1. An Introduction to Compressive Sensing Speaker: Ying-Jou Chen Advisor: Jian-Jiun Ding

2. Compressive Compressed Sensing Sampling CS

3. Outline Conventional Sampling & Compression Compressive Sensing Why it is useful? Framework When and how to use Recovery Simple demo

4. Review… Sampling and Compression

5. Nyquist’s Rate

6. Transform Coding

7. Compressive Sensing

9. Comparison Nyquist’s SamplingCompressive Sensing Sampling Frequency RecoveryLow pass filterConvex Optimization

10. Some Applications ECG One-pixel Camera Medical Imaging: MRI

11. Framework N M N M

12. When? How? Two things you must know…

13. When…. Signal is compressible, sparse… N M N M

14. Example… ECG

15. How… How to design the sampling matrix? How to decide the sampling rate (M)? M N

16. Sampling Matrix Low coherence

17. Coherence

18. Example: Time and Frequency

19. Fortunately…

20. More about low coherence… Random Sampling

21. Sampling Rate Can be exactly recovered with high probability. Theorem S: sparsity n: signal length

22. Recovery BUT…. M N M N N

23. Many related research… – GPSR (Gradient projection for sparse reconstruction) – L1-magic – SparseLab – BOA (Bound optimization approach) …..

24. Total Procedure f Sampling (Assume f is spare somewhere) Recovering

25. Sum up 1. 有 size 為 nx1 在某 domain 上 sparse 的訊號 2. 用 size 為 mxn 的 random matrix 做 sampling (m<n) 3. 得到 size 為 mx1 的 measurement y 4. 將 y 做 L1 norm recovery 還原得到 x_recovery

26. Demo Time

27. Reference Candes, E. J. and M. B. Wakin (2008). "An Introduction To Compressive Sampling." Signal Processing Magazine, IEEE 25(2): 21-30. Baraniuk, R. (2008). Compressive sensing. Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference on. Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An Introduction to Compressive Sensing. https://sites.google.com/site/igorcarron2/cs#sparse http://videolectures.net/mlss09us_candes_ocsssrl1m/

28. Thanks a lot!

29. Key Points 1.Nyquist’s Rate 2.CS and Transform coding… 3.Sampling in time V.S. Sampling as inner products 4.About compressibility 5.About designing sampling matrix 6.About L1 norm explanation by geometry! 7.Application( MRI, One-pixel camera…)