A Tool for Signal Synthesis & Analysis Robert J. Marks II, Distinguished Professor of Electrical and...
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A Tool for Signal Synthesis & Analysis Robert J. Marks II, Distinguished Professor of Electrical and Computer Engineering Baylor University [email protected]
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
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
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