Collaborative Research: Statistical Analysis on Manifolds: A Nonparametric Approach for Shapes and Images

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The shape or image of an object may be recorded digitally by a finite number k of landmarks or positions on the object, called a k-ad. The space of orbits under rotation and translation of k-ads is the size-and-shape space. If one includes scaling with rotation and translation, then the space of orbits under the resulting group of transformations on the k-adsis the shape space. These are examples of Riemannian manifolds, some with singularities. One approach proposed here for inference about a distribution on a general differentiable or Riemannian manifold M, based on a random sample from it, is to carry out a multivariate analysis of the image under the so-called Log map on one or more tangent spaces to M. Apart from this intrinsic approach, less computation intensive extrinsic procedures based on embeddings in Euclidean spaces are investigated. The project explores consistency and asymptotic distribution theory on manifolds for robust tests and confidence regions from both points of view--intrinsic and extrinsic. One motivation for this study comes from the need to identify deformations or shape changes for purposes of medical diagnosis and biomorphology. Immediate applications also arise to machine vision and pattern recognition. There are significant impacts of this research on these and many other fields. Another important goal of this project is to train students in the newly developed methodologies, leading to the dissemination of knowledge gained through this research and the creation of a body of technicians and experts to apply it.