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

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Introduction to Compressive Sensing

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Introduction to compressive sensing

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


Outline

Outline

  • Introduction to compressive sensing (CS)

    • First CS theory

    • Concepts and applications

    • Theory

      • Compression

      • Reconstruction


Introduction

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

Compressive Sensing: concept and applications


Compression reconstruction

Compression/Reconstruction

Transmit

X RNx1

CS sampling

yRMx1

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 technology compression

Theory and Core Technologycompression

  • K-sparse

    • most of the energy is at low frequencies

    • Knon-zero wavelet (DCT) coefficients


Compression

Compression

Measurement matrix


Compression1

Compression

transform basis

coefficient


Compression2

Compression

transform basis

coefficient


Reconstruction

Reconstruction


Reconstruction optimization

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

Compressive sensing: significant parameters

  • What measurement matrix  should we use?

  • How many measurements? (M=?)

  • K-sparse?


Measurement matrix incoherence

Measurement Matrix  Incoherence

(1) Correlation between  and 


Examples

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

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.


How many measurements

How many measurements ?


Single pixel cs camera baraniuk and kelly et al

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


Introduction to compressive sensing

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)


Network scenario setting

Network Scenario Setting

X

Irregular network setting [4]

Graph wavelet

Diffusion wavelet

Example of the considered multi-hop topology.


Measurement matrix built on routing path

Measurement matrix Built on routing path

Routing path

……………………

………………

……………………

……………………


Measurement matrix

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

Transform basis

  • T1: DCT

  • T2: Haar Wavelet

  • T3: Horizontal difference

  • T4: Vertical difference + Horizontal difference


Degree of sparsity

Degree of sparsity

H-diff

VH-diff

Haar

DCT


Incoherence

Incoherence

DCT

Haar

H-diff

VH-diff


Performance comparison

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.

  • RS-CS

    • the data values are combined

      with that of any other node

      encountered along the path.

Routing path


Reconstruction error

Reconstruction Error


Reconstruction error pre distribution for t 3 and t 4 5

Reconstruction Errorpre-distribution for T3 and T4 [5]


Research issues when applying cs in sensor networks

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

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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