<s>
Thin	B-Algorithm
plate	I-Algorithm
splines	I-Algorithm
(	O
TPS	O
)	O
are	O
a	O
spline-based	O
technique	O
for	O
data	O
interpolation	B-Algorithm
and	O
smoothing	B-Application
.	O
</s>
<s>
They	O
were	O
introduced	O
to	O
geometric	B-Algorithm
design	I-Algorithm
by	O
Duchon	O
.	O
</s>
<s>
They	O
are	O
an	O
important	O
special	O
case	O
of	O
a	O
polyharmonic	O
spline	B-Algorithm
.	O
</s>
<s>
The	O
name	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
refers	O
to	O
a	O
physical	O
analogy	O
involving	O
the	O
bending	O
of	O
a	O
thin	O
sheet	O
of	O
metal	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
is	O
two	O
dimensional	O
,	O
for	O
interpolation	B-Algorithm
,	O
the	O
TPS	O
fits	O
a	O
mapping	O
function	O
between	O
corresponding	O
point-sets	O
and	O
that	O
minimizes	O
the	O
following	O
energy	O
function	O
:	O
</s>
<s>
The	O
smoothing	B-Application
variant	O
,	O
correspondingly	O
,	O
uses	O
a	O
tuning	O
parameter	O
to	O
control	O
the	O
rigidity	O
of	O
the	O
deformation	O
,	O
balancing	O
the	O
aforementioned	O
criterion	O
with	O
the	O
measure	O
of	O
goodness	O
of	O
fit	O
,	O
thus	O
minimizing	O
:	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
discretization	O
of	O
this	O
variational	O
problem	O
,	O
the	O
method	O
of	O
elastic	B-Algorithm
maps	I-Algorithm
,	O
is	O
used	O
for	O
data	B-Application
mining	I-Application
and	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
The	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
has	O
a	O
natural	O
representation	O
in	O
terms	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
This	O
is	O
a	O
desirable	O
property	O
,	O
especially	O
when	O
compared	O
to	O
other	O
splines	B-Algorithm
,	O
since	O
the	O
global	O
pose	O
parameters	O
included	O
in	O
the	O
affine	O
transformation	O
are	O
not	O
penalized	O
.	O
</s>
<s>
An	O
additional	O
application	O
is	O
the	O
analysis	O
and	O
comparisons	O
of	O
archaeological	O
findings	O
in	O
3D	O
and	O
was	O
implemented	O
for	O
triangular	B-Algorithm
meshes	I-Algorithm
in	O
the	O
GigaMesh	B-Language
Software	I-Language
Framework	I-Language
.	O
</s>
<s>
The	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
has	O
a	O
number	O
of	O
properties	O
which	O
have	O
contributed	O
to	O
its	O
popularity	O
:	O
</s>
<s>
However	O
,	O
note	O
that	O
splines	B-Algorithm
already	O
in	O
one	O
dimension	O
can	O
cause	O
severe	O
"	O
overshoots	O
"	O
.	O
</s>
