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.

Abstract

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 Transform/Principal Component 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 complicated domain.

  • 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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