Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Simulation Results “To err is human. To really screw up, you need a computer.” The Murphy Institute Assume τ = 1. Estimates and their standard deviations are based on averaging over 100 Monte-Carlo runs n = number of data points, iter = average number of iterations required for convergence, and % miss = average number of missing observations Estimates of τ are labeled � τ , and std ( � τ ) is the experimental standard deviation Threshold value of η 0 = 0 . 35 τ = 0 . 35 was used Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Simulation Results, Cont’d 1.) Noise-free estimation. Results from simulating noise-free estimation of τ . n M % miss iter τ 2 τ 3 τ > 3 τ 10 1 10 81 . 69 3 . 3 100% 0 0 0 10 2 10 97 . 92 10 . 5 100 0 0 0 10 3 10 99 . 80 46 . 5 100 0 0 0 10 4 10 99 . 98 316 . 2 100 0 0 0 10 5 10 99 . 998 2638 . 7 100 0 0 0 10 2 4 97 . 84 15 . 2 82% 12 4 2 10 2 6 97 . 82 14 . 2 97 3 0 0 10 2 8 97 . 80 10 . 2 98 1 1 0 10 2 10 97 . 78 10 . 2 99 1 0 0 10 2 12 97 . 76 8 . 6 100 0 0 0 10 2 14 97 . 75 7 . 4 100 0 0 0 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Simulation Results, Cont’d 2.) Uniformly distributed noise. Results from estimation of τ from noisy measurements. n M ∆ % miss iter τ std ( � τ ) � 10 1 10 − 3 10 81 . 37 4 . 35 0 . 9987 0 . 0005 10 2 10 − 3 10 97 . 88 9 . 67 0 . 9980 0 . 0010 10 3 10 − 3 50 99 . 80 16 . 0 0 . 9969 0 . 0028 10 1 10 − 2 10 80 . 85 4 . 38 0 . 9888 0 . 0046 10 1 10 − 2 10 81 . 94 4 . 45 0 . 9883 0 . 0051 10 1 10 − 1 10 81 . 05 4 . 33 0 . 8857 0 . 0432 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability Let P = { p 1 , p 2 , p 3 , . . . } = { 2 , 3 , 5 , . . . } be the set of all prime numbers. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability Let P = { p 1 , p 2 , p 3 , . . . } = { 2 , 3 , 5 , . . . } be the set of all prime numbers. “God gave us the integers. The rest is the work of man.” Kronecker Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability Let P = { p 1 , p 2 , p 3 , . . . } = { 2 , 3 , 5 , . . . } be the set of all prime numbers. “God gave us the integers. The rest is the work of man.” Kronecker “ . . . the Euler formulae (1736) ∞ ∞ � � 1 n − z = ζ ( n ) = 1 − ( p j ) − z , ℜ ( z ) > 1 n =1 j =1 was introduced to us at school, as a joke.” Littlewood Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d “Euclid’s algorithm is found in Book 7, Proposition 1 and 2 of his Elements (c.300 B.C.). We might call it the grand daddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.” Knuth Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d “Euclid’s algorithm is found in Book 7, Proposition 1 and 2 of his Elements (c.300 B.C.). We might call it the grand daddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.” Knuth The Euclidean algorithm is a division process for the integers Z . The algorithm is based on the property that, given two positive integers a and b , a > b , there exist two positive integers q and r such that a = q · b + r , 0 ≤ r < b . If r = 0, we say that b divides a . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d “Euclid’s algorithm is found in Book 7, Proposition 1 and 2 of his Elements (c.300 B.C.). We might call it the grand daddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.” Knuth The Euclidean algorithm is a division process for the integers Z . The algorithm is based on the property that, given two positive integers a and b , a > b , there exist two positive integers q and r such that a = q · b + r , 0 ≤ r < b . If r = 0, we say that b divides a . The Euclidean algorithm yields the greatest common divisor of two (or more) elements of Z . The greatest common divisor of two integers a and b , denoted by gcd( a , b ), is the the largest integer that evenly divides both integers. