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(
HKS
). More precisely, given a point
x
on a manifold
M
they define its heat kernel signature
HKS.x/
to be a function over the temporal domain:
C
!R; HKS.x;t/Dh
t
.x;x/:
HKS.x/WR
In practice, they sample HKS uniformly over the logarithmic scaled temporal domain and obtain
an
n
-dimensional descriptor vector to represent the HKS for each point:
p.x/Dc.x/.h
t
1
.x;x/;:::;h
t
n
.x;x//
where
c.x/
is a scaling factor.
Figure 5.5:
A color-coded (red for higher values, blue for lower values) visualization of the
HKS
function of isometric transformations of the same shape (left) and of a model having different topology
(glued fingers, right).
e
HKS
descriptor has many advantages, which make it a favorable choice for shape
description and matching. First, since the heat kernel is intrinsic (i.e., expressible solely in terms
of the Riemannian structure of
M
) it is invariant under isometric deformations of the manifold
(Figure
5.5
, left). Second, such a descriptor captures information about the neighborhood of
a point
x
on the shape at a scale defined by
t
: it captures differential information in a small
neighborhood of
x
for small
t
, and global information about the shape for large values of
t
. us,
the
n
-dimensional feature descriptor vector
p.x/
can be seen as analogous to the multi-scale
feature descriptors used in the computer vision community. ird, for small scales
t
, the
HKS
descriptor takes into account local information, which makes topological noise have only local
effect (Figure
5.5
, right). Note that this is a main difference with respect to the behavior of the
integral geodesic distance discussed before.
Sun et al. [
190
] proposed different applications of the
HKS
. First, the local maxima of
the function
k
t
.x;x/
for a large
t
can be used to find salient feature points. en, by computing
the
L
2
-norm of the difference between
HKS
vectors at feature points, it is possible to perform
shape correspondence between different models. Finally, the
HKS
can be used for multi-scale
self-matching: for a given point on a model, other points on the same model having
HKS
values
within a threshold identify repeated structure on the object, possibly at different scales.
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