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ABH I JI TH R K AS H YAP
0 7 E C 0 1
Sparse Signal Recovery
Seminar on :
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Organization
Introduction
Brief Timeline
Sparse Signals
Compressed Sensing Numerical Example
Recovery of CS Signal L1 Minimization
Orthogonal Matching Pursuit
Applications Summary
References
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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.
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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.
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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.
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Compressed Sampling
y is n x 1 Measurement Matrix is the n x m Sampling matrix x is the m x 1 sparse signal
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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
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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
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Comparison of Norms
L1 norm can be solved in many ways Linear Programming prob
Inner point methodSimplex Algorithm
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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.
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Applications
Can be directly extended to all signals that are sparsein transform domain.
EEG/MEG localizations
Single sensor camera Speech Coding
Spectral Estimation
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Single Sensor Camera
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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.
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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
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Thank you
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