<s>
A	O
heat	B-Algorithm
kernel	I-Algorithm
signature	I-Algorithm
(	O
HKS	O
)	O
is	O
a	O
feature	O
descriptor	O
for	O
use	O
in	O
deformable	O
shape	B-Algorithm
analysis	I-Algorithm
and	O
belongs	O
to	O
the	O
group	O
of	O
spectral	B-Algorithm
shape	I-Algorithm
analysis	I-Algorithm
methods	O
.	O
</s>
<s>
For	O
each	O
point	O
in	O
the	O
shape	O
,	O
HKS	O
defines	O
its	O
feature	B-Algorithm
vector	I-Algorithm
representing	O
the	O
point	O
's	O
local	O
and	O
global	O
geometric	O
properties	O
.	O
</s>
<s>
Shape	B-Algorithm
analysis	I-Algorithm
is	O
the	O
field	O
of	O
automatic	O
digital	O
analysis	O
of	O
shapes	O
,	O
e.g.	O
,	O
3D	O
objects	O
.	O
</s>
<s>
For	O
many	O
shape	B-Algorithm
analysis	I-Algorithm
tasks	O
(	O
such	O
as	O
shape	O
matching/retrieval	O
)	O
,	O
feature	B-Algorithm
vectors	I-Algorithm
for	O
certain	O
key	O
points	O
are	O
used	O
instead	O
of	O
using	O
the	O
complete	O
3D	O
model	O
of	O
the	O
shape	O
.	O
</s>
<s>
For	O
rigid	B-Algorithm
transformations	I-Algorithm
,	O
commonly	O
used	O
feature	O
descriptors	O
include	O
shape	B-General_Concept
context	I-General_Concept
,	O
spin	O
images	O
,	O
integral	O
volume	O
descriptors	O
and	O
multiscale	O
local	O
features	O
,	O
among	O
others	O
.	O
</s>
<s>
HKS	O
allows	O
isometric	O
transformations	O
which	O
generalizes	O
rigid	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
The	O
heat	O
diffusion	O
equation	O
over	O
a	O
compact	O
Riemannian	B-Architecture
manifold	I-Architecture
(	O
possibly	O
with	O
a	O
boundary	O
)	O
is	O
given	O
by	O
,	O
</s>
<s>
The	O
heat	O
kernel	O
fully	O
characterizes	O
a	O
surface	O
up	O
to	O
an	O
isometry	O
:	O
For	O
any	O
surjective	B-Algorithm
map	I-Algorithm
between	O
two	O
Riemannian	B-Architecture
manifolds	I-Architecture
and	O
,	O
if	O
then	O
is	O
an	O
isometry	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
The	O
HKS	O
can	O
be	O
computed	O
if	O
a	O
mesh	B-Algorithm
representation	O
of	O
the	O
manifold	O
is	O
available	O
,	O
by	O
using	O
a	O
discrete	O
approximation	O
to	O
and	O
using	O
the	O
discrete	O
analogue	O
of	O
the	O
heat	O
equation	O
.	O
</s>
<s>
where	O
is	O
a	O
positive	O
diagonal	O
matrix	O
with	O
entries	O
corresponding	O
to	O
the	O
area	O
of	O
the	O
triangles	O
in	O
the	O
mesh	B-Algorithm
sharing	O
the	O
vertex	O
,	O
and	O
is	O
a	O
symmetric	O
semi-definite	O
weighting	O
matrix	O
.	O
</s>
<s>
The	O
discrete	O
heat	O
kernel	O
requires	O
eigendecomposition	O
of	O
a	O
matrix	O
of	O
size	O
,	O
where	O
is	O
the	O
number	O
of	O
vertices	O
in	O
the	O
mesh	B-Algorithm
representation	O
of	O
the	O
manifold	O
.	O
</s>
<s>
The	O
(	O
continuous	O
)	O
HKS	O
at	O
a	O
point	O
,	O
on	O
the	O
Riemannian	B-Architecture
manifold	I-Architecture
is	O
related	O
to	O
the	O
scalar	O
curvature	O
by	O
,	O
</s>
<s>
SGWS	O
provides	O
a	O
general	O
form	O
for	O
spectral	B-Algorithm
descriptors	I-Algorithm
,	O
where	O
one	O
can	O
obtain	O
HKS	O
by	O
specifying	O
the	O
filter	O
function	O
.	O
</s>
<s>
Alternatively	O
,	O
scale-invariant	O
version	O
of	O
the	O
HKS	O
can	O
also	O
be	O
constructed	O
by	O
generating	O
a	O
Scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
In	O
the	O
scale-space	B-Algorithm
,	O
the	O
HKS	O
of	O
a	O
scaled	O
shape	O
corresponds	O
to	O
a	O
translation	O
up	O
to	O
a	O
multiplicative	O
factor	O
.	O
</s>
<s>
The	O
HKS	O
is	O
defined	O
for	O
a	O
boundary	O
surface	O
of	O
a	O
3D	O
shape	O
,	O
represented	O
as	O
a	O
2D	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
The	O
scale-invariant	O
HKS	O
features	O
can	O
be	O
used	O
in	O
the	O
bag-of-features	B-General_Concept
model	O
for	O
shape	O
retrieval	O
applications	O
.	O
</s>
