We discuss our recently developed method to analyze data recorded on a domain of complicated shape in R^d 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 conditions, computing our eigenfunctions via the integral operator is simple and is amenable to fully utilizing modern fast algorithms (e.g., Fast Multipole Method) to accelerate the computation.
At the workshop, we hope to present our preliminary study on the analysis, feature extraction, and clustering of dendrite network patterns of mouse's retinal ganglion cells using these eigenfunctions and the corresponding eigenvalues.
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