a tool for signal synthesis & analysis robert j. marks ii, distinguished professor of electrical and...
TRANSCRIPT
- Slide 1
- A Tool for Signal Synthesis & Analysis Robert J. Marks II, Distinguished Professor of Electrical and Computer Engineering Baylor University [email protected]
- Slide 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
- Slide 3
- POCS: What is a convex set? In a vector space, a set C is convex iff and, C
- Slide 4
- POCS: Example convex sets Convex Sets Sets Not Convex
- Slide 5
- POCS: Example convex sets in a Hilbert space Bounded Signals a box then, if 0 1 x ( t )+( 1 - ) y ( t ) u. If x ( t ) u and y ( t ) u, x(t)x(t) t u
- Slide 6
- POCS: Example convex sets in a Hilbert space Identical Middles a plane t c(t)c(t) x(t)x(t) t c(t)c(t) y(t)
- Slide 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)
- Slide 8
- POCS: Example convex sets in a Hilbert space Fixed Area Signals a plane x(t)x(t) t A y(t) t A
- Slide 9
- POCS: Example convex sets in a Hilbert space Identical Tomographic Projections a plane y p(y) p(y) y x
- Slide 10
- POCS: Example convex sets in a Hilbert space Identical Tomographic Projections a plane y p(y) p(y) y x
- Slide 11
- POCS: Example convex sets in a Hilbert space Bounded Energy Signals a ball 2E2E
- Slide 12
- POCS: Example convex sets in a Hilbert space Bounded Energy Signals a ball 2E2E
- Slide 13
- 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 P C x, is the projection of x onto C: C x P C x y P C y zP C z=
- Slide 14
- Example Projections Bounded Signals a box y(t) t u P C y(t) y(t)y(t)
- Slide 15
- Example Projections Identical Middles a plane y(t)y(t) c(t)c(t) t P C y(t) t y(t) C P C y(t)
- Slide 16
- y(t) y(t) Example Projections Bandlimited Signals a plane (subspace) y(t)y(t) C P C y(t) Low Pass Filter: Bandwidth B PC y(t)PC y(t)
- Slide 17
- Example Projections Constant Area Signals a plane (linear variety) x[1] x[2]x[1]+ x[2]=A A A Continuous Discrete
- Slide 18
- Example Projections Constant Area Signals a plane (linear variety) x[1] x[2] x[1]+ x[2]=A A A Same value added to each element. y(t)y(t) C P C y(t)
- Slide 19
- y p(y) p(y) y x Example Projections Identical Tomographic Projections a plane g(x,y) P C g(x,y) C
- Slide 20
- Example Projections Bounded Energy Signals a ball 2E2E P C y(t) y(t)y(t)
- Slide 21
- The intersection of convex sets is convex C1C1 C2C2 } C 1 C 2 C1C1 C2C2 C 1 C 2
- Slide 22
- Alternating POCS will converge to a point common to both sets C1C1 C2C2 = Fixed Point The Fixed Point is generally dependent on initialization
- Slide 23
- Lemma 1: Alternating POCS among N convex sets with a non-empty intersection will converge to a point common to all. C2C2 C3C3 C1C1
- Slide 24
- Lemma 2: Alternating POCS among 2 nonintersecting convex sets will converge to the point in each set closest to the other. C2C2 C1C1 Limit Cycle
- Slide 25
- Lemma 3: Alternating POCS among 3 or more convex sets with an empty intersection will converge to a limit cycle. C1C1 C3C3 C2C2 Different projection operations can produce different limit cycles. 1 2 3 Limit Cycle 3 2 1 Limit Cycle
- Slide 26
- Lemma 3: (cont) POCS with non-empty intersection. C1C1 C3C3 1 2 3 Limit Cycle 3 2 1 Limit Cycle C2C2
- Slide 27
- Simultaneous Weighted Projections C1C1 C3C3 C2C2
- Slide 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
- Slide 29
- Application: The Papoulis-Gerchberg Algorithm Restoration of lost data in a band limited function: g(t) g(t) ? ? f(t) f(t) BL ? Convex Sets: (1) Identical Tails and (2) Bandlimited.
- Slide 30
- Application: The Papoulis-Gerchberg Algorithm Example
- Slide 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
- Slide 32
- A Plane! Application: Neural Network Associative Memory Placing the Library in Hilbert Space
- Slide 33
- Application: Neural Network Associative Memory Associative Recall Identical Pixels
- Slide 34
- Application: Neural Network Associative Memory Associative Recall by POCS Identical PixelsColumn Space
- Slide 35
- Application: Neural Network Associative Memory Example 23 5 1 1019 4
- Slide 36
- Application: Neural Network Associative Memory Example 12 4 0 1019 3
- Slide 37
- A Library of Mathematicians What does the Average Mathematician Look Like?
- Slide 38
- Convergence As a Function of Percent of Known Image
- Slide 39
- Convergence As a Function of Library Size
- Slide 40
- Convergence As a Function of Noise Level
- Slide 41
- Combining Euler & Hermite Not Of This Library
- Slide 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
- Slide 43
- Application: Combining Low Resolution Sub-Pixel Shifted Images http://www.stecf.org/newsletter/stecf-nl-22/adorf/adorf.html Hans-Martin Adorf http://www.stecf.org/newsletter/stecf-nl-22/adorf/adorf.html Problem: Multiple Images of Galazy Clusters with subpixel shifts. POCS Solution: 1.Sets of high resolution images giving rise to lower resolution images. 2.Positivity 3.Resolution Limit 2 4
- Slide 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.
- Slide 45
- Application: Subpixel Resolution Randomly chosen 8x8 pixel blocks. Over 6 Million projections.
- Slide 46
- Slow (7:41) Fast (3:00) Faster (1:30) Fastest (0:56)SlowFastFasterFastest Application: Subpixel Resolution Randomly chosen 8x8 pixel blocks. Over 6 Million projections.
- Slide 47
- Application: Subpixel Resolution Blocks of random dimension