Illinois/Missouri Applied
Harmonic Analysis Seminar
September 27, 2008
Akram Aldroubi - Vanderbilt University
Recent research and new paradigms in mathematics, engineering, and science assume non-linear signal models of the form M = [i2IVi consisting of a union of subspaces Vi instead of a single subspace M = V . For example, these models have been used in 1) Sampling and reconstruction of signals with nite rate of innovation; 2) The Generalized Principle Component Analysis and the subspace segmentation problem in computer vision; and 3) Problems related to sparsity, compressed sensing, and dictionary design. In this talk, we will present a mathematical framework and computational schemes that unify, extend, and complement some of the techniques used in sampling theory, the Generalized Principle Components Analysis, and the dictionary design problem. We will also show how this new frame- work can be applied to several problems in engineering and biomedicine including data classication and segmentation (e.g., face recognition, brain morphology, DNA sequence comparison, movement tracking), and signal modeling (e.g., for signals with nite rate of innovation).
Kanghui Guo - Missouri State University
It is known that the continuous wavelet transform has the ability to signal the location of the singularities of a function or distribution from its asymptotic decay at fine scales. However, the wavelet approach only yields a limited information since it is unable to provide a description of the geometry of discontinuities in dimensions larger than one. On the contrary, we show that the continuous shearlet transform can identify both the location and orientation of piecewise smooth edges for a planar object. In particular, this approach can be applied to exactly detect and analyze the corner points on the edges.
This work fully extends the result of Candes and Donoho, who showed that the continuous curvelet transform can detect the location and orientation of both the smooth edge (circle) for a disk and the piecewise smooth edges (segments) for a polygon.
Jeff Hogan - University of Arkansas
In this talk we attempt to synthesize recent progress made in the mathematical and electrical engineering communities on topics in Clifford analysis and the processing of color images (for example), in particular the construction and application of Clifford-Fourier transforms designed to treat vector-valued signals. Emphasis will be placed on the two-dimensional setting where the appropriate underlying Clifford algebra is the set of quaternions. We'll conclude with some results and problems in the construction of discrete wavelet bases for color images, and the difficulties encountered in constructing the correct Fourier kernels in dimensions 3 and higher.
Abstract in pdf
Ka Lung Law - University of Illinois at Urbana-Champaign
We study the invertibility of M-variate polynomial (respectively : Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multiple-input multiple-output systems, and multirate systems. Given an N x P polynomial matrix H(z1, ..., zM) of degree at most k, we want to find a P x N polynomial (resp. : Laurent polynomial) left inverse matrix G(z) of H(z) such that G(z)H(z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. Once a particular inverse is found, we can characterize all inverses and find an optimal inverse according to a design criterion. The main result of this paper is to prove that when N гн P >= M, then H(z) is generically invertible; whereas when N гн P < M, then H(z) is generically noninvertible. As a result, we propose a faster algorithm to find a particular inverse of a Laurent polynomial matrix.
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Southern Illinois University Edwardsville
Department of Mathematics and Statistics Campux Box 1653 Edwardsville, IL 62026 Phone: (618) 650-2385 FAX: (618) 650-3771. |