1. An Introduction to Compressed Sensing Student : Shenghan TSAI Advisor : Hsuan-Jung Su and Pin-Hsun Lin Date : May 02, 2014 1

2. Outline Introduction Signal---Sparse and Compressible - Sparse & Compressible - Power law - The p-norm in finite dimensions Sensing Matrices - NSP(Null space conditions) - RIP(Restricted isometry proerty) Sparse Signal Recovery Conclusion Conclusion 2

3. Introduction 3

4. Compressed Sensing Compressed sensing is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems.This takes advantage of the signal's sparseness or compressibility in some domain, allowing the entire signal to be determined from relatively few measurements. 4

5. History “If we sample a signal at twice its highest frequency, then we can recover it exactly.” Whittaker-Nyquist-Kotelnikov-Shannon Emmanuel Candès, Terence Tao, and David Donoho proved that given knowledge about a signal's sparsity(2004) Emmanuel CandèsTerence TaoDavid Donohosparsity 5

6. Motivation

8. How it work? 8

9. Deterministic compressive sensing Signal---Sparse and Compressible Sensing Matrices—NSP and RIP 9

10. Sparse & Compressible Sparse Model: Signals of interest are often sparse or compressible, i.e., very few large coefficients, many close to zero. Sparse signals: have few non-zero coefficients. i.e. K-sparse mean it has at most K nonzeros. Compressible signals: have few significant coefficients; coefficients decay as a power law. 10

11. Power law A signal is compressible if its sorted coefficient magnitudes in decay rapidly. x be a signal The signal should observe a power law decay : s=1,2,… q decay faster, more compressible 11

12. The p-norm in finite dimensions Lp mean norm p EX: P=2 Ex: p=0 Ex: p= ∞

13. The p-norm in finite dimensions The grid distance between two points is never shorter than the length of the line segment between them. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm Using Cauchy–Schwarz inequality.Cauchy–Schwarz inequality

15. Sensing Matrices NSP(Null space conditions) RIP(Restricted isometry proerty) 15

16. Sensing Matrices 16

17. Sensing Matrices 17

18. How to design Sensing Matrices If we sure our date is sparse and compressible, then we want to design Φ with M<<N and want it can recover To ensure Φ can recover there are two property NSP & RIP need to follow and we got a measurement bound 18

19. The Null space property If we want it can recover K-sparse signals it is that require Φx1 ≠ Φx2 for all K-sparse x1 ≠ x2 So we necessary that Φ must have at least 2K rows otherwise there exist K-sparse x1,x2 s.t. Φ(x1-x2)=0 Spark are almost the same meaning M >=2K 19

20. The Null space property Null space property (NSP) of order K if there exists a constant C > 0 holds for all and for all such that 20

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22. The restricted isometry property 22

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24. The RIP and NSP(relationship between RIP & NSP) Suppose that Φ satisfies the RIP of order 2K with. Then Φ satisfies the NSP of order 2K with constant ( 接第 X 幾頁 ) Suppose that Φ satisfies the RIP of order 2K, 24

25. Sparse Signal Recovery NSP(Null space conditions) RIP(Restricted isometry proerty) 25

26. Sparse Signal Recovery with noiseless in noise 26

27. Cont. 27

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29. Recovery in noise 29

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32. Conclusion If signals are sparse and compressible then can use CS to compressed. Signals can be perfect recovered,if satisfies NSP & RIP. 32

33. References 1. [ E. Cand[U+FFFD] The restricted isometry property and its implications for compressed sensing. Comptes rendus de l'Acad[U+FFFD]e des Sciences, S[U+FFFD]e I, 346(9-10):5898211;592, 2008.. 2.[Yu TSP 11] G. Yu and Guillermo Sapiro, “Statistical Compressed Sensing of Gaussian Mixture Models,” IEEE Trans of Signal Processing, vol. 59, no. 12, pp. 5842–5857, Dec. 2011. 3.[R. Baraniuk, M.A. Davenport, M.F. Duarte, C. Hegde], An Introduction to Compressive Sensing, CONNEXIONS, Rice University, Houston, Texas, 2010. 4. [R.G. Baraniuk,] “Compressive sensing,” IEEE Signal Processing Mag., vol. 24, no. 4, pp. 118–120, 124, 2007. 5. http://en.wikipedia.org/wiki/Lp_spacehttp://en.wikipedia.org/wiki/Lp_space 6. http://en.wikipedia.org/wiki/Compressed_sensinghttp://en.wikipedia.org/wiki/Compressed_sensing 33

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