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Robust Compressive Sensing: Theory and Applications in Signal Processing

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This work provides an overview of robust compressive sensing, highlighting its theoretical underpinning and practical applications. We explore how compressed sensing techniques are applied to various domains, including network tomography and graph-constrained group testing. The concept of approximate sparsity and handling measurement noise are discussed, along with advancements such as the nearly optimal sparse Fourier transform and innovative one-pixel camera technology. Insights from key studies and recent developments in the field are also reviewed, emphasizing the importance of efficient reconstruction and recovery of signals.

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Robust Compressive Sensing: Theory and Applications in Signal Processing

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  1. m ? n m<n

  2. Compressive sensing ? m ? n k k ≤ m<n

  3. Robust compressive sensing ? e z y=A(x+z)+e Approximate sparsity Measurement noise

  4. Apps: 1. Compression W(x+z) x+z BW(x+z) = A(x+z) M.A. Davenport, M.F. Duarte, Y.C. Eldar, and G. Kutyniok, "Introduction to Compressed Sensing,"inCompressed Sensing: Theory and Applications, Cambridge University Press, 2012. 

  5. Apps: 2. Network tomography Weiyu Xu; Mallada, E.; Ao Tang; , "Compressive sensing over graphs," INFOCOM, 2011 M. Cheraghchi, A. Karbasi, S. Mohajer, V.Saligrama: Graph-Constrained Group Testing. IEEE Transactions on Information Theory 58(1): 248-262 (2012)

  6. Apps: 3. Fast(er) Fourier Transform H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. InProceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA, 563-578.

  7. Apps: 4. One-pixel camera http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

  8. y=A(x+z)+e

  9. y=A(x+z)+e

  10. y=A(x+z)+e

  11. y=A(x+z)+e

  12. y=A(x+z)+e (Information-theoretically) order-optimal

  13. (Information-theoretically) order-optimal • Support Recovery

  14. SHO(rt)-FA(st)O(k) meas., O(k) steps

  15. SHO(rt)-FA(st)O(k) meas., O(k) steps

  16. SHO(rt)-FA(st)O(k) meas., O(k) steps

  17. 1. Graph-Matrix A d=3 ck n

  18. 1. Graph-Matrix A d=3 ck n

  19. 1. Graph-Matrix

  20. 2. (Most) x-expansion ≥2|S| |S|

  21. 3. “Many” leafs L+L’≥2|S| ≥2|S| |S| 3|S|≥L+2L’ L≥|S| L+L’≤3|S| L/(L+L’) ≥1/2 L/(L+L’) ≥1/3

  22. 4. Matrix

  23. Encoding – Recap. 0 1 0 1 0

  24. Decoding – Initialization

  25. Decoding – Leaf Check(2-Failed-ID)

  26. Decoding – Leaf Check (4-Failed-VER)

  27. Decoding – Leaf Check(1-Passed)

  28. Decoding – Step 4 (4-Passed/STOP)

  29. Decoding – Recap. 0 0 0 0 0 0 0 0 1 0 ? ? ?

  30. Decoding – Recap. 0 1 0 1 0

  31. Noise/approx. sparsity

  32. Meas/phase error

  33. Correlated phase meas.

  34. Correlated phase meas.

  35. Correlated phase meas.

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