# POCS:

## POCS:

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##### Presentation Transcript

1. POCS: A Specialized Application Tool for Signal Synthesis & AnalysisRobert J. Marks II, CIA LabDepartment of Electrical EngineeringUniversity of WashingtonSeattle, WA 98036marks@ee.washington.edu Prof. Paul Cho, Dr. Jai Choi, Prof. Mohamed El-Sharkawi, Mark Goldberg, Dr. Shinhak Lee, Sreeram Narayanan, Dr. Seho Oh, Dr. Dong-Chul Park, Dr. Jiho Park, Prof. Ceon Ramon, Dennis Sarr, Dr. John Vian & Ben Thompson.

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

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

10. 2E POCS: Example convex sets in a Hilbert space • Bounded Energy Signals– a ball

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

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

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

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

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

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

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

22. 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.

23. 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

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

25. 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

26. 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.

27. 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

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

29. Identical Pixels Application:Neural Network Associative MemoryAssociative Recall

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

31. 1 2 3 4 5 10 19 Application:Neural Network Associative MemoryExample

32. 0 1 2 3 4 10 19 Application:Neural Network Associative MemoryExample

33. 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

34. 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

35. 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

36. Application:Radiation Oncology Conventional Therapy Illustration

38. Application:Radiation Oncology • Convex sets for dosage optimization CN = Set of beam patterns giving dosage in N less than TN. CC = Set of beam patterns giving dosage in C less than TC <TN. CT = Set of beam patterns giving dosage in T between TLow and Thi (no cold spots or hot spots). CC = Set of nonnegative beam patterns. N = normal tissue C = critical organ T = tumor

39. T C Application:Radiation Oncology N Example POCS Solution

40. 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

41. Missing 8x8 block of pixels ApplicationBlock Loss Recovery Techniques for Image and Video CommunicationsTransmission & Error • During jpg or mpg transmission, some packets may be lost due to bit error, congestion of network, noise burst, or other reasons. • Assumption: No Automatic Retransmission Request (ARQ)

42. ApplicationProjections based Block Recovery – Algorithm • Two Steps Edge orientation detection POCS using Three Convex Sets  

43. Application • Recovery vectors are extracted to restore missing pixels. • Two positions of recovery vectors are possible according to the edge orientation. • Recovery vectors consist of known pixels(white color) and missing pixels(gray color). • The number of recovery vectors, rk, is 2. Vertical line dominating area Horizontal line dominating area

44. ApplicationForming a Convex Constraint Using Surrounding Vectors • Surrounding Vectors, sk, are extracted from surrounding area of a missing block by N x N window. • Each vector has its own spatial and spectral characteristic. • The number of surrounding vectors, sk, is 8N.

45. Convex Cone Hull ApplicationProjections based Block Recovery –Projection operator P1 Surrounding Blocks

46. r P1 r Convex Cone Hull ApplicationProjection operator P1

47. Keep This Part P1 Replace this part with the known image ApplicationProjection operator P2Identical Middles

48. ApplicationProjection operator P3Smoothness Constraint • Convex set C3 acts as a convex constraint between missing pixels and adjacent known pixels, (fN-1 fN). The rms difference between the columns is constrained to lie below a threshold. fN-1 fN

49. ApplicationSimulation Results – Test Data and Error • Peak Signal to Noise Ratio