Differences Between Observation and Sampling Error in Sparse Signal Reconstruction

The field of Compressed Sensing has shown that a relatively small number of random projections provide sufficient information to accurately reconstruct sparse signals. Inspired by applications in sensor networks in which each sensor is likely to observe a noisy version of a sparse signal and subsequently add sampling error through computation and communication, we investigate how the distortion differs depending on whether noise is introduced before sampling (observation error) or after sampling (sampling error). We analyze the optimal linear estimator (for known support) and an l1 constrained linear inverse (for unknown support). In both cases, observation noise is shown to be less detrimental than sampling noise and low sampling rates. We also provide sampling bounds for a non-stochastic l¿ bounded noise model.

[1]  Robert D. Nowak,et al.  Signal Reconstruction From Noisy Random Projections , 2006, IEEE Transactions on Information Theory.

[2]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[3]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[4]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[5]  J. W. Silverstein The Limiting Eigenvalue Distribution of a Multivariate F Matrix , 1985 .

[6]  M. Rabbat,et al.  Decentralized compression and predistribution via randomized gossiping , 2006, 2006 5th International Conference on Information Processing in Sensor Networks.

[7]  J. Haupt,et al.  Compressive wireless sensing , 2006, 2006 5th International Conference on Information Processing in Sensor Networks.

[8]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[9]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[10]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[11]  Martin J. Wainwright,et al.  Sharp thresholds for high-dimensional and noisy recovery of sparsity , 2006, ArXiv.

[12]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.