Laplace-Beltrami Operator
Laplace Beltrami operator ($\Delta$) takes a function and gives out another function. It is the generalisation of Euclidean Laplacian Operator on manifolds. Laplacian eigenfunctions depend on each shape and thus coefficients or learned filters from one shape are not trivially transferable to another.
# Properties of LB operator
- Invariant under isometric deformation.
- Has countable eigen decomposition $\Delta \phi_i = \lambda_i\phi_i$ that forms orthonormal basis for $L^2(\mathcal{M})$
- Characterises geodesic distance.
# Eigen functions
Eigen functions on the LB of a shape ( or manifold $\mathcal{M}$) has desirable properties. Any function on $\mathcal{M}$ can be expressed as a linear sum of the eigen functions on the LB of $\mathcal{M}$. It is very similar to Fourier transforms.
NOTE: Eigenfunctions of Laplace-Beltrami operator on a sphere correspond to basis function of spherical harmonics.