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Richard Baraniuk Rice University dsp.rice/cs

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

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Richard Baraniuk Rice University dsp.rice/cs

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

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

  3. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find sparsesignal measurements nonzeroentries

  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 solutionis 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 ofnonzeroentries: ie: find sparsest potential solution

  8. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0correct, slowonly 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 • L1correct, 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

  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 cameraM = 11000 M = 1300 measurementsmeasurements

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