Geometric harmonics as a statistical image processing tool for images on irregularly-shaped domains

We propose a new method to analyze and represent stochastic data recorded on a domain of general shape by computing the eigen-functions of Laplacian defined over there (also called "geometric harmonics") and expanding the data into these eigenfunctions. In essence, what our Laplacian eigenfunctions do for data on a general domain is roughly equivalent to what the Fourier cosine basis functions do for data on a rectangular domain. Instead of directly solving the Laplacian eigenvalue problem on such a domain (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and then diagonalize that operator. We then show that our method is better suited for small sample data than the Karhunen-Loeve transform. In fact, our Laplacian eigenfunctions depend only on the shape of the domain, not the statistics (e.g. covariance) of the data. We also discuss possible approaches to reduce the computational burden of the eigenfunction computation