Compressive Signal Processing

1. CompressiveSignal Processing Richard Baraniuk Rice University

2. Better, Stronger, Faster

3. Analog sensing Digital sensing

4. Sense by Sampling sample

5. Sense by Sampling too much data! sample

6. 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

7. 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 comm rate Available transmission bandwidth

9. Sense then Compress sample compress JPEG JPEG2000 … decompress

10. Sparsity largewaveletcoefficients (blue = 0) pixels

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

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

13. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspacesaligned w/ coordinate axes sparsesignal nonzeroentries sorted index

14. 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-lawapproximately sparse power-lawdecay sorted index

15. 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

16. 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

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

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

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

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

21. 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

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

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

24. 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

25. 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 MxKsubmatricesare full rank (ideally close to orthobasis) • Restricted Isometry Property (RIP) columns

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

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

28. 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-Hard design problem 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! Signal Recovery “find sparsest vectorin translated nullspace”

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

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

40. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Correct! Polynomial time alg(linear programming) Much recent alg progress greedy, Bayesian approaches, … Signal Recovery

43. 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)

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

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

47. Analog sensing Digital sensing Computational sensing

48. Compressive SensingIn Action

49. Gerhard Richter 4096 Farben / 4096 Colours 1974254 cm X 254 cmLaquer on CanvasCatalogue Raisonné: 359 Museum Collection:Staatliche Kunstsammlungen Dresden (on loan)

50. 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