POCS:

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.