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

Sampling and Compression Review… Sampling and Compression

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

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

Compressive Sensing

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

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

Some Applications ECG One-pixel Camera Medical Imaging: MRI

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

When? How? Two things you must know…

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

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

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

Sampling Matrix Low coherence Low coherence 𝑦 𝑥 Φ = Ψ

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]

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

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

More about low coherence… Random Sampling

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

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

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

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

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

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. 42nd Annual Conference on. Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An Introduction to Compressive Sensing. https://sites.google.com/site/igorcarron2/cs#sparse http://videolectures.net/mlss09us_candes_ocsssrl1m/

Thanks a lot!