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Examples Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Examples gcd(3 , 7) = 1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Examples gcd(3 , 7) = 1 gcd(3 , 6) = 3 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Examples gcd(3 , 7) = 1 gcd(3 , 6) = 3 gcd(35 , 21) = 7 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem (Fundamental Theorem of Arithmetic) Every positive integer can be written uniquely as the product of primes, with the prime factors in the product written in the order of nondecreasing size. If gcd( a , b ) = 1, we say that the numbers are relatively prime. This means that a and b share no common prime factors in their prime factorization. gcd( k 1 , . . . , k n ) is the greatest common divisor of the set { k j } , i.e., the product of the powers of all prime factors p that divide each k j . Examples gcd(3 , 7) = 1 gcd(3 , 6) = 3 gcd(35 , 21) = 7 gcd(35 , 21 , 15) = 1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem Given n ( n ≥ 2) “randomly chosen” positive integers { k 1 , . . . , k n } , P { gcd( k 1 , . . . , k n ) = 1 } = [ ζ ( n )] − 1 . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Heuristic argument for this “theorem.” Given randomly distributed positive integers, by the Law of Large Numbers, about 1/2 of them are even, 1/3 of them are multiples of three, and 1 / p are a multiple of some prime p . Thus, given n independently chosen positive integers, P { p | k 1 , p | k 2 , . . . , and p | k n } = ( Independence ) P { p | k 1 } · P { p | k 2 } · . . . · P { p | k n } = 1 / ( p ) · 1 / ( p ) · . . . · 1 / ( p ) = 1 / ( p ) n . Therefore, P { p � | k 1 , p � | k 2 , . . . , and p � | k n } = 1 − 1 / ( p ) n . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d By the Fundamental Theorem of Arithmetic, every integer has a unique representation as a product of primes. Combining that theorem with the definition of gcd, we get ∞ � 1 − 1 / ( p j ) n , P { gcd( k 1 , . . . , k n ) = 1 } = j =1 where p j is the j th prime. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d By the Fundamental Theorem of Arithmetic, every integer has a unique representation as a product of primes. Combining that theorem with the definition of gcd, we get ∞ � 1 − 1 / ( p j ) n , P { gcd( k 1 , . . . , k n ) = 1 } = j =1 where p j is the j th prime. But, by Euler’s formula, ∞ � 1 ζ ( z ) = 1 − ( p j ) − z , ℜ ( z ) > 1 . j =1 Therefore, P { gcd( k 1 , . . . , k n ) = 1 } = 1 / ( ζ ( n )) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d This argument breaks down on the first line. Any uniform distribution on the positive integers would have to be identically zero. The merit in the argument lies in the fact that it gives an indication of how the zeta function plays a role in the problem. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d This argument breaks down on the first line. Any uniform distribution on the positive integers would have to be identically zero. The merit in the argument lies in the fact that it gives an indication of how the zeta function plays a role in the problem. Let card {·} denote cardinality of the set {·} , and let { 1 , . . . , ℓ } n denote the sublattice of positive integers in R n with coordinates c such that 1 ≤ c ≤ ℓ . Therefore, N n ( ℓ ) = card { ( k 1 , . . . , k n ) ∈ { 1 , . . . , ℓ } n : gcd( k 1 , . . . , k n ) = 1 } is the number of relatively prime elements in { 1 , . . . , ℓ } n . