Disciplines/fields: Mathematics / Applied Harmonic Analysis, Mathematical Sparsity, Compressed Sensing

Duration: 4 sessions

Course Content

The first session will give an introduction to some of the most influential transforms developed and investigated within the field of harmonic analysis, such as the Fourier transform, Gabor transform, wavelet transform and transforms specifically designed to analyse higher dimensional data. We will discuss their historical roots as well some of their mathematical properties but most importantly, we will talk about how they work and what they are good for.

In the second lecture, we will define a mathematical framework conceptually unifying many of the transforms from the first session. This will lead to a mathematical notion of sparsity which can then be used to assess how efficiently a specific transform is capable of representing certain kinds of data.  In fact, signals from many classes of interest can be sparsely represented by applying the right transform, which may be defined a priori or learned from training data. The session will close with an introduction to the theory of compressed sensing, which leverages such sparse representations to determine a desired signal with far fewer measurements than would be normally required.

The third session will be devoted to practical applications of the theory developed so far in the realm of signal and image processing. These applications will include the removal of noise and the restoration of missing parts in images and videos as well as the separation of two distinct components in a signal (e.g. texture and contour) by considering two different transforms at the same time. Furthermore, we will outline how compressed sensing can be applied to significantly reduce the necessary amount of measurements in Magnetic Resonance Imaging (MRI). 

In the fourth and final session, we will take a look at how tools and ideas from applied harmonic analysis can improve the understanding and modelling of human perception and vice versa.

Objectives

This course aims to provide people with little to no mathematical background with a good understanding of the basic concepts and applications of widely used tools such as the Fourier transform or wavelet-like transforms. Furthermore, it should give an idea of how these transforms and their properties as well the concept of sparsity itself can be formulated mathematically and how these formulations can be turned into practical applications.

Literature

Terence Tao, “Compressed sensing and single-pixel cameras.” https://terrytao.wordpress.com/2007/04/13/compressed-sensing-and-single-pixel-cameras/

Michael Elad, “Sparse and Redundant Representation Modeling — What Next?”  IEEE Signal Process. Lett. 19(12) (Dec. 2012): 922-8. 

Martin Vetterli, “Wavelets, Approximation, and Compression.” IEEE Signal Process. Mag (Sep. 2001): 59-73.

Jianwei Ma and Gerlinde Plonka. “The Curvelet Transform: A review of recent applications.” IEEE Signal Process. Mag (Mar. 2010): 118-133.

John G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters."  J Opt Soc Am A 2(7) (1985):1160-9.

Jeremy Freeman and Eero P. Simoncelli, "Metamers of the ventral stream."  Nature Neuroscience 14(9) (2011): 1195-201

Vita

Emily J. King is a junior professor in mathematics at the University of Bremen.  As a Humboldt Fellow, she worked at the Technical University of Berlin, the University of Bonn, and the University of Osnabrück, focussing on pure and applied harmonic analysis.  Prior to that, she was an IRTA Postdoctoral Fellow at the National Institutes of Health in the U.S., where she worked in various fields of mathematical biology.  She earned her Ph.D. from the University of Maryland her B.S./M.S. from Texas A&M University, all in mathematics or applied mathematics.  Her current research involves bridging the gap between pure and applied harmonic analysis, from integrating geometric methods with finite frame theory to using sparsity-based methods to analyse atmospheric data.

Rafael Reisenhofer is a doctoral student at the Department of Mathematics at the University of Bremen. Besides a master’s degree in mathematics from TU Berlin, he also received a bachelor's degree in cognitive science from the University of Osnabrück. His current research mostly revolves around image processing applications of a complex-valued wavelet-like transform, the so-called complex shearlet transform. In general, he hopes that a profound understanding of mathematical tools and concepts might be helpful in gaining a deeper insight into the mysteries of human cognition.

Link

http://www.math.uni-bremen.de/zetem/cms/detail.php?template=parse_title&person=EmilyKing&language=en