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Introduction to Compressive Sensing

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### Introduction to Compressive Sensing

### On the Interplay Between Routing and SignalRepresentation for Compressive Sensing inWireless Sensor Networks

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

Outline

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

Introduction

- 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

Compressive Sensing: concept and applications

Compression/Reconstruction

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)

Theory and Core Technologycompression

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

Compression

Measurement matrix

Reconstruction: optimization

(1)

NP-hard problem

(2)

Minimum energy ≠ k-sparse

(3)

Linear programming [1][2]

Orthogonal matching pursuit (OMP)

(4)

Greedy algorithm [3]

Compressive sensing: significant parameters

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

Measurement Matrix Incoherence

(1) Correlation between and

Examples

= 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)

Restricted Isometry Property (RIP)preserving length

- 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.

Single-Pixel CS Camera[Baraniuk and Kelly, et al.]

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)

Network Scenario Setting

X

Irregular network setting [4]

Graph wavelet

Diffusion wavelet

Example of the considered multi-hop topology.

Measurement matrix

- 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]

Transform basis

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

Performance Comparison

- 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

Reconstruction Errorpre-distribution for T3 and T4 [5]

Research issues when applying CS in Sensor Networks

- 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

References

[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.

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