Illinois/Missouri Applied Harmonic Analysis Seminar


Title: “Data Acquisition Schemes for Compressed Sensing in Magnetic Resonance Imaging”

Justin Haldar

University of Illinois, Urbana-Champaign

Abstract: Magnetic resonance (MR) imaging is a powerful tool for the non-invasive investigation of biological systems. With conventional MR acquisition, data sampling occurs in the Fourier domain. The large number of Fourier samples required for accurate high-resolution image reconstruction leads to relatively long experimental durations.

Compressed sensing has drawn recent interest due to the surprising result that sparse/compressible signals can often be recovered using a small set of non-adaptive measurements. This has the potential to significantly improve the speed of MRI data acquisition, and promising results have been demonstrated with quasi-random sampling in the Fourier domain. However, the design of optimal sampling schemes is still an important open problem.

This talk describes the design of MR acquisition schemes, leveraging theoretical results from the compressed sensing literature. It is demonstrated that non-Fourier encoding based on the use of specialized signal excitation can work significantly better than more traditional Fourier schemes. Noise sensitivity results are also presented.

Title: “On the nonexistence of AB scaling functions satisfying certain decay and smoothness constraints when B is a shear group with an odd number of generators”

Robert Houska

Washington University in St. Louis

Abstract: Let B be a countable shear group, and suppose that V is a closed subspace of L2(Rn) . Consider the following two properties:

1. There is a function φ є L2( Rn) satisfying certain decay and smoothness constraints such that the collection {DbTk φ: b є B, k є Zn } forms a frame for V, where Db represents dilation by b and Tk translation by k.
2. There is a matrix a є GLn(R) that “interacts well” with B such that V is contained in Da-1V.

We will show that for many such B, properties 1 and 2 above cannot both be satisfied when B has an odd number of generators. One consequence of this is the nonexistence of AB MRA wavelets with well-behaved scaling functions when B has an odd number of generators and the matrix a is as in 2 above.

Title: “Signal estimation from noisy frame coefficients”

Alexander Powell

Vanderbilt University

Abstract: We shall discuss the problem of estimating a signal from noisy versions of its frame coefficients. We prove refined mean squared error (MSE) bounds for existing estimation algorithms in the setting of random frames. We also derive new MSE bounds in deterministic settings and prove that frame ordering issues play an important role.

Title: “Analysis of Fractals, Image Compression and Entropy Encoding”

Myung-Sin Song

Department of Mathematics and Statistics, Southern Illinois University Edwardsville


Palle E. T. Jorgensen

Department of Mathematics and Statistics, The University of Iowa

Abstract: In this talk we show that algorithms in a diverse set of applications may be cast in the context of relations on a finite set of operators in Hilbert space. The Cuntz relations for a finite set of isometries form a prototype of these relations. Such applications as entropy encoding, analysis of correlation matrices (Karhunen-Loe`ve), fractional Brownian motion, and fractals more generally, admit multi-scales. In signal/image processing, this may be implemented with recursive algorithms using subdivisions of frequency-bands; and in fractals with scale similarity.

Title: “The basic properties of principal shift invariant spaces”

Guido Weiss

Washington University in St. Louis

Abstract: The material presented in the talk is not new; however, the methods used are new and lead to results that can be stated in a very simple way. The consequence of this approach is that it naturally leads to a considerable generalization and a theory that involves generating systems of functions associated with general locally compact Abelian Groups and associated unitary representations.

Illinois Wesleyan University
Mathematics and Computer Science Department
PO Box 2900
Bloomington, IL 61702-2900
Tel: (309) 556-3089   Fax: (309) 556-3864.