Richard Baraniuk Rice Universitydsp.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

measurementssparsesignal

sparsein some

basis

[Candes-Romberg-Tao, Donoho, 2004]

• Reconstruction/decoding: given(ill-posed inverse problem) find

CS Signal Recovery

measurementssparsesignal

nonzeroentries

• Reconstruction/decoding: given(ill-posed inverse problem) find

• L2 fast

CS Signal Recovery

• Reconstruction/decoding: given(ill-posed inverse problem) find

• L2 fast, wrong

CS Signal Recovery

Why L2 Doesn’t Work

least squares,minimum L2 solutionis almost never sparse null space of

translated to(random angle)

• Reconstruction/decoding: given(ill-posed inverse problem) find

• L2 fast, wrong

• L0

CS Signal Recovery

number ofnonzeroentries:

ie: find sparsestpotential solution

• Reconstruction/decoding: given(ill-posed inverse problem) find

• L2 fast, wrong

• L0 correct, slowonly M=K+1 measurements required to perfectly reconstruct K-sparse signal[Bresler; Rice]

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]

CS Signal Recovery

linear program

Why L1 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 measurementsM = 150 random measurements/second (6.8x sub-Nyquist)

MSE < 2% of signal energy

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

Ex: Sub-Nyquist Sampling• Nyquist rate samples of image

(N = 65536 pixels)

• Reconstruction from measurements from a compressive cameraM = 11000 M = 1300 measurements measurements