Spectral Methods for Shape Analysis - Mathematical Tools for Shape Analysis and Description - page 37

In order to clarify the form assumed by 5.1 and 5.2 , let us introduce some notations.

• fWD.f.p 1 /;:::;f.p n // T is the vector of the function values at the vertices;

• W WD.w ij / the weighted adjacency matrix coding the vertex adjacency in the mesh;

P
j2N.i/ w ij ;

• VWD diag .v 1 ;:::;v n / the diagonal matrix whose elements are v i D

• AWDVW is the stiffness matrix ;

• DWD diag .d 1 ;:::;d n / the lumped mass matrix ;

• LWDD

1 A the Laplace matrix (generally not symmetric).

With these notations, then the problem ( 5.1 ) can be expressed as LfDf or better, as a
generalized symmetric problem AfDDf .
Depending on the different choices of the edge weights and masses, the discrete Laplacian
operators are distinguished between geometric operators and finite-element operators [ 169 ]. A deep
analysis of different discretizations of the Laplace-Beltrami operator (geometric Laplacians, linear
and cubic FEM operators) in terms of the correctness of their eigenfunctions with respect to the
continuous case is shown in [ 169 ].
Except for some special cases (e.g., [ 14 , 27 , 100 , 183 ]), the discrete Laplacian is not guar-
anteed to converge to the Laplace-Beltrami operator. In addition, when dealing with intrinsic
shape properties, the Laplacian must be independent or at least minimally dependent on the tri-
angular mesh and thus the discrete approximation must preserves the geometric properties of the
Laplace-Beltrami operator. Unfortunately, Wardetzky et al. in [ 209 ] showed that for a general
mesh, it is theoretically impossible to satisfy all properties of the Laplace-Beltrami operator at
the same time, and thus the ideal discretization does not exist. is result also explains why there
exists such a large diversity of discrete Laplacians, each having a subset of the properties that make
it suitable for certain applications and unsuitable for others [ 35 ].

5.1.1 CONCEPTS IN ACTION
In the field of shape analysis and segmentation, spectral methods are extremely promising, as
they naturally provide a set of tools that are intrinsically multi-scale and defined by the shape
itself. Indeed, the eigenfunctions of the Laplace-Beltrami operator yield a family of real valued
functions that provide interesting insights in the structure and morphology of shapes.

Shape segmentation An example of use of the Laplace-Beltrami eigenfunctions to provide a
multiscale segmentation of the shape has been proposed in [ 169 ]. ere, the focus is on the nodal
sets and nodal domains of the Laplace-Beltrami eigenfunctions, showing that they induce a shape
decomposition which captures features at different scales, generally well-aligned with perceptually
relevant shape features. e set of decompositions induced by the eigenfunctions yields the sought
library of intrinsic shape segmentations.