Collaborative Research: Nonparametric Theory on Manifolds of Shapes and Images, with Applications to Biology, Medical Imaging and Machine Vision.

Florida State University, Tallahassee FL

Investigators

Abstract

Much of the focus of this collaborative project is on the analysis of landmark based shapes in which a k-ad, i.e., a set of k points or landmarks on an object or a scene are observed in 2-D or 3-D, usually with expert help, for purposes of identification, discrimination, or diagnostics. Depending on the way the data are collected or recorded, the appropriate shape of an object is the maximal invariant specified by the space of orbits under a group G of transformations. In particular, Kendall's shape spaces of k-ads are invariant under scaling and Euclidean rigid motions. While this is a proper choice for many problems in biology and medical imaging, other notions of shape such as affine shape and projective shape are important in machine vision and bioinformatics. All these spaces are differentiable manifolds, often with natural Riemannian structures for measuring lengths and angles. The statistical analysis based on Riemannian structures is said to be intrinsic. In other cases, proper distances are sought via an equivariant embedding of the manifold M in a vector space E. Corresponding statistical analysis is called extrinsic. Finding proper Riemannian structures and equivariant embeddings is one of the objectives of this project, which is crucial for the statistical inference proposed. Establishing broad conditions for the existence of the Fre´chet mean, as the unique minimizer of the Fre´chet function the expected squared distance from a Q-distributed random shape is important for statistical inference; and it is a goal of the project to pursue, especially for intrinsic analysis where it has remained an outstanding open problem from the inception of shape theory. Reconstruction of a scene from two (or more) aerial photographs taken from a plane is one of the research problems in affine shape analysis. Potential applications of projective shape analysis proposed here include face recognition and robotics-for robots to visually recognize a scene. Statistical analysis of data on geometric objects, or manifolds, is an exciting and challenging field of research, where statistical theory and differential geometry are inextricably intertwined, and implementation requires innovative algorithms and high speed computation. The project proposed here deals with the development of nonparametric methodology in this context, which must also resolve associated geometric issues and problems of implementation. Past progress in this field by the PIs has laid the foundation for the present project. The statistical analysis proposed has wide ranging applications, especially in biology and bioinformatics, health sciences, and machine vision. Under this project, the PIs plan to train both undergraduate and graduate students, as well as at least one postdoctoral fellow, in theory and in its practical implementation. This continues much further the present activities of the PIs in this regard. In addition, computational algorithms and codes are made available on websites to create and disseminate this research and its applications.