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Applications on Signal Recovering

Miguel ArgáezCarlos A. Quintero

Computational Science Program

El Paso, Texas, USA

April 16, 2009

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Abstract

Recent theoretical developments have generated a great deal of interest in sparse signal representation. A full-rank matrix

generates an underdetermined system of linear equations

Our purpose is to find the sparsest solution. i.e., the one with the fewest nonzero entries. Finding sparse representations ultimately requires solving for the sparsest solution of an underdetermined system of linear equations. Some recently works had shown that the minimum l1-norm solution to an underdetermined linear system is also the sparsest solution to that system under some conditions.

,mxnRAnm

.bAx

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Objective

We develop an algorithm using a fixed point method to solve the l1 minimization problem

And for solving the linear system associated to the problem we use a conjugate gradient method.

Our principal purpose for this work is to show that our algorithm is capable to work efficiently to recover the reflection coefficients from seismic data in the area of seismic reflection and separating two speakers in a single channel recording in the audio separation problem.

bAxxx s.t min arg1

*

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Seismic Reflection

Seismic reflection is a method of exploration geophysics to estimate the properties of the Earth's subsurface from reflected seismic waves.

The method requires a controlled seismic source of energy (such as dynamite or vibrators).

Using the time it takes for a reflection to arrive at a receiver, it is possible to estimate the depth of the feature that generated the reflection.

The reflected signal is detected on surface using

an array of high frequency geophones.

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Seismic Reflection

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Seismic Reflection

How can we obtain the reflectivity function from the

recorded signal? Problems:The seismic trace is the result of a convolution of

theinput pulse and the reflectivity function.The recorded signal has noise.

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We can express the recorded signal as:

where is the convolution between the signals and

represents the reflectivity function we want to recover.The convolution kernel is a “wavelet” which depends on the

pressure wave sent in the underground.

ε is noise that has entered the recorded signal.

(1) x y

Sparse-spike deconvolution

y

x .x

x

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Sparse-spike deconvolution

Clearbout and Mouir [3] proposed in 1973 to use l1 minimization to recover x from the recorder signal y.

Santosa and Symes (Rice University) [4] implemented this idea in 1986 with an l1 relaxed minimization.

The resulting sparse spike deconvolution algorithm defines the solution as: )2(

2

1minarg

1

2

2

ffyx

f

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To invert the convolution equation (1), we can model the reflectivity x as a sum of Diracs, that is:

Each Dirac is located at a depth i.

Using (3) we can express y as a function of the depth z

Sparse-spike deconvolution

)3(

Si

iiax

i

Si

i izazxzy )()()(

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Sparse-spike deconvolution

If we introduce a dictionary constructed by translating the wavelet at all locations

This is a matrix whose columns vectors are:

We solve the problem using a equivalent problem given by

niAAAAA iin ,...,1 , where, , 21

i

)4( s.t. minarg1

yAffxf

Seismic Reflection. Sparco Problem (903): m=n=1024

http://www.cs.ubc.ca/labs/scl/sparco/index.php/Main/problem903.html

Numerical Experimentations

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The Speech Separation Problem  

Separate a single-channel mixture of speech from known speakers

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Non-negative Sparse Coding

We assume an additive mixing modeland we can represent the signal as where A and x are non-negative and x is sparse

Axy 21 yyy

2

121 ,

x

xAAAxy

Dictionary, A Source dependent (over-complete) basis Learned from data

Sparse Code, x Time and amplitude for each dictionary element Sparseness: Only a few dictionary elements active

simultaneously

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Non-negative Sparse Coding

1. Learn a dictionary for each source

2. Compute sparse coding x of mixture

3. Reconstruct each source separately

21 , AA

yAffxf

s.t. minarg1

, , 222111 xAyxAy

Numerical Experimentation

Problem Sparco (401): m=29166, n=57344http://www.cs.ubc.ca/labs/scl/sparco/index.php/Main/problem401.html

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References and Acknowledgements

1) Stochastic sparse-spike deconvolution, Danilo R. Velis.

2) Deconvolution with curvelet-domain sparsity, Vishal Kumar, EOS-UBC and Felix J. Herrmann.

3) Robust modeling of erratic data, J.F. Clearbout and F. Muir. Geophyscis, 38

4) Linear inversion of band limited reflection seismograms. F. Santosa and W.W. Symes. SIAM J. Sci. Statistic.Comput

5) Sparse coding and NMF, J. Eggert and E. Körner, Proceedings of Neural Networks

The authors thank the financial support from:

ARL Grant No. W911NF-07-2-0027.

Computational Science Program,

NSF CyberShARE grant No. NSF HRD-0734825. (Some partial support)

We also acknowledge the office space provided by the Department of Mathematical Sciences.