EE 520: Special Topics class on Compressive Sensing
The key goal of Compressive Sensing (CS) is: how and when can I reconstruct a signal using fewer measurements than the signal length, but using the fact that the signal is "sparse", in some domain. It answers the following question: if I am given a set of measurements y := Ax  where A is a fat matrix, and x is the unknown sparse signal, under what assumptions on A can I exactly reconstruct x from y using efficient (polynomial time) recovery algorithms? Extensions to noisy measurements and approximately sparse signals are considered.

CS uses the well known fact that has always been exploited by lossy compression algorithms such as JPEG and JPEG2000 that natural signals and images are approximately sparse (compressible) in some domain. JPEG assumes compressibility in the DCT domain, JPEG2000 assumes compressibility in the wavelet transform domain (valid for piecewise smooth signals/images). But traditional compression techniques first acquire all the data, then compress it. The question is do we need to acquire all the data at all?
CS is particularly useful for  MRI or CT applications since both acquire measurements one at a time (or one radial line at a time) and in the Fourier domain (CT measurements can be converted to Fourier domain). Thus their scan time is directly proportional to number of measurements needed. Thus if we can reconstruct accurately with less measurements, we would reduce scan time!
A new type of camera, the single-pixel camera, has been built at Rice that tries to acquire measurements sequentially!


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