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An Introduction to Compressive Sensing

An Introduction to Compressive Sensing. Speaker: Ying- Jou Chen Advisor: Jian-Jiun Ding. Compressive Compressed. Sensing Sampling. CS. Outline. Conventional Sampling & Compression Compressive Sensing Why it is useful? Framework When and how to use Recovery Simple demo. Review…

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An Introduction to Compressive Sensing

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

  2. CompressiveCompressed SensingSampling 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 • Perfect recovery

  6. Transform Coding • Assume: signal is sparse in some domain… • e.g. JPEG, JPEG2000, MPEG… • Sample with frequency . Get signal of length N • Transform signal  K (<< N) nonzero coefficients • Preserve K coefficients and their locations

  7. Compressive Sensing

  8. Compressive Sensing • Sample with rate lower than !! • Can be recoveredPERFECTLY!

  9. Comparison

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

  11. Framework N M M N N: length for signal sampled with Nyquist’s rate M: length for signal with lower rate Sampling matrix

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

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

  14. Example… ECG : 心電圖訊號:DCT(discrete cosine transform)

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

  16. Sampling Matrix • Low coherence Low coherence

  17. Coherence • Describe similarity • High coherence  more similar Low coherence  more different

  18. Example: Time and Frequency • For example, • ,

  19. Fortunately… • Random Sampling • iid Gaussian N(0,1) • Random • Low coherence with deterministic basis.

  20. More about low coherence… Random Sampling

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

  22. Recovery N M M N N BUT….

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

  24. Total Procedure Sampling (Assume f is spare somewhere) Find an incoherent matrix e.g. random matrix f Sample signal 已知: Recovering

  25. Demo Time

  26. Reference • Candes, E. J. and M. B. Wakin (2008). "An Introduction To Compressive Sampling." Signal Processing Magazine, IEEE25(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, ChinmayHegde. An Introduction to Compressive Sensing. • https://sites.google.com/site/igorcarron2/cs#sparse • http://videolectures.net/mlss09us_candes_ocsssrl1m/

  27. Thanks a lot!

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

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