1. Richard Baraniuk Rice University dsp.rice.edu/cs Compressive Signal Processing

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

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

4. Reconstruction/decoding:given (ill-posed inverse problem) find L 2 fast CS Signal Recovery

5. Reconstruction/decoding:given (ill-posed inverse problem) find L 2 fast, wrong CS Signal Recovery

6. Why L 2 Doesn’t Work least squares, minimum L 2 solution is almost never sparse null space of translated to (random angle)

7. Reconstruction/decoding:given (ill-posed inverse problem) find L 2 fast, wrong L 0 CS Signal Recovery number of nonzero entries: ie: find sparsest potential solution

8. Reconstruction/decoding:given (ill-posed inverse problem) find L 2 fast, wrong L 0 correct, slow only M = K +1 measurements required to perfectly reconstruct K -sparse signal [Bresler; Rice] CS Signal Recovery

9. Reconstruction/decoding:given (ill-posed inverse problem) find L 2 fast, wrong L 0 correct, slow L 1 correct, mild oversampling [Candes et al, Donoho] CS Signal Recovery linear program

10. Why L 1 Works minimum L1 solution = sparsest solution (with high probability) if

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

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

13. Ex: Sub-Nyquist Sampling Nyquist rate samples of wideband signal (sum of 20 wavelets) N = 1024 samples/second Reconstruction from compressive measurements M = 150 random measurements/second (6.8x sub-Nyquist) MSE < 2% of signal energy

14. Ex: Sub-Nyquist Sampling Nyquist rate samples of image (N = 65536 pixels) Reconstruction from M = 20000 compressive measurements (3.2x sub-Nyquist) MSE < 3% of signal energy

15. Ex: Sub-Nyquist Sampling Nyquist rate samples of image (N = 65536 pixels) Reconstruction from measurements from a compressive camera M = 11000 M = 1300 measurements measurements