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. Sampling and Compression
Review…
Sampling and Compression

5. Nyquist’s Rate
Perfect recovery
𝑓 𝑠 ≥2 𝑓 𝑏

6. Transform Coding Assume: signal is sparse in some domain…
e.g. JPEG, JPEG2000, MPEG…
Sample with frequency 𝑓 𝑠 .
Get signal of length N
Transform signal  K (<< N) nonzero coefficients
Preserve K coefficients and their locations
畫圖講一下

7. Compressive Sensing

8. Compressive Sensing Sample with rate lower than 𝒇 𝒔 !!
Can be recovered PERFECTLY!
單單with lower rate 不厲害 能夠還原才是真的很屌

9. Comparison Nyquist’s Sampling Compressive Sensing Sampling Frequency
≥ 2𝑓 𝑏
< 2𝑓 𝑏
Recovery
Low pass filter
Convex Optimization

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

11. Φ Framework 𝐲= 𝚽𝐟 = 𝑦 𝑓 N M N M
N: length for signal sampled with Nyquist’s rate
M: length for signal with lower rate
Φ: Sampling matrix

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

13. When….
Signal is compressible, sparse… 𝑦 𝑓 N 𝑥 Φ = M M N Ψ

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

15. Φ Ψ How… How to design the sampling matrix?
How to decide the sampling rate (M)? 𝑦 N 𝑥 Φ = M Ψ

16. Sampling Matrix
Low coherence
Low coherence 𝑦 𝑥 Φ = Ψ

17. Coherence Describe similarity 𝛍 𝚽,𝚿 =𝐧∙ 𝐦𝐚𝐱 𝛗 𝐤 , 𝛙 𝐣 𝟐
𝛍 𝚽,𝚿 =𝐧∙ 𝐦𝐚𝐱 𝛗 𝐤 , 𝛙 𝐣 𝟐
High coherence  more similar
Low coherence  more different
𝛍 𝚽,𝐇 ∈[1,𝑛]

18. Example: Time and Frequency
For example, 𝑺𝒑𝒊𝒌𝒆 𝒃𝒂𝒔𝒊𝒔 and 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒃𝒂𝒔𝒊𝒔
𝜑 𝑘 =𝛿(𝑡−𝑘), ℎ 𝑗 = 1 𝑛 𝑒 𝑖 2𝜋 𝑗𝑡/𝑛

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

20. More about low coherence…
Random Sampling

21. Sampling Rate Theorem 𝐦≥𝐂∙ 𝛍 𝟐 𝚽,𝚿 ∙𝐒∙ 𝐥𝐨𝐠 𝐧
Can be exactly recovered with high probability.
Theorem
𝐦≥𝐂∙ 𝛍 𝟐 𝚽,𝚿 ∙𝐒∙ 𝐥𝐨𝐠 𝐧
C : constant
μ: coherence
S: sparsity
n: signal length

22. BUT…. Φ Ψ Recovery y= Φf 𝐒𝐨𝐥𝐯𝐞 𝐟𝐨𝐫 x f= Φ −1 y s.t. y= ΦΨx = 𝑦 𝑓 N 𝑥 M

23. ℓ 1 Recovery Many related research… GPSR
(Gradient projection for sparse reconstruction)
L1-magic
SparseLab
BOA
(Bound optimization approach)
…..

24. Total Procedure 已知: 𝐲 , 𝚽 Sampling (Assume f is spare somewhere)
Find an incoherent matrix Φ
e.g. random matrix f
Sample signal y=Φf
已知: 𝐲 , 𝚽
𝒂𝒓𝒈 𝒔 𝒎𝒊𝒏 𝒔 𝟏 𝑠.𝑡. 𝐲=𝚯 𝐬
𝐱 =𝐇 𝐬
Recovering

25. Sum up 有 size 為 nx1 在某domain 上 sparse的訊號
用size為 mxn 的 random matrix 做sampling
(m<n)
得到 size 為 mx1 的measurement y
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):
Baraniuk, R. (2008). Compressive sensing. Information Sciences and Systems, CISS nd Annual Conference on.
Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An Introduction to Compressive Sensing.

28. Thanks a lot!