Compressive Sensing
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An Introduction to Compressive Sensing Speaker: Ying-Jou Chen Advisor: Jian-Jiun Ding 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 • Perfect recovery • ππ ≥ 2ππ Transform Coding • Assume: signal is sparse in some domain… • e.g. JPEG, JPEG2000, MPEG… 1. Sample with frequency ππ . Get signal of length N 2. Transform signal ο K (<< N) nonzero coefficients 3. Preserve K coefficients and their locations Compressive Sensing Compressive Sensing • Sample with rate lower than ππ !! • Can be recovered PERFECTLY! Comparison Nyquist’s Sampling Compressive Sensing Sampling Frequency ≥ 2ππ < 2ππ Recovery Low pass filter Convex Optimization Some Applications • ECG • One-pixel Camera • Medical Imaging: MRI Framework π² = π½π π¦ M N = M Φ N: length for signal sampled with Nyquist’s rate M: length for signal with lower rate Φ: Sampling matrix π N When? How? Two things you must know… When…. • Signal is compressible, sparse… π¦ M N = M Φ π π₯ N Ψ Example… ECG π: εΏι»εθ¨θ Ψ: DCT (discrete cosine transform) How… • How to design the sampling matrix? • How to decide the sampling rate (M)? π¦ π₯ N =M Φ Ψ Sampling Matrix • Low coherence Low coherence π¦ = π₯ Φ Ψ Coherence • Describe similarity π π½, πΏ = π§ β π¦ππ± ππ€ , ππ£ – High coherence ο more similar Low coherence ο more different – π π½, π ∈ [1, π] π Example: Time and Frequency • For example, πΊππππ πππππ and πππππππ πππππ 1 • ππ = πΏ(π‘ − π), βπ = π π 2π ππ‘/π π 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Fortunately… • Random Sampling – iid Gaussian N(0,1) – Random ±1 • Low coherence with deterministic basis. More about low coherence… Random Sampling Sampling Rate Theorem π π¦ ≥ π β π π½, πΏ β π β π₯π¨π π§ C : constant μ: coherence S: sparsity n: signal length • Can be exactly recovered with high probability. Recovery y = Φf f = Φ−1 y π¦ M ππ¨π₯π―π ππ¨π« x s. t. y = ΦΨx N =M Φ BUT…. π π₯ N Ψ N β1 Recovery • Many related research… – GPSR (Gradient projection for sparse reconstruction) – L1-magic – SparseLab – BOA (Bound optimization approach) ….. Total Procedure Sampling (Assume f is spare somewhere) f Find an incoherent matrix Φ e.g. random matrix Sample signal y = Φf ε·²η₯: π² , π½ ππππ πππ π Recovering π π . π‘. π² = π―π¬ π± = ππ¬ 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! Key Points 1. Nyquist’s Rate 2. CS and Transform coding… 3. Sampling in time V.S. Sampling as inner products 4. About compressibility 5. About designing sampling matrix 6. About L1 norm explanation by geometry! 7. Application( MRI, One-pixel camera…)
