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Randolph E. Bank
Philip E. Gill
Michael Holst
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Justin Romberg
Applied & Computational Mathematics, Caltech
Abstract:
A common problem in applied science is to recover a signal orimage from a set of indirect measurements. For example, inMagnetic Resonance Imaging (and many other imaging modes widelyused in medicine, astronomy, and other fields) we wish to reconstructan image from samples of its 2D Fourier spectrum. A naturalquestion arises: How many measurements do we need to recoverthe object of interest? In this talk, we will discuss some recentresearch that addresses this fundamental question, and present arecovery framework that performs surprisingly well (both in theoryand in practice).
A special instance of this theory yields a novel sampling theorem:Suppose that a finite N dimensional signal f has only B nonzeroFourier coefficients at unknown frequencies. Then we can recover fperfectly from a small number of samples (on the order of B log N)in the time domain.
The recovery procedure is nonlinear, and consists of solving atractable convex program. Despite its nonlinearity, the recoveryprocedure is exceptionally stable against measurement error.
The theory (and the recovery algorithm) can be extended to signalsand images that are sparse in a known representation, and "sampled"using a specified set of measurement signals. We will show how ourability to recover a sparse signal depends on the representation andthe measurements system obeying an uncertainty principle.
We will close by showing how these ideas can be applied to problemsin tomographic imaging and data compression.Tuesday, February 7, 2006
11:00AM AP&M 2402