O
.	O
</s>
<s>
This	O
algorithm	O
was	O
introduced	O
in	O
2013	O
by	O
H	O
.	O
Assalih	O
to	O
accommodate	O
sonar	O
image	B-Algorithm
registration	I-Algorithm
.	O
</s>
<s>
SCS	O
can	O
match	O
rigid	O
and	O
non-rigid	O
transformations	O
,	O
and	O
performs	O
best	O
when	O
the	O
target	O
transformation	B-Algorithm
is	O
between	O
three	O
and	O
six	O
degrees	O
of	O
freedom	O
.	O
</s>