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d This argument breaks down on the first line. Any uniform distribution on the positive integers would have to be identically zero. The merit in the argument lies in the fact that it gives an indication of how the zeta function plays a role in the problem. Let card {·} denote cardinality of the set {·} , and let { 1 , . . . , ℓ } n denote the sublattice of positive integers in R n with coordinates c such that 1 ≤ c ≤ ℓ . Therefore, N n ( ℓ ) = card { ( k 1 , . . . , k n ) ∈ { 1 , . . . , ℓ } n : gcd( k 1 , . . . , k n ) = 1 } is the number of relatively prime elements in { 1 , . . . , ℓ } n . Theorem (MEA Theorem, C (1998), ...) Let N n ( ℓ ) = card { ( k 1 , . . . , k n ) ∈ { 1 , . . . , ℓ } n : gcd( k 1 , . . . , k n ) = 1 } . For n ≥ 2 , we have that N n ( ℓ ) = [ ζ ( n )] − 1 . lim ℓ n ℓ →∞ Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Brief Discussion of Proof : Let ⌊ x ⌋ denote the floor function of x , namely ⌊ x ⌋ = max k ≤ x { k : k ∈ Z } . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Brief Discussion of Proof : Let ⌊ x ⌋ denote the floor function of x , namely ⌊ x ⌋ = max k ≤ x { k : k ∈ Z } . �� ℓ �� n �� �� n �� �� n � � � ℓ ℓ N n ( ℓ ) = ℓ n − + − + · · · . p i p i · p j p i · p j · p k p i p i < p j p i < p j < p k Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Brief Discussion of Proof : Let ⌊ x ⌋ denote the floor function of x , namely ⌊ x ⌋ = max k ≤ x { k : k ∈ Z } . �� ℓ �� n �� �� n �� �� n � � � ℓ ℓ N n ( ℓ ) = ℓ n − + − + · · · . p i p i · p j p i · p j · p k p i p i < p j p i < p j < p k Convergence is demonstrated by a sequence of careful estimates, use of M¨ obius Inversion, and more careful estimates. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d The counting formula is seen as follows. Choose a prime number p i . The number of integers in { 1 , . . . , ℓ } such that p i divides an element of that � � � � ℓ ℓ set is . (Note that is possible to have p i > ℓ , because = 0.) p i p i Therefore, the number of n -tuples ( k 1 , . . . , k n ) contained in the lattice { 1 , . . . , ℓ } n such that p i divides every integer in the n -tuple is �� ℓ �� n . p i Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d The counting formula is seen as follows. Choose a prime number p i . The number of integers in { 1 , . . . , ℓ } such that p i divides an element of that � � � � ℓ ℓ set is . (Note that is possible to have p i > ℓ , because = 0.) p i p i Therefore, the number of n -tuples ( k 1 , . . . , k n ) contained in the lattice { 1 , . . . , ℓ } n such that p i divides every integer in the n -tuple is �� ℓ �� n . p i Next, if p i · p j divides an integer k , then p i | k and p j | k . Therefore, the number of n -tuples ( k 1 , . . . , k n ) contained in the lattice { 1 , . . . , ℓ } n such that p i or p j or their product divide every integer in the n -tuple is �� ℓ �� ℓ �� n �� n �� �� n ℓ + − , p i p j p i · p j where the last term is subtracted so that we do not count the same numbers twice (in a simple application of the inclusion-exclusion principle). Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Each term is convergent – �� �� n � � n � � 1 ℓ ≤ 1 ℓ ℓ n ℓ n p i · · · · p k p i · p j · · · · · p k p i <...< p k p i <...< p k ≤ ℓ k � � n � � 1 1 = = p n p i · · · · p k p i <...< p k ≤ ℓ p ≤ ℓ k k ∞ � � 1 1 ≤ ≤ . p n j n p prime j =2 Since n ≥ 2, this series is convergent. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Now, let k � ∞ 1 M k = , for k = 2 , 3 , . . . . j n j =2 By noting that since n ≥ 2 and the sum is over j ∈ N \ { 1 } , we get � π 2 � � 1 0 < j n ≤ 6 − 1 < 1 . j Since the k th term in the expansion of N n ( ℓ ) /ℓ n is dominated by M k and since � π 2 � k ∞ ∞ � � 6 M k ≤ 6 − 1 = (12 − π 2 ) k =0 k =0 is convergent, the series converges absolutely. