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POCS:

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  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. Application:The Papoulis-Gerchberg Algorithm Example

  28. 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

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

  30. Identical Pixels Application:Neural Network Associative MemoryAssociative Recall

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

  32. 1 2 3 4 5 10 19 Application:Neural Network Associative MemoryExample

  33. 0 1 2 3 4 10 19 Application:Neural Network Associative MemoryExample

  34. 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

  35. 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

  36. 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

  37. Application:Radiation Oncology Conventional Therapy Illustration

  38. Application:Radiation Oncology Intensity Modulated Radiotherapy

  39. 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

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

  41. 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

  42. 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)

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

  44. 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

  45. 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.

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

  47. r P1 r Convex Cone Hull ApplicationProjection operator P1

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

  49. 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

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