Spectral signatures of point clouds and applications to detection of Alzheimer´s Disease through Neuroimaging

We introduce a class of spectral shape signatures constructed from symmetric functions on the eigenfunctions of the Laplacian exponentially weighted by their eigenvalues. Such a construction is motivated by problems that arise in the use of the eigenfunctions for shape comparison, such as indeterminacies in the choice of signs and the particular ordering in which the eigenfunctions are presented. The spectral invariants are applied to the analysis of Alzheimer´s disease (AD) data collected by the Alzheimer´s Disease Neuroimaging Initiative, in particular, to the problem of determining whether the signatures can aid in early detection of AD through morphology and imaging.