Richard Baraniuk Rice University

1. compressivenonsensing Richard Baraniuk Rice University

2. Chapter 1 The Problem

3. challenge 1data too expensive

4. Case in Point: MR Imaging • Measurements very expensive • $1-3 million per machine • 30 minutes per scan

5. Case in Point: IR Imaging

6. challenge 2too much data

7. Case in Point: DARPA ARGUS-IS • 1.8 Gpixel image sensor • video rate output: 444 Gbits/s • comm data rate: 274 Mbits/sfactor of 1600x way out of reach ofexisting compressiontechnology • Reconnaissancewithout conscience • too much data to transmit to a ground station • too much data to make effective real-time decisions

8. Chapter 2 The Promise

9. COMPRESSIVESENSING

10. innovation 1sparse signal models

11. Sparsity largewaveletcoefficients (blue = 0) largeGabor (TF)coefficients pixels widebandsignalsamples frequency time

12. Sparsity largewaveletcoefficients (blue = 0) pixels sparsesignal nonlinear signal model nonzeroentries

13. innovation 2dimensionalityreduction for sparse signals

14. Dimensionality Reduction • When data is sparse/compressible, can directly acquire a compressed representation with no/little information loss through linear dimensionality reduction sparsesignal measurements nonzero entries

15. Stable Embedding An information preserving projection preserves the geometry of the set of sparse signals SE ensures that K-dim subspaces

16. Stable Embedding An information preserving projection preserves the geometry of the set of sparse signals SE ensures that

17. Random Embedding is Stable • Measurements = random linear combinations of the entries of • No information loss for sparse vectors whp sparsesignal measurements nonzero entries

18. innovation 3sparsity-basedsignal recovery

19. Signal Recovery • Goal: Recover signal from measurements • Problem: Randomprojection not full rank(ill-posed inverse problem) • Solution: Exploit the sparse/compressiblegeometry of acquired signal • Recovery via (convex) sparsitypenalty or greedy algorithms[Donoho; Candes, Romberg, Tao, 2004]

20. Signal Recovery • Goal: Recover signal from measurements • Problem: Randomprojection not full rank(ill-posed inverse problem) • Solution: Exploit the sparse/compressiblegeometry of acquired signal • Recovery via (convex) sparsitypenalty or greedy algorithms[Donoho; Candes, Romberg, Tao, 2004]

21. “Single-Pixel” CS Camera scene single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array w/ Kevin Kelly

22. “Single-Pixel” CS Camera scene single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array … • Flip mirror array M times to acquire M measurements • Sparsity-based recovery

23. Random Demodulator • Problem: In contrast to Moore’s Law, ADC performance doubles only every 6-8 years A2Isampling rate number oftones /window Nyquistbandwidth • CS enables sampling near signal’s (low) “information rate” rather than its (high) Nyquist rate

24. Example: Frequency Hopper • Sparse in time-frequency 20x sub-Nyquist sampling Nyquist rate sampling sparsogram spectrogram

25. challenge 1data too expensive means fewer expensive measurements needed for the same resolution scan

26. challenge 2too much data means we compress on the fly as we acquire data

27. EXCITING!!!

28. 2004—2014 9797 citations 6640 citations dsp.rice.edu/cs archive >1500 papers nuit-blanche.blogspot.com> 1 posting/sec

29. Chapter 3 The Hype

30. CS is Growing Up

31. Gerhard Richter4096 Colours

32. muralsoflajolla.com/roy-mcmakin-mural

35. “L1 is the new L2” - Stan Osher

36. Exponential Growth

38. ?

39. Chapter 4 The Fallout

40. “L1 is the new L2” - Stan Osher

41. CS for “Face Recognition”

42. From: M. V. Subject: Interesting application for compressed sensing Date: June 10, 2011 at 11:37:31 PM EDT To: candes@stanford.edu, jrom@ece.gatech.edu Drs. Candes and Romberg, Youmay have alreadybeenapproachedaboutthis, but I feel I shouldsaysomething in case youhaven't. I'mwriting to youbecauseI recentlyread an article in WiredMagazine aboutcompressedsensing I'mexcitedabout the applications CS could have in manyfields, but today I wasreminded of a specificapplicationwhere CS couldconceivablysettle an area of disputebetween mainstream historians and Roswell UFO theorists.  As outlined in the linked video below, Dr. Rudiak has analyzedphotos from 1947 in which a General Rameyappearsholding a typewritten letter from whichRudiakbelieveshe has beenable to discern a number of wordswhichhebelievessubstantiate the extraterrestrialhypothesis for the Roswell Incident).  For your perusal, I'velocated a "hi-res" copy of the cropped image of the letter in Ramey'shand. I hope to hear back from you. Is this an applicationwherecompressedsensingcouldbeuseful?  Any chance youwouldconsidertrying it? Thankyou for your time, M. V. P.S. - Out of personalcuriosity, aretherecurrentlyanycommercialentitiesinvolved in developing CS-based software for use by the general public? --

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45. Chapter 5 Back to Reality

46. Back to Reality • “There's no such thing as a free lunch” • “Something for Nothing” theorems • Dimensionality reductionis no exception • Result: CompressiveNonsensing

47. Nonsense1 Robustness

48. MeasurementNoise • Stable recoverywith additive measurement noise • Noise is added to • Stability: noise only mildly amplified in recovered signal

49. SignalNoise • Often seek recoverywith additive signal noise • Noise is added to • Noise folding: signal noise amplified in by 3dB for every doubling of • Same effect seen in classical “bandpass subsampling” [Davenport, Laska, Treichler, B 2011]

50. Noise Folding in CS slope = -3 CS recovered signal SNR