<s>
Spectral	B-Algorithm
shape	I-Algorithm
analysis	I-Algorithm
relies	O
on	O
the	O
spectrum	O
(	O
eigenvalues	O
and/or	O
eigenfunctions	B-Algorithm
)	O
of	O
the	O
Laplace	O
–	O
Beltrami	O
operator	O
to	O
compare	O
and	O
analyze	O
geometric	O
shapes	O
.	O
</s>
<s>
It	O
can	O
be	O
defined	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
as	O
the	O
divergence	B-Application
of	O
the	O
gradient	O
of	O
a	O
real-valued	O
function	O
f	O
:	O
</s>
<s>
The	O
solutions	O
are	O
the	O
eigenfunctions	B-Algorithm
(	O
modes	O
)	O
and	O
corresponding	O
eigenvalues	O
,	O
representing	O
a	O
diverging	O
sequence	O
of	O
positive	O
real	O
numbers	O
.	O
</s>
<s>
For	O
the	O
sphere	O
,	O
for	O
example	O
,	O
the	O
eigenfunctions	B-Algorithm
are	O
the	O
spherical	O
harmonics	O
.	O
</s>
<s>
The	O
most	O
important	O
properties	O
of	O
the	O
eigenvalues	O
and	O
eigenfunctions	B-Algorithm
are	O
that	O
they	O
are	O
isometry	O
invariants	O
.	O
</s>
<s>
The	O
resulting	O
shapes	O
are	O
called	O
near-isometric	O
and	O
can	O
be	O
compared	O
using	O
spectral	B-Algorithm
shape	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Geometric	O
shapes	O
are	O
often	O
represented	O
as	O
2D	O
curved	O
surfaces	O
,	O
2D	O
surface	B-Algorithm
meshes	I-Algorithm
(	O
usually	O
triangle	B-Algorithm
meshes	I-Algorithm
)	O
or	O
3D	O
solid	O
objects	O
(	O
e.g.	O
</s>
<s>
using	O
voxels	B-Algorithm
or	O
tetrahedra	O
meshes	B-Algorithm
)	O
.	O
</s>
<s>
Several	O
discretizations	O
of	O
the	O
Laplace	O
operator	O
exist	O
(	O
see	O
Discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
)	O
for	O
the	O
different	O
types	O
of	O
geometry	O
representations	O
.	O
</s>
<s>
However	O
,	O
the	O
eigenvalues	O
are	O
global	O
descriptors	O
,	O
therefore	O
the	O
shapeDNA	O
and	O
other	O
global	O
spectral	O
descriptors	O
cannot	O
be	O
used	O
for	O
local	O
or	O
partial	O
shape	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
The	O
global	O
point	O
signature	O
at	O
a	O
point	O
is	O
a	O
vector	O
of	O
scaled	O
eigenfunctions	B-Algorithm
of	O
the	O
Laplace	O
–	O
Beltrami	O
operator	O
computed	O
at	O
(	O
i.e.	O
</s>
<s>
The	O
spectral	O
decomposition	O
of	O
the	O
graph	O
Laplacian	O
associated	O
with	O
complex	O
shapes	O
(	O
see	O
Discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
)	O
provides	O
eigenfunctions	B-Algorithm
(	O
modes	O
)	O
which	O
are	O
invariant	O
to	O
isometries	O
.	O
</s>
<s>
Computational	O
efficiency	O
now	O
enables	O
dense	O
correspondences	O
on	O
full	O
meshes	B-Algorithm
,	O
for	O
instance	O
between	O
cortical	O
surfaces	O
.	O
</s>
<s>
Spectral	O
matching	O
could	O
also	O
be	O
used	O
for	O
complex	O
non-rigid	O
image	B-Algorithm
registration	I-Algorithm
,	O
which	O
is	O
notably	O
difficult	O
when	O
images	O
have	O
very	O
large	O
deformations	O
.	O
</s>
<s>
Such	O
image	B-Algorithm
registration	I-Algorithm
methods	O
based	O
on	O
spectral	O
eigenmodal	O
values	O
indeed	O
capture	O
global	O
shape	O
characteristics	O
,	O
and	O
contrast	O
with	O
conventional	O
non-rigid	O
image	B-Algorithm
registration	I-Algorithm
methods	O
which	O
are	O
often	O
based	O
on	O
local	O
shape	O
characteristics	O
(	O
e.g.	O
,	O
image	O
gradients	O
)	O
.	O
</s>
