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An Introduction to Compressive SensingPowerPoint Presentation

An Introduction to Compressive Sensing

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Outline

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

Sampling and Compression

Nyquist’s Rate

- Perfect recovery

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

- Sample with rate lower than !!
- Can be recoveredPERFECTLY!

Some Applications

- ECG
- One-pixel Camera
- Medical Imaging: MRI

Framework

N

M

M

N

N: length for signal sampled with Nyquist’s rate

M: length for signal with lower rate

Sampling matrix

Two things you must know…

Example… ECG

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

Coherence

- Describe similarity
- High coherence more similar
Low coherence more different

- High coherence more similar

Example: Time and Frequency

- For example,
- ,

Fortunately…

- Random Sampling
- iid Gaussian N(0,1)
- Random

- Low coherence with deterministic basis.

More about low coherence…

Random Sampling

Sampling Rate

- Can be exactly recovered with high probability.

C : constant

S: sparsity

n: signal length

Recovery

- Many related research…
- GPSR
(Gradient projection for sparse reconstruction)

- L1-magic
- SparseLab
- BOA
(Bound optimization approach)

…..

- GPSR

Total Procedure

Sampling (Assume f is spare somewhere)

Find an incoherent matrix

e.g. random matrix

f

Sample signal

已知:

Recovering

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

- 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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