Applications on Signal Recovering

1. Applications on Signal Recovering Miguel Argáez Carlos A. Quintero Computational Science Program El Paso, Texas, USA April 16, 2009

2. 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. 2

3. 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. 3

4. 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. 4

5. Seismic Reflection

6. How can we obtain the reflectivity function from the recorded signal? Problems: The seismic trace is the result of a convolution of the input pulse and the reflectivity function. The recorded signal has noise. Seismic Reflection

7. 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. Sparse-spike deconvolution 7

8. 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: 8

9. 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 9

10. 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 10

11. Numerical Experimentations Seismic Reflection. Sparco Problem (903): m=n=1024 http://www.cs.ubc.ca/labs/scl/sparco/index.php/Main/problem903.html 11

12. The Speech Separation Problem   Separate a single-channel mixture of speech from known speakers 12

13. Non-negative Sparse Coding We assume an additive mixing model and we can represent the signal as where A and x are non-negative and x is sparse • 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 13

14. Non-negative Sparse Coding Learn a dictionary for each source Compute sparse coding xof mixture Reconstruct each source separately 14

15. Numerical Experimentation Problem Sparco (401): m=29166, n=57344 http://www.cs.ubc.ca/labs/scl/sparco/index.php/Main/problem401.html 15

16. 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. 16