Nonparametric Statistical Image Analysis: Theory and Applications

University Of Arizona, Tucson AZ

Investigators

Abstract

Non-Euclidean data are ubiquitous. They arise in many forms such as digital images for (1) medical diagnostics (MRI, CT scans, DTI of the brain), (2) scene recognition from satellite images, (3) identifying defects in manufactured products, (4) artificial intelligence (robotic identification of objects), etc. Proper geometric descriptions of these require tools from modern differential geometry. Classical statistical methods are inadequate for their analysis; parametric models, which assume the form of the distribution of the underlying data modulo a finite number of unknown parameters, are often misspecified. A model-independent methodology developed by the PI and others has been shown to be very effective in analyzing such data. The present project aims at vastly broadening the scope of this methodology for applications. A basic component of the methodology proposed is the notion of the Fre'chet mean of a probability Q on a metric space, which minimizes the expected squared distance from a point. The metric space is generally a differential manifold, often provided with a natural Riemannian metric. But it may also be a so called geodesic space of non-positive curvature, including many graphical spaces as well as stratified spaces made up of manifolds of different dimensions glued together. For the methodology to work one must establish (a) the uniqueness of the Fre'chet minimizer and (b) the asymptotic distribution of the sample Fre'chet mean. It is one of the goals of the present project to significantly extend the earlier theory in this regard, opening the way to many new applications. Another important objective is to extend to such spaces the nonparametric Bayes theory of density estimation, classification and regression. One special aim here is to explore an intriguing phenomenon: in simulation studies with moderate sample sizes, the nonparametric Bayes estimator of the density of Q far outperforms not only the kernel density estimator, but also the MLE when the data are simulated from a parametric model! An understanding of this is expected to lead to a wider and more effective use of the nonparametric Bayes methodology. Finally, the PI proposes to develop a graphical method for robotic vision of objects, with much less computational complexity than that of other methods. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.