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To add

To add. Relation with JL and JLCS Volkan for UMd:

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  1. To add • Relation with JL and JLCS • Volkan for UMd: • At UMD, there will be quite a few skeptical professors about CS. If you just present camera compression results, they might give the response we had from the CVPR review. I would put a hardcore 1D signal recovery example from A2I or an A/D converter example. Then, I'd present the BS and 3D recon work as exploring possibilities. I have to say that there will be quite a bit of interest.

  2. Compressive Sensing Richard Baraniuk Rice University

  3. Petros BoufounosVolkan CevherMona SheikhMatthew Moravec JustinRombergMikeWakinDrorBaron Marco DuarteMark DavenportShri Sarvotham

  4. Sensing

  5. Digital Revolution

  6. Pressure is on Digital Sensors • Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV, x-ray, gamma ray, …

  7. Pressure is on Digital Sensors • Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV = deluge of data • how to acquire, store, fuse, process efficiently?

  8. Digital Data Acquisition • Foundation: Shannon sampling theorem “if you sample densely enough(at the Nyquist rate), you can perfectly reconstruct the original analog data” time space

  9. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) sample

  10. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) too much data! sample

  11. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) • compress data sample compress transmit/store JPEG JPEG2000 … receive decompress

  12. Sparsity / Compressibility largewaveletcoefficients (blue = 0) largeGabor (TF)coefficients pixels widebandsignalsamples frequency time

  13. What’s Wrong with this Picture? • Long-established paradigm for digital data acquisition • uniformly sample data at Nyquist rate • compress data sample compress transmit/store sparse /compressiblewavelettransform receive decompress

  14. What’s Wrong with this Picture? • Why go to all the work to acquire N samples only to discard all but K pieces of data? sample compress transmit/store sparse /compressiblewavelettransform receive decompress

  15. What’s Wrong with this Picture? nonlinear processing nonlinear signal model (union of subspaces) linear processing linear signal model (bandlimited subspace) sample compress transmit/store sparse /compressiblewavelettransform receive decompress

  16. Compressive Sensing • Directly acquire “compressed” data • Replace samples by more general “measurements” compressive sensing transmit/store receive reconstruct

  17. Sampling • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain sparsesignal nonzeroentries

  18. Sampling • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Samples sparsesignal measurements nonzeroentries

  19. Compressive Data Acquisition • When data is sparse/compressible, can directly acquire a condensed representation with no/little information lossthrough dimensionality reduction sparsesignal measurements nonzero entries

  20. How Can It Work? • Projection not full rank…… and so loses information in general • Ex: Infinitely many ’s map to the same

  21. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors • Design so that its MxK submatrices are full rank columns

  22. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors • Design so that its MxK submatrices are full rank • even better: difference between 2 sparse vectors is 2K sparse in general • design so that its Mx2K submatrices are full rank columns

  23. How Can It Work? • Goal: Design so that its Mx2K submatrices are full rank(Restricted Isometry Property – RIP) • Unfortunately, a combinatorial, NP-complete design problem columns

  24. Insight from the 80’s [Kashin, Gluskin] • Draw at random • iid Gaussian • iid Bernoulli … • Then has the RIP with high probability • Mx2K submatrices are full rank • Suggests randomized data acquisition protocol columns

  25. Compressive Data Acquisition • Measurements = random linear combinations of the entries of • WHP does not distort structure of sparse signals • no information loss sparsesignal measurements nonzero entries

  26. CS Signal Recovery • Random projection not full rank but preserves structureand informationin sparse/compressible signals models with high probability • To recover from measurementsexploit sparse signal structure • only K out of N coordinates nonzero

  27. CS Signal Recovery • Random projection not full rank but preserves structureand informationin sparse/compressible signals models with high probability • To recover from measurementsexploit sparse signal structure • only K out of N coordinates nonzero K-dim hyperplanesaligned withcoordinate axes

  28. Aside: Compressibility • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces • Compressible signal: sorted coordinates decay rapidly to zero • model: ball, power-lawdecay coefficients sorted index

  29. CS Signal Recovery • Random projection not full rank but preserves structureand informationin sparse/compressible signals models with high probability K-sparsemodel recovery

  30. CS Signal Recovery • Random projection not full rank • Recovery problem:givenfind • Null space • So search in null space for the “best” according to some criterion • ex: least squares (M-N)-dim hyperplaneat random angle

  31. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • fast

  32. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • fast, wrong

  33. Why Doesn’t Work least squares,minimum solutionis almost never sparse for signals sparse in the space/time domain null space of translated to(random angle)

  34. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • fast, wrong number ofnonzeroentries “find sparsestin translated nullspace”

  35. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • fast, wrong • correct:only M=2K measurements required to perfectly reconstruct K-sparse signal stablyslow: NP-complete algorithm number ofnonzeroentries

  36. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • fast, wrong • correct, slow • correct, efficient mild oversampling[Candes, Romberg, Tao; Donoho] linear program

  37. Why Works minimum solution= sparsest solution (with high probability) if for signals sparse in the space/time domain

  38. Universality • Random measurements can be used for signals sparse in any basis

  39. Universality • Random measurements can be used for signals sparse in any basis

  40. Universality • Random measurements can be used for signals sparse in any basis sparsecoefficient vector nonzero entries

  41. Compressive Sensing • Directly acquire “compressed” data • Replace samples by more general “measurements” random measurements transmit/store … receive linear pgm

  42. CS Hallmarks • CS changes the rules of the data acquisition game • exploits a priori signal sparsity information • Stable • acquisition/recovery process is numerically stable • Universal • same random projections / hardware can be used forany compressible signal class (generic) • Asymmetrical(most processing at decoder) • conventional: smart encoder, dumb decoder • CS: dumb encoder, smart decoder • Random projections weakly encrypted

  43. CS Hallmarks • Democratic • each measurement carries the same amount of information • robust to measurement loss and quantization simple encoding • Ex: wireless streaming application with data loss • conventional: complicated (unequal) error protection of compressed data • DCT/wavelet low frequency coefficients • CS: merely stream additional measurements and reconstruct using those that arrive safely (fountain-like)

  44. Compressive SensinginAction

  45. Gerhard Richter 4096 Farben / 4096 Colours 1974254 cm X 254 cmLaquer on CanvasCatalogue Raisonné: 359 Museum Collection:Staatliche Kunstsammlungen Dresden (on loan) Sales history: 11 May 2004Christie's New York Post-War and Contemporary Art (Evening Sale), Lot 34US$3,703,500  

  46. Gerhard Richter Cologne CathedralStained Glass

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

  48. “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 • Linear programming recovery

  49. Single Pixel Camera Object LED (light source) Lens 2 Lens 1 Photodiode circuit DMD+ALP Board

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