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A Tool for Signal Synthesis & Analysis Robert J. Marks II, Distinguished Professor of Electrical and Computer Engineering Baylor University RobertMarks@RobertMarks.org

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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

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POCS: What is a convex set? In a vector space, a set C is convex iff and, C

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POCS: Example convex sets Convex Sets Sets Not Convex

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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

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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)

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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)

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

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

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

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POCS: Example convex sets in a Hilbert space Bounded Energy Signals a ball 2E2E

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POCS: Example convex sets in a Hilbert space Bounded Energy Signals a ball 2E2E

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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=

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Example Projections Bounded Signals a box y(t) t u P C y(t) y(t)y(t)

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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)

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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)

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Example Projections Constant Area Signals a plane (linear variety) x[1] x[2]x[1]+ x[2]=A A A Continuous Discrete

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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)

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y p(y) p(y) y x Example Projections Identical Tomographic Projections a plane g(x,y) P C g(x,y) C

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Example Projections Bounded Energy Signals a ball 2E2E P C y(t) y(t)y(t)

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The intersection of convex sets is convex C1C1 C2C2 } C 1 C 2 C1C1 C2C2 C 1 C 2

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Alternating POCS will converge to a point common to both sets C1C1 C2C2 = Fixed Point The Fixed Point is generally dependent on initialization

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Lemma 1: Alternating POCS among N convex sets with a non-empty intersection will converge to a point common to all. C2C2 C3C3 C1C1

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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

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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

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Lemma 3: (cont) POCS with non-empty intersection. C1C1 C3C3 1 2 3 Limit Cycle 3 2 1 Limit Cycle C2C2

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Simultaneous Weighted Projections C1C1 C3C3 C2C2

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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

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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.

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Application: The Papoulis-Gerchberg Algorithm Example

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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

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A Plane! Application: Neural Network Associative Memory Placing the Library in Hilbert Space

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Application: Neural Network Associative Memory Associative Recall Identical Pixels

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Application: Neural Network Associative Memory Associative Recall by POCS Identical PixelsColumn Space

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Application: Neural Network Associative Memory Example 23 5 1 1019 4

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Application: Neural Network Associative Memory Example 12 4 0 1019 3

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A Library of Mathematicians What does the Average Mathematician Look Like?

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Convergence As a Function of Percent of Known Image

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Convergence As a Function of Library Size

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Convergence As a Function of Noise Level

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Combining Euler & Hermite Not Of This Library

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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

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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

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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.

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Application: Subpixel Resolution Randomly chosen 8x8 pixel blocks. Over 6 Million projections.

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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.

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Application: Subpixel Resolution Blocks of random dimension