1. Compressive Signal Processing
Richard Baraniuk
Rice University
dsp.rice.edu/cs
Came out of my personal experience with 301 – fourier analysis and linear systems

2. Compressive Sensing (CS)
When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss
Random projection will work
sparse signal
measurements
sparse in some basis
[Candes-Romberg-Tao, Donoho, 2004]

3. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
sparse signal
measurements
nonzero entries

4. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
L2 fast

5. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
L2 fast, wrong

6. Why L2 Doesn’t Work
least squares, minimum L2 solution is almost never sparse
null space of translated to (random angle)

7. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
L2 fast, wrong L0
number of nonzero entries:
ie: find sparsest
potential solution

8. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
L2 fast, wrong
L0 correct, slow only M=K+1 measurements required to perfectly reconstruct K-sparse signal [Bresler; Rice]

9. CS Signal Recovery
Reconstruction/decoding: given (ill-posed inverse problem) find
L2 fast, wrong
L0 correct, slow
L1 correct, mild oversampling [Candes et al, Donoho]
linear program

10. Why L1 Works
minimum L1 solution = sparsest solution (with high probability) if

11. Universality
Gaussian white noise basis is incoherent with any fixed orthonormal basis (with high probability)
Signal sparse in time domain:

12. Universality
Gaussian white noise basis is incoherent with any fixed orthonormal basis (with high probability)
Signal sparse in frequency domain:
Product remains white Gaussian