Introduction to Compressive Sensing. Richard Baraniuk , Compressive sensing . IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007) Emmanuel Candès and Michael Wakin , An introduction to compressive sampling . IEEE Signal Processing Magazine, 25(2), pp. 21 - 30, March 2008
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Introduction to Compressive Sensing
Richard Baraniuk, Compressive sensing. IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007)
Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), pp. 21 - 30, March 2008
A course on compressive sensing, http://w3.impa.br/~aschulz/CS/course.html
Cand`es
Romberg
Tao
Transmit
X RNx1
CS sampling
yRMx1
Quantization
human coding
RMxN
Measurement matrix
CS Reconstruction
Optimization
Inverse transform
(e.g., IDCT)
X’
s
Inverse
Quantization
human coding
y’
: transform basis (e.g., DCT basis)
Measurement matrix
transform basis
coefficient
transform basis
coefficient
(1)
NP-hard problem
(2)
Minimum energy ≠ k-sparse
(3)
Linear programming [1][2]
Orthogonal matching pursuit (OMP)
(4)
Greedy algorithm [3]
(1) Correlation between and
= noiselet, = Haar wavelet (,)=2
= noiselet, = Daubechies D4 (,)=2.2
= noiselet, = Daubechies D8 (,)=2.9
= White noise (random Gaussian)
For each integer k = 1, 2, …, define the isometry constant k of a matrix A as the smallest number such that
signals
(2) Imply that k-sparse vectors cannot be in the nullspace of A
(3) All subsets of s columns taken from A are in fact nearly orthogonal
On the Interplay Between Routing and SignalRepresentation for Compressive Sensing inWireless Sensor Networks
G. Quer, R. Masiero, D. Munaretto, M. Rossi, J. Widmer and M. Zorzi
University of Padova, Italy.
DoCoMo Euro-Labs, Germany
Information Theory and Applications Workshop (ITA 2009)
X
Irregular network setting [4]
Graph wavelet
Diffusion wavelet
Example of the considered multi-hop topology.
Routing path
……………………
………………
……………………
……………………
from {+1, -1}
from(0, 1]
H-diff
VH-diff
Haar
DCT
DCT
Haar
H-diff
VH-diff
the data packets are not processed at internal nodes but simply forwarded.
with that of any other node
encountered along the path.
Routing path
[1] Bloomfield, P., Steiger, W., Least Absolute Deviations:
Theory, Applications, and Algorithms. Progr. Probab. Statist.
6, Birkhäuser, Boston, MA, 1983.
[2] Chen, S. S., Donoho, D. L., Saunders, M. A, Atomic
decomposition by basis pursuit. SIAM J. Sci. Comput. 20
(1999), 33–61.
[3] J. Tropp and A. C. Gilbert, “Signal recovery from partial
information via orthogonal matching pursuit,” Apr. 2005,
Preprint.
[4] J. Haupt, W.U. Bajwa, M. Rabbat, and R. Nowak, “Compressed
sensing for networked data,” IEEE Signal Processing Mag., vol. 25,
no. 2, pp. 92-101, Mar. 2008.
[5] M. Rabbat, J. Haupt, A. Singh, and R. Novak, “Decentralized Compression and Predistribution via Randomized Gossiping,” in IPSN, 2006.