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Introduction to Compressive Sensing

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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

- Introduction to compressive sensing (CS)
- First CS theory
- Concepts and applications
- Theory
- Compression
- Reconstruction

- Compressive sensing
- Compressed sensing
- Compressive sampling

- First CS theory
- E. Cand`es, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.

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)

- K-sparse
- most of the energy is at low frequencies
- Knon-zero wavelet (DCT) coefficients

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]

- What measurement matrix should we use?
- How many measurements? (M=?)
- K-sparse?

(1) Correlation between and

= noiselet, = Haar wavelet (,)=2

= noiselet, = Daubechies D4 (,)=2.2

= noiselet, = Daubechies D8 (,)=2.9

- Noiselets are also maximally incoherent with spikes and incoherent with the Fourier basis
= White noise (random Gaussian)

- RIP:
For each integer k = 1, 2, …, define the isometry constant k of a matrix A as the smallest number such that

- A approximately preserves the Euclidean length of k-sparse
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

- To design a sensing matrix , so that any subset of columns of size k be approximately 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

……………………

………………

……………………

……………………

- R1: is built according to routing protocol,
- randomly selected from {+1, -1}

- R2: is built according to routing protocol
- randomly selected from (0, 1]

- R3: has all coefficients in randomly selected
from {+1, -1}

- R4: has all coefficients in randomly selected
from(0, 1]

- T1: DCT
- T2: Haar Wavelet
- T3: Horizontal difference
- T4: Vertical difference + Horizontal difference

H-diff

VH-diff

Haar

DCT

DCT

Haar

H-diff

VH-diff

- Random sampling (RS)
- each node sends its data with probability P= M/N,
the data packets are not processed at internal nodes but simply forwarded.

- each node sends its data with probability P= M/N,
- RS-CS
- the data values are combined
with that of any other node

encountered along the path.

- the data values are combined

Routing path

- How to construct measurement matrix
- Incoherent with transform basis
- Distributed
- M=?

- How to choose transformation basis
- Sparsity
- Incoherent with measurement matrix

- Irregular sensor deployment
- Graph wavelet
- Diffusion wavelet

[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.