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Ron DeVore University of South Carolina Richard Baraniuk Rice University

Compressive Sensing Theory and Applications. Ron DeVore University of South Carolina Richard Baraniuk Rice University. Agenda. Introduction to Compressive Sensing (CS) [richb] motivation basic concepts CS Theoretical Foundation [ron] sparsity encoding and decoding

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Ron DeVore University of South Carolina Richard Baraniuk Rice University

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  1. Compressive Sensing Theory and Applications Ron DeVore University of South Carolina Richard Baraniuk Rice University

  2. Agenda • Introduction to Compressive Sensing (CS) [richb] • motivation • basic concepts • CS Theoretical Foundation [ron] • sparsity • encoding and decoding • restricted isometry principle (RIP) • recovery algorithms • CS Applications [richb]

  3. Compressive SensingIntroduction and Background

  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 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 (signal-dependent, nonlinear) sample compress transmit/store sparsewavelettransform receive decompress

  12. Sparsity / Compressibility largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples frequency time

  13. What’s Wrong with this Picture? • Long-established paradigm for digital data acquisition • sampledata at Nyquist rate (2x bandwidth) • compressdata (signal-dependent, nonlinear) • brick wall to resolution/performance sample compress transmit/store sparse /compressiblewavelettransform receive decompress

  14. Beyond Uniform Sampling • Recall Shannon/Nyquist theorem • Shannon was a pessimist • 2x oversampling Nyquist rate is a worst-case bound for any bandlimited data • sparsity/compressibility irrelevant • Shannon sampling is a linear process while compression is a nonlinear process

  15. Compressive Sensing (CS) • Recall Shannon/Nyquist theorem • Shannon was a pessimist • 2x oversampling Nyquist rate is a worst-case bound for any bandlimited data • sparsity/compressibility irrelevant • Shannon sampling is a linear process while compression is a nonlinear process • Compressive sensing • new sampling theory that leverages compressibility • based on new uncertainty principles • randomness plays a key role

  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. Compressive Data Acquisition • When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss • Random projection will work sparsesignal measurements nonzero entries [Candes-Romberg-Tao, Donoho, 2004]

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

  22. Why Does It Work? • Random projection not full rank…… and so loses information in general

  23. Why Does It Work? • Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability

  24. Why Does It Work? • Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability K-dim hyperplanesaligned with coordinate axes K-sparsemodel

  25. Why Does It Work? • Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability K-sparsemodel

  26. CS Signal Recovery • Random projection not full rank… … but is invertiblefor sparse/compressible signals models with high probability(solves ill-posed inverse problem) K-sparsemodel recovery

  27. 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

  28. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast

  29. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong

  30. Why L2 Doesn’t Work least squares,minimum L2 solutionis almost never sparse for signals sparse in the space/time domain null space of translated to(random angle)

  31. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 number ofnonzeroentries

  32. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0correct, slowonly M=2K measurements required to perfectly reconstruct K-sparse signal stably[Bresler; Rice] number ofnonzeroentries

  33. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 correct, slow • L1correct, efficient mild oversampling[Candes, Romberg, Tao; Donoho] linear program

  34. Why L1 Works minimum L1solution= sparsest solution (with high probability) if for signals sparse in the space/time domain

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

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

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

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

  39. CS Hallmarks • CS changes the rules of the data acquisition game • exploits a priori signal sparsity information • Universal • same random projections / hardware can be used forany compressible signal class (generic) • Democratic • each measurement carries the same amount of information • simple encoding • robust to measurement loss and quantization • Asymmetrical(most processing at decoder) • Random projections weakly encrypted

  40. Theoretical Foundations ofCompressive Sensing[Ron DeVore]

  41. Applications of Compressive Sensing[Richard Baraniuk]

  42. Recall: CS Hallmarks • CS changes the rules of the data acquisition game • exploits a priori signal sparsity information • Universal • same random projections / hardware can be used forany compressible signal class (generic) • Democratic • each measurement carries the same amount of information • simple encoding • robust to measurement loss and quantization • Asymmetrical(most processing at decoder) • Random projections weakly encrypted

  43. 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  

  44. Gerhard Richter Cologne CathedralStained Glass

  45. Rice Single-Pixel CS Camera single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array

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

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

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

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

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