1 / 100

POCS:

POCS:. A Tool for Signal Synthesis & Analysis Robert J. Marks II, Distinguished Professor of Electrical and Computer Engineering Baylor University RobertMarks@RobertMarks.org. Outline. POCS: What is it? Convex Sets Projections POCS Applications The Papoulis-Gerchberg Algorithm

whitby
Download Presentation

POCS:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. POCS: A Tool for Signal Synthesis & AnalysisRobert J. Marks II, Distinguished Professor of Electrical and Computer EngineeringBaylor University RobertMarks@RobertMarks.org

  2. Outline • POCS: What is it? • Convex Sets • Projections • POCS • Applications • The Papoulis-Gerchberg Algorithm • Neural Network Associative Memory • Resolution at sub-pixel levels • Radiation Oncology / Tomography • JPG / MPEG repair • Missing Sensors

  3. C POCS: What is a convex set? In a vector space, a set C is convex iff  and,

  4. POCS: Example convex sets • Convex Sets • Sets Not Convex

  5. u x(t) t  POCS: Example convex sets in a Hilbert space • Bounded Signals – a box • If   x(t)  u and   y(t)  u, then, if 0   1   x(t)+(1- ) y(t)  u.

  6. c(t) x(t) t  c(t) y(t) t  POCS: Example convex sets in a Hilbert space • Identical Middles– a plane

  7. POCS: Example convex sets in a Hilbert space • Bandlimited Signals – a plane (subspace) B = bandwidth If x(t) is bandlimited and y(t) is bandlimited, then so is z(t) =  x(t) + (1- ) y(t)

  8. y(t) x(t) A A t t POCS: Example convex sets in a Hilbert space • Fixed Area Signals– a plane

  9. y y p(y) x POCS: Example convex sets in a Hilbert space • Identical Tomographic Projections– a plane

  10. y y p(y) x POCS: Example convex sets in a Hilbert space • Identical Tomographic Projections– a plane

  11. 2E POCS: Example convex sets in a Hilbert space • Bounded Energy Signals– a ball

  12. 2E POCS: Example convex sets in a Hilbert space • Bounded Energy Signals– a ball

  13. C y x z POCS: What is a projection onto a convex set? • For every (closed) convex set, C, and every vector, x, in a Hilbert space, there is a unique vector in C, closest to x. • This vector, denoted PC x, is the projection of x onto C: PC y PC x PC z=

  14. y(t) PC y(t) u t  PC y(t) y(t) Example Projections • Bounded Signals – a box

  15. c(t) y(t) C t  PC y(t) PC y(t) y(t) t  Example Projections • Identical Middles– a plane

  16. C Low Pass Filter: BandwidthB PCy(t) y(t) PC y(t) y(t) Example Projections • Bandlimited Signals – a plane (subspace)

  17. Continuous x[2] x[1]+ x[2]=A A Discrete A Example Projections • Constant Area Signals • – a plane (linear variety) x[1]

  18. Same value added to each element. C PC y(t) y(t) Example Projections • Constant Area Signals • – a plane (linear variety) x[1]+ x[2]=A x[2] A A x[1]

  19. p(y) y y g(x,y) C PC g(x,y) x Example Projections • Identical Tomographic Projections– a plane

  20. y(t) PC y(t) 2E Example Projections • Bounded Energy Signals– a ball

  21. C2 C1 The intersection of convex sets is convex C1 C1 C2 } C1 C2 C2

  22. C2 = Fixed Point Alternating POCS will converge to a point common to both sets C1 The Fixed Point is generally dependent on initialization

  23. C2 C3 C1 Lemma 1: Alternating POCS among N convex sets with a non-empty intersection will converge to a point common to all.

  24. C2 C1 Limit Cycle Lemma 2: Alternating POCS among 2 nonintersecting convex sets will converge to the point in each set closest to the other.

  25. C3 3 2  1 Limit Cycle C1 C2 Lemma 3: Alternating POCS among 3 or more convex sets with an empty intersection will converge to a limit cycle. Different projection operations can produce different limit cycles. 1 2  3 Limit Cycle

  26. C3 C2 C1 1 2  3 Limit Cycle 3 2  1 Limit Cycle Lemma 3: (cont) POCS with non-empty intersection.

  27. C3 C1 C2 Simultaneous Weighted Projections

  28. Outline • POCS: What is it? • Convex Sets • Projections • POCS • Diverse Applications • The Papoulis-Gerchberg Algorithm • Neural Network Associative Memory • Resolution at sub-pixel levels • Radiation Oncology / Tomography • JPG / MPEG repair • Missing Sensors

  29. g(t) f(t) ? ? Application:The Papoulis-Gerchberg Algorithm • Restoration of lost data in a band limited function: BL  ? Convex Sets: (1) Identical Tails and (2) Bandlimited.

  30. Application:The Papoulis-Gerchberg Algorithm Example

  31. Outline • POCS: What is it? • Convex Sets • Projections • POCS • Applications • The Papoulis-Gerchberg Algorithm • Neural Network Associative Memory • Resolution at sub-pixel levels • Radiation Oncology / Tomography • JPG / MPEG repair • Missing Sensors

  32. A Plane! Application:Neural Network Associative MemoryPlacing the Library in Hilbert Space

  33. Identical Pixels Application:Neural Network Associative MemoryAssociative Recall

  34. Identical Pixels Application:Neural Network Associative MemoryAssociative Recall by POCS Column Space

  35. 1 2 3 4 5 10 19 Application:Neural Network Associative MemoryExample

  36. 0 1 2 3 4 10 19 Application:Neural Network Associative MemoryExample

  37. A Library of Mathematicians • What does the Average Mathematician Look Like?

  38. Convergence As a Function of Percent of Known Image

  39. Convergence As a Function of Library Size

  40. Convergence As a Function of Noise Level

  41. Combining Euler & Hermite Not Of This Library

  42. Outline • POCS: What is it? • Convex Sets • Projections • POCS • Applications • The Papoulis-Gerchberg Algorithm • Neural Network Associative Memory • Resolution at sub-pixel levels • Radiation Oncology / Tomography • JPG / MPEG repair • Missing Sensors

  43. 2 4 Application:Combining Low Resolution Sub-Pixel Shifted Imageshttp://www.stecf.org/newsletter/stecf-nl-22/adorf/adorf.htmlHans-Martin Adorf • Problem: Multiple Images of Galazy Clusters with subpixel shifts. • POCS Solution: • Sets of high resolution images giving rise to lower resolution images. • Positivity • Resolution Limit

  44. Application:Subpixel Resolution • Object is imaged with an aperture covering a large area. The average (or sum) of the object is measured as a single number over the aperture. • The set of all images with this average value at this location is a convex set. • The convex set is constant area.

  45. Application:Subpixel Resolution Randomly chosen 8x8 pixel blocks. Over 6 Million projections.

  46. Application:Subpixel Resolution Slow (7:41) Fast (3:00) Faster (1:30) Fastest (0:56) Randomly chosen 8x8 pixel blocks. Over 6 Million projections.

  47. Application:Subpixel Resolution Slow (0.58) Fast (0:29) Faster (0:14) Blocks of random dimension <33 chosen at random locations. 2.7 Million blocks.

  48. Outline • POCS: What is it? • Convex Sets • Projections • POCS • Applications • The Papoulis-Gerchberg Algorithm • Neural Network Associative Memory • Resolution at sub-pixel levels • Radiation Oncology / Tomography • JPG / MPEG repair • Missing Sensors

  49. Application:TomographySame procedure as subpixel resolution.

  50. Application:TomographySame procedure as subpixel resolution.

More Related