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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

An Introduction to Compressive Sensing

Speaker: Ying-Jou Chen

Advisor: Jian-Jiun Ding


CompressiveCompressed

SensingSampling

CS


Outline
Outline

  • Conventional Sampling & Compression

  • Compressive Sensing

  • Why it is useful?

  • Framework

  • When and how to use

  • Recovery

  • Simple demo


Review…

Sampling and Compression


Nyquist s rate
Nyquist’s Rate

  • Perfect recovery


Transform coding
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



Compressive sensing
Compressive Sensing

  • Sample with rate lower than !!

  • Can be recoveredPERFECTLY!



Some applications
Some Applications

  • ECG

  • One-pixel Camera

  • Medical Imaging: MRI


Framework
Framework

N

M

M

N

N: length for signal sampled with Nyquist’s rate

M: length for signal with lower rate

Sampling matrix


When? How?

Two things you must know…


When….

  • Signal is compressible, sparse…

N

M

M

N


Example ecg
Example… ECG

: 心電圖訊號:DCT(discrete cosine transform)


How…

  • How to design the sampling matrix?

  • How to decide the sampling rate(M)?

N

M


Sampling matrix
Sampling Matrix

  • Low coherence

Low coherence


Coherence
Coherence

  • Describe similarity

    • High coherence  more similar

      Low coherence  more different



Fortunately
Fortunately…

  • Random Sampling

    • iid Gaussian N(0,1)

    • Random

  • Low coherence with deterministic basis.



Sampling rate
Sampling Rate

  • Can be exactly recovered with high probability.

C : constant

S: sparsity

n: signal length


Recovery
Recovery

N

M

M

N

N

BUT….


Recovery1
Recovery

  • Many related research…

    • GPSR

      (Gradient projection for sparse reconstruction)

    • L1-magic

    • SparseLab

    • BOA

      (Bound optimization approach)

      …..


Total procedure
Total Procedure

Sampling (Assume f is spare somewhere)

Find an incoherent matrix

e.g. random matrix

f

Sample signal

已知:

Recovering



Reference
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/



Key points
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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