Linear representations are ubiquitous in all areas of computational sciences. Many applications in image analysis and computer vision involve analysis of large-dimensional data by projecting them linearly to low-dimensional subspaces. In view of their computational efficiency such linear projections have become standard in certain applications. However, for recognizing objects from their images, there is seldom a discussion on finding "optimal" linear representations. Many societal, commercial, and scientific operations, such as homeland security and biometrics, rely heavily on image-based recognition and the recognition performance becomes a vital factor. This project aims to provide efficient algorithms for finding linear and non-linear representations that perform optimally in the context of object recognition. We propose to achieve this goal by: (i) formulating the search for optimal linear representations as that of optimization on Grassmann manifolds, (ii) using the geometry of Grassmannians to develop algorithms for finding optimal linear representations, (iii) analyzing the convergence properties and the theoretical limits of the proposed optimization techniques, and (iv) demonstrating the potential significant performance improvement on problems wherever linear representations are applicable. Since there are numerous applications utilizing dimension reduction using linear projections, including object recognition, image and text retrieval, subspace tracking, and nonlinear filtering, the potential benefits of the proposed research are tremendous. This research will be built on tools for stochastic optimization and statistical inferences on nonlinear manifolds, tools that will prove beneficial in many other applications.
This research will also enhance significantly the learning and research environment on computer vision at the Florida State University. Utilization of geometric and statistical approaches to applications in computer vision makes this a multidisciplinary effort to the benefit of all participants, including both graduate and undergraduate students. Outcomes of this research will be incorporated in recently designed courses on Computer Vision and Computational Statistics.
