# High Dimensional Signal Analysis and Fast Computation using Spares Transforms

## Abstract

A growing number of applications, ranging from network analysis to hyperspectral imaging, require the analysis, modeling, and processing of very high dimensional signals. For many of these applications, the traditional methods of statistics break down because the number of dimensions, p, can be much larger than the number of observations, n. So for example, if we need to model a high dimensional signal, such as an image or sensor network observation, from just a few snap shots of, then even estimating the data covariance is an ill-posed problem.

The objective of this talk is to introduce a framework of the analysis and processing of high dimensional signals which is based on a family of sparse transforms which we call the sparse matrix transform (SMT). The SMT is formed by a product of pair-wise coordinate rotations known as Givens rotations, and it can be viewed as a generalization of the Fast Fourier Transform (FFT) and the paraunitary wavelet transform. However, unlike the FFT, the SMT is appropriate for the analysis of time-varying systems and nonstationary random processes. In particular, the SMT is a fast transform that can be used to decorrelate or estimate the spectrum of a non-stationary random process, and it can also be used to dramatically reduce the computation of space-varying convolution.

The talk starts with a quick overview of research in model based image processing. Then we introduce the SMT and show how it can be used to estimate the covariance of high dimensional data with limited training data. Next we show how this same approach can be used to sparsify general matrix-vector operations, and we show how the technique can be applied to the problem of scatter reduction in digital photography.

******Biography******

Charles A. Bouman is the Michael J. and Katherine R. Birck Professor of Electrical and Computer Engineering at Purdue University where he also holds a courtesy appointment in the School of Biomedical Engineering and serves has a co-director of Purdue’s Magnetic Resonance Imaging Facility. He received his B.S.E.E. degree from the University of Pennsylvania, M.S. degree from the University of California at Berkeley, and Ph.D. from Princeton University in 1989.

Professor Bouman's research focuses on inverse problems, stochastic modeling, and their application in a wide variety of imaging problems including tomographic reconstruction and image processing and rendering. Prof. Bouman is a Fellow of the IEEE, AIMBE, IS&T, and SPIE and is a member of the IEEE Signal Processing Society’s Board of Governors. He has also served as the Editor-in-Chief of the IEEE Transactions on Image Processing and the Vice President of Publications for the IS&T Society.