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Compressive Sensing and Signal Processing Towards Analog-to-Information Conversion

Compressive Sensing and Signal Processing Towards Analog-to-Information Conversion. Richard Baraniuk Rice University. Accelerating Data Deluge. 1250 billion gigabytes generated in 2010 # digital bits > # stars in the universe growing by a factor of 10 every 5 years

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Compressive Sensing and Signal Processing Towards Analog-to-Information Conversion

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  1. CompressiveSensing and Signal Processing Towards Analog-to-Information Conversion Richard Baraniuk Rice University

  2. Accelerating Data Deluge • 1250 billion gigabytes generated in 2010 • # digital bits > # stars in the universe • growing by a factor of 10 every 5 years • Total data generated > total storage • Increases in generation rate >>increases in transmission rate Available transmission bandwidth

  3. Agenda • What’s wrong with today’s multimedia sensor systems? why go to all the work to acquire massive amounts of multimedia data only to throw much/most of it away? • One way out: dimensionality reduction (compressive sensing)enables the design of radically new sensors and systems • Theory: mathematics of sparsitynew nonlinear signal models and recovery algorithms • Practice: compressive sensing in actionnew cameras, imagers, ADCs, …

  4. Sense by Sampling sample

  5. Sense by Sampling too much data! sample

  6. Sense then Compress sample compress JPEG JPEG2000 … decompress

  7. Sparsity largewaveletcoefficients (blue = 0) pixels

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

  9. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero sorted index

  10. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces aligned w/ coordinate axes sorted index

  11. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces • Compressible signal: sorted coordinates decay rapidly with power-law power-lawdecay sorted index

  12. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces • Compressible signal: sorted coordinates decay rapidly with power-law • model: ball: power-lawdecay sorted index

  13. 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 decompress

  14. What’s Wrong with this Picture? • nonlinear processing • nonlinear signal model • (union of subspaces) linear processing linear signal model (bandlimited subspace) sample compress decompress

  15. Compressive Sensing • Directly acquire “compressed” data via dimensionality reduction • Replace samples by more general “measurements” compressive sensing recover

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

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

  18. Compressive Sampling • When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss through linear dimensionality reduction sparsesignal measurements nonzero entries

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

  20. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors columns

  21. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors • is effectively MxK 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 each of its MxK submatrices are full rank (ideally close to orthobasis) • Restricted Isometry Property (RIP) columns

  23. Geometrical Interpretation of RIP • An information preserving projection preserves the geometry of the set of sparse signals sparsesignal nonzero entries

  24. Geometrical Interpretation of RIP • An information preserving projection preserves the geometry of the set of sparse signals • Model: union of K-dimensional subspaces aligned w/ coordinate axes (highly nonlinear!) sparsesignal nonzero entries

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

  26. RIP = Stable Embedding An information preserving projection preserves the geometry of the set of sparse signals RIP ensures that

  27. How Can It Work? • Projection not full rank…… and so loses information in general • Design so that each of its MxK submatrices are full rank (RIP) • Unfortunately, a combinatorial, NP-complete design problem columns

  28. Insight from the 70’s [Kashin, Gluskin] • Draw at random • iid Gaussian • iid Bernoulli … • Then has the RIP with high probability provided columns

  29. Insight from the 70’s [Kashin, Gluskin] • Draw at random • iid Gaussian • iid Bernoulli … • Then has the RIP with high probability provided columns

  30. Randomized Sensing • Measurements = random linear combinations of the entries of • No information loss for sparse vectors whp sparsesignal measurements nonzero entries

  31. CS Signal Recovery • Goal: Recover signal from measurements • Problem: Randomprojection not full rank(ill-posed inverse problem) • Solution: Exploit the sparse/compressiblegeometry of acquired signal

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

  33. Signal Recovery • Recovery: given(ill-posed inverse problem) find (sparse) • Optimization: • Closed-form solution:

  34. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Closed-form solution: Wrong answer! Signal Recovery

  35. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Closed-form solution: Wrong answer! Signal Recovery

  36. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Signal Recovery “find sparsest vectorin translated nullspace”

  37. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Correct! But NP-Complete alg Signal Recovery “find sparsest vectorin translated nullspace”

  38. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Signal Recovery Candes Romberg Tao Donoho

  39. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Correct! Polynomial time alg(linear programming) Signal Recovery

  40. Compressive Sensing Signal recovery via optimization [Candes, Romberg, Tao; Donoho] sparsesignal random measurements nonzero entries

  41. Compressive Sensing Signal recovery via iterative greedy algorithm (orthogonal) matching pursuit [Gilbert, Tropp] iterated thresholding [Nowak, Figueiredo; Kingsbury, Reeves; Daubechies, Defrise, De Mol; Blumensath, Davies; …] CoSaMP [Needell and Tropp] sparsesignal random measurements nonzero entries

  42. CS Hallmarks Stable acquisition/recovery process is numerically stable Asymmetrical(most processing at decoder) conventional: smart encoder, dumb decoder CS: dumb encoder, smart decoder Democratic each measurement carries the same amount of information robust to measurement loss and quantization “digital fountain” property Random measurements encrypted Universal same random projections / hardware can be used forany sparse signal class (generic)

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

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

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

  46. Summary So Far: CS Encoding: = random linear combinations of the entries of Decoding: Recover from via optimization sparsesignal measurements nonzero entries

  47. Compressive SensingIn ActionCameras

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

  49. “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 (linear programming) recovery

  50. First Image Acquisition target 65536 pixels 1300 measurements (2%) 11000 measurements (16%)

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