Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc.
It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function f: Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem): The solutions are the eigenfunctions
For some shapes, the spectrum can be computed analytically (e.g. rectangle, flat torus, cylinder, disk or sphere).
In other words, if the shape is not stretched (e.g. a sheet of paper bent into the third dimension), the spectral values will not change.
Bendable objects, like animals, plants and humans, can move into different body postures with only minimal stretching at the joints.
The GPS is a global feature in the sense that it cannot be used for partial shape matching.
The IWKS[9] improves the WKS for non-rigid shape retrieval by introducing a new scaling function to the eigenvalues and aggregating a new curvature term.
SGWS is a local descriptor that is not only isometric invariant, but also compact, easy to compute and combines the advantages of both band-pass and low-pass filters.
An important facet of SGWS is the ability to combine the advantages of WKS and HKS into a single signature, while allowing a multiresolution representation of shapes.
[10] The spectral decomposition of the graph Laplacian associated with complex shapes (see Discrete Laplace operator) provides eigenfunctions (modes) which are invariant to isometries.
Computational efficiency now enables dense correspondences on full meshes, for instance between cortical surfaces.