EXACT SIGNAL RECOVERY FROM SPARSELY CORRUPTED MEASUREMENTS

Transcription

1 EXACT SIGNAL RECOVERY FROM SPARSELY CORRUPTED MEASUREMENTS THROUGH THE PURSUIT OF JUSTICE Jason Laska, Mark Davenport, Richard Baraniuk SSC 2009

2 Collaborators Mark Davenport Richard Baraniuk

3 Compressive Sensing measurements sparse signal nonzero entries Hallmarks fuses sensing and compression non-adaptive measurements democratic asymmetric sensing and signal reconstruction randomized measurement systems [Candes, Romberg, Tao]

4 Compressive Sensing Setup measurements sparse signal nonzero entries Restricted Isometry Property (RIP) for all -sparse If elements of are drawn from subgaussian distribution, then has RIP with high probability for [Candes, Romberg, Tao; Baraniuk et. al.]

5 Stable Reconstruction Basis Pursuit (BP) Basis Pursuit Denoising (BPDN) for bounded noise: (no noise) Reconstruction Error best K-term approximation error due to measurement noise error due to K-term approximation [Candes, Romberg, Tao; Donoho]

6 Corrupted Measurements measurements sparse signal corrupted measurements noise-free measurements nonzero entries bursts of high noise (e.g., hardware power-supply spikes) measurements dropped during transmission malfunctioning sensors in network * magnitudes of corruptions are potentially unbounded or is large

7 Corrupted Measurements measurements sparse signal discard corrupted measurements nonzero entries Suppose location of the corruptions is known: democracy: we can discard corrupted measurements [Laska, Boufounos, Davenport, Baraniuk]

8 Corrupted Measurements measurements sparse signal discard corrupted measurements discard rows of nonzero entries Suppose location of the corruptions is known: democracy: we can discard corruption measurements exact reconstruction of signal BPDN guarantees are suboptimal in this setup [Laska, Boufounos, Davenport, Baraniuk]

9 Corrupted Measurements measurements sparse signal nonzero entries democracy exact recovery from, [Laska, Boufounos, Davenport, Baraniuk] Problem: in general, do not know locations of corruptions

10 Noise Model Sparse noise model measurement matrix noise basis or subspace ( ) -sparse signal -sparse noise (potentially unbounded) nonzeros In previous examples, Other Examples 60Hz/50Hz Hum [and harmonics] Abrupt changes in DC bias

11 Justice Pursuit In justice is all virtues found in [random] sum -Aristotle since -sparse vector Justice Pursuit (JP) Example

12 Justice Pursuit Justice Pursuit (JP) Hallmarks exact signal recovery exact noise recovery justice is blind to the location and magnitude of corruption (error can be unbounded) no user-defined parameters standard algorithms can be trivially modified, i.e., justified Requirements justified matrix must have RIP of order

13 RIP of Justified Matrices Theorem : matrix with entries : matrix with orthonormal columns If then has RIP of order with probability (where, depend on RIP constant )

14 Related Work Wright, et. al. Dense error correction via -minimization - proposes and analyzes JP in face recognition setting where consists of highly correlated training vectors and does not have RIP Carrillo, et. al. Robust sampling and reconstruction methods for sparse signals in the presence of impulsive noise - considers sparse noise model - uses probabilistic approach Others Uses Wojtaszczyk Harmany, Marcia, and Willett Rish and Grabarnik Candes and Tao to recover higher dimension intermediate vector Unbounded and/or alternative noise models Decoding with sparse errors

15 Average Error: Noise Norm Parameters: Average Error Fixed: Noise Norm=0.01 Noise Norm=0.2 Noise Norm=0.3 JP BPDN M/N

16 Average Error: Noise Sparsity Parameters: Fixed: Average Error !=10!=40!=70 JP BPDN M/N

17 Recovery with Hum (simulated) Simulation: 60Hz tone added to measurements, M/N = 0.2 BPDN JP

18 Denoising CS Camera Data CS camera data [Kelly Lab] BPDN with M/N = 0.1

19 Denoising CS Camera Data CS camera data [Kelly Lab] 1)Recover, with JP 2)Create by choosing largest 15 entries from 3)BPDN from with

20 Denoising CS Camera Data CS camera data [Kelly Lab] 1)Recover, with JP 2)Create by choosing largest 15 entries from 3)BPDN from with

21 Justice and Democracy Revisiting Democracy measurements sparse signal exact recovery from, nonzero entries Relationship democracy was original motivation for justice justified can be used to prove the democracy property [Davenport, Laska, Boufounos, Baraniuk]

22 Conclusion Justice Pursuit extends capabilities of noisy CS systems by exploiting the structure of the added noise Hallmarks of Justice exact sparse signal and sparse noise recovery handles unbounded sparse errors no user-defined parameters Applications measurements dropped during transmission malfunctioning sensors in sensor network 60Hz/50Hz Hum Abrupt changes in DC bias dsp.rice.edu/cs

23 EXTRA SLIDES

24 Unbounded Recovery Parameters: 1 Probability of exact reconstruction !=10!=20!= M