8/6/2019 Sparse Signal Recovery

1/16

ABH I JI TH R K AS H YAP

0 7 E C 0 1

Sparse Signal Recovery

Seminar on :

8/6/2019 Sparse Signal Recovery

2/16

Organization

Introduction

Brief Timeline

Sparse Signals

Compressed Sensing Numerical Example

Recovery of CS Signal L1 Minimization

Orthogonal Matching Pursuit

Applications Summary

References

8/6/2019 Sparse Signal Recovery

3/16

Introduction

Sparse signal Most samples are zero.

Compressed Sampling (CS) Intelligently samplesparse signals.

Sampling rate < Nyquist rate Recovery from CS Combinatorially Complex

1 Optimization

Orthogonal Matching Pursuit.

8/6/2019 Sparse Signal Recovery

4/16

Brief Timeline

1967 R.R. Hocking and R.N. Leslie : Selection ofthe best subset in Regression Analysis

1991 S.D. Cabrera and T.W. Parks : Extrapolation

and spectral estimation with iterative weighted normmodification

2006 Donoho, D. L. : Compressed Sensing,IEEE Transactions on Information Theory

Many new theories related to Sparse signals after2006.

8/6/2019 Sparse Signal Recovery

5/16

Sparse Signals

x is known to be S-sparse for some 1 < S < n, which means that atmost S of the samples of x can be non-zero.

Here n = 100 and S = 5.

8/6/2019 Sparse Signal Recovery

6/16

Compressed Sampling

y is n x 1 Measurement Matrix is the n x m Sampling matrix x is the m x 1 sparse signal

8/6/2019 Sparse Signal Recovery

7/16

Numerical Example

1

0

2

0

0

2

5

3

1

4

12233

32412

61141

3

5

4

0xx

y

Non SparseSoln.

SparseSoln.

Measurements Sampling Matrix

8/6/2019 Sparse Signal Recovery

8/16

Recovery from CS Signals - LP

Norm LP Norm is defined as

For p>1

L0 norm

L1 norm

Ideally solve for x given y and such that L0

norm isminimum

Donoho, Candes, Terence proved L1 norm minimizationis sufficient for sparse signals.

0

10 n

i ixx

n

i ixx

11

8/6/2019 Sparse Signal Recovery

9/16

Comparison of Norms

L1 norm can be solved in many ways Linear Programming prob

Inner point methodSimplex Algorithm

8/6/2019 Sparse Signal Recovery

10/16

Recovery from CS Signals - OMP

Regression basedoptimization

Select a column that ismost correlated with the

current residual. Remove contribution of

that column to form newresidual.

Loop until results aresatisfactory.

8/6/2019 Sparse Signal Recovery

11/16

Applications

Can be directly extended to all signals that are sparsein transform domain.

EEG/MEG localizations

Single sensor camera Speech Coding

Spectral Estimation

8/6/2019 Sparse Signal Recovery

12/16

Single Sensor Camera

8/6/2019 Sparse Signal Recovery

13/16

8/6/2019 Sparse Signal Recovery

14/16

Summary

Sparse signal recovery is an interesting area withmany potential applications.

Methods developed are valuable tools in Signal

Processing. Widely applicable Many naturally occurring

signals are sparse.

Expectation that there will be continued growth in

the application and algorithm development.

8/6/2019 Sparse Signal Recovery

15/16

References

Bhaskar Rao and David Wipf, Sparse Signal Recovery:Theory, Application and Algorithms, IEEE SPCOM,July 2010

Donoho, D. L., Compressed Sensing, IEEETransactions on Information Theory, V. 52(4), 12891306, 2006

Cands, E.J., & Wakin, M.B., An Introduction ToCompressiveSampling, IEEE Signal ProcessingMagazine, V.21, March 2008

Rice University Web Resource,http://dsp.rice.edu/cscamera Terence Tao, Compressed sensing Or: the equation

Ax = b, revisited ,Mahler Lecture Series

8/6/2019 Sparse Signal Recovery

16/16

Thank you

Questions