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Figure 5.1:
Segmentations induced by the nodal domains of some eigenfunctions selected among the
first 15 eigenfunctions (in order of increasing eigenvalues). Blue regions correspond to regions where
the eigenfunctions have negative values, while red regions correspond to positive values.
Figure 5.2:
Segmentations induced by the nodal domains of some eigenfunctions selected among the
first 15 eigenfunctions.
More formally, the
nodal sets
i
are the zero sets of the eigenfunctions of the Laplacian
operator on a Riemannian manifold, i.e.,
i
WD
1
i
.0/
. A
nodaldomain
is a connected component
of the complement of the nodal sets. ๎en, for each eigenfunction
i
, the nodal sets decompose
a surface into regions where
i
has constant sign. In other words, each eigenfunction induces a
shape segmentation, with segments corresponding to regions of positive or negative values. ๎e
use of nodal sets and nodal domains to segment 3D shapes was addressed in [
125
]. ๎e first
k
Laplacian eigenfunctions, ordered according to increasing frequencies, provide a family of shape
segmentations, each capturing different shape properties (see Figure
5.1
and Figure
5.2
).
๎e quality of these segmentations was based on the type and correctness of the segmen-
tation; the quality of boundaries; the definition of a multi-scale segmentation; the invariance to
pose; the sensitivity to noise and tessellation; the computational complexity and the use of control
parameters. Details of the discussions can be found in [
169
]. We simply remark that the nodal
domains related to the first eigenfunctions subdivide the input surface into patches which have
almost the same weight, measured as the sum of the edge weights associated with the
1
-star of
each vertex. In this case, the nodal sets often identify privileged directions, related to the symme-
tries of the objects (see also [
154
]). For articulated objects, the first eigenvectors define patches
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