An Introduction to Compressive Sensing Speaker YingJou Chen

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

e v i s s e r p m d Co e s s e r p m o C CS Sensi ng Samp ling

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 •

Transform Coding •

Compressive Sensing

Compressive Sensing •

Comparison Nyquist’s Sampling Compressive Sensing Low pass filter Convex Optimization Sampling Frequency Recovery

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

Framework N M M N

When? How? Two things you must know…

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

Example… ECG

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

Sampling Matrix • Low coherence

Coherence •

Example: Time and Frequency

Fortunately… •

More about low coherence… Random Sampling

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

Recovery N M M N BUT…. N

• Many related research… – GPSR (Gradient projection for sparse reconstruction) – L 1 -magic – Sparse. Lab – BOA (Bound optimization approach) …. .

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

Demo Time

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. 42 nd Annual Conference on. Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An Introduction to Compressive Sensing. https: //sites. google. com/site/igorcarron 2/cs#sparse http: //videolectures. net/mlss 09 us_candes_ocsssrl 1 m/

Thanks a lot!