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

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Presentation transcript:

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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