Illinois/Missouri Applied Harmonic Analysis Seminar

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”

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

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

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