Data analysis and representation on a general domain using
eigenfunctions of Laplacian, Applied and Computational Harmonic Analysis, vol.25, no.1, pp.68-97, 2008.
We propose a new method to analyze and represent data recorded on a
domain of general shape in Rd
by computing the eigenfunctions of Laplacian defined over there and
expanding the data into these eigenfunctions. Instead of directly
solving the eigenvalue problem on such a domain
via the Helmholtz equation (which can be quite complicated and costly),
we find the integral operator commuting with the Laplacian and
diagonalize that operator. Although our eigenfunctions satisfy neither the
Dirichlet nor the Neumann boundary condition, computing our eigenfunctions
via the integral operator is simple and has a potential to utilize modern fast
algorithms to accelerate the computation. We also show that our method
is better suited for small sample data than the Karhunen-Loève
Analysis. In fact, our eigenfunctions depend only on the shape of the domain,
not the statistics of the data. As a further application, we demonstrate the
use of our Laplacian eigenfunctions for solving the heat equation on a
Get the full paper (revised version as of 08/27/07): PDF file.
Get the official version via doi:10.1016/j.acha.2007.09.005.
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