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Euler showed that � � � 1 1 1 1 − + ( p i · p j ) n − ( p i · p j · p k ) n + · · · p n i p i p i < p j p i < p j < p k � µ ( m ) = [ ζ ( n )] − 1 . = m n m where the last sum is over m ∈ N . For n ≥ 2, this series is absolutely convergent. ✷ Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem Let ω ∈ (1 , ∞ ) . Then lim ω →∞ [ ζ ( ω )] − 1 = 1 , converging to 1 from below faster than 1 / (1 − 2 1 − ω ) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem Let ω ∈ (1 , ∞ ) . Then lim ω →∞ [ ζ ( ω )] − 1 = 1 , converging to 1 from below faster than 1 / (1 − 2 1 − ω ) . Proof : Since ζ ( ω ) = � ∞ n =1 n − ω and ω > 1, 1 ≤ ζ ( ω ) = 1 + 1 2 ω + 1 3 ω + 1 4 ω + 1 5 ω + · · · 1 + 1 2 ω + 1 2 ω + 1 4 ω + · · · + 1 + 1 8 ω + · · · + 1 ≤ + · · · 4 ω 8 ω � �� � � �� � 4 − times 8 − times � 2 � k ∞ � 1 1 = = = 1 − 2 1 − ω . 2 ω 2 1 − 2 ω k =0 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function π , the Primes, and Probability, Cont’d Theorem Let ω ∈ (1 , ∞ ) . Then lim ω →∞ [ ζ ( ω )] − 1 = 1 , converging to 1 from below faster than 1 / (1 − 2 1 − ω ) . Proof : Since ζ ( ω ) = � ∞ n =1 n − ω and ω > 1, 1 ≤ ζ ( ω ) = 1 + 1 2 ω + 1 3 ω + 1 4 ω + 1 5 ω + · · · 1 + 1 2 ω + 1 2 ω + 1 4 ω + · · · + 1 + 1 8 ω + · · · + 1 ≤ + · · · 4 ω 8 ω � �� � � �� � 4 − times 8 − times � 2 � k ∞ � 1 1 = = = 1 − 2 1 − ω . 2 ω 2 1 − 2 ω k =0 → 1 + . Thus, [ ζ ( ω )] − 1 − → 1 − as ω − → ∞ , (1 − 2 1 − ω ) − As ω − → ∞ . ✷ Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA) S = { s j } n j =1 , with s j = k j τ + ϕ + η j Let � τ denote the value the algorithm gives for τ , and let “ ← − ” denote replacement . Initialize: Sort the elements of S in descending order. Set iter = 0 . 1.) [Adjoin 0 after first iteration.] If iter > 0 , then S ← − S ∪ { 0 } . 2.) [Form the new set with elements ( s j − s j +1 ).] Set s j ← − ( s j − s j +1 ). 3.) [Sort.] Sort the elements in descending order. 4.) [Eliminate zero(s).] If s j = 0, then S ← − S \ { s j } . 5.) The algorithm terminates if S has only one element s 1 . Declare � τ = s 1 . If not, iter ← − ( iter + 1 ). Go to 1.) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Euclidean algorithm for { k j } n j =1 ⊂ N , τ > 0 – Lemma gcd( k 1 τ, . . . , k n τ ) = τ gcd( k 1 , . . . , k n ) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Euclidean algorithm for { k j } n j =1 ⊂ N , τ > 0 – Lemma gcd( k 1 τ, . . . , k n τ ) = τ gcd( k 1 , . . . , k n ) . What if “integers are noisy?” Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Euclidean algorithm for { k j } n j =1 ⊂ N , τ > 0 – Lemma gcd( k 1 τ, . . . , k n τ ) = τ gcd( k 1 , . . . , k n ) . What if “integers are noisy?” Remainder terms could be noise, and thus could be non-zero numbers arbitrarily close to zero. Subsequent steps in the procedure may involve dividing by such numbers, which would result in arbitrarily large numbers. The standard algorithm is unstable under perturbation by noise. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Euclidean algorithm for { k j } n j =1 ⊂ N , τ > 0 – Lemma gcd( k 1 τ, . . . , k n τ ) = τ gcd( k 1 , . . . , k n ) . What if “integers are noisy?” Remainder terms could be noise, and thus could be non-zero numbers arbitrarily close to zero. Subsequent steps in the procedure may involve dividing by such numbers, which would result in arbitrarily large numbers. The standard algorithm is unstable under perturbation by noise. Solution : Replace division with subtraction, and threshold/average/filter/transform to eliminate noise. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Lemma gcd( k 1 , . . . , k n ) = gcd(( k 1 − k 2 ) , ( k 2 − k 3 ) , . . . , ( k n − 1 − k n ) , k n ) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Lemma gcd( k 1 , . . . , k n ) = gcd(( k 1 − k 2 ) , ( k 2 − k 3 ) , . . . , ( k n − 1 − k n ) , k n ) . Lemma gcd(( k 1 − k 2 ) , ( k 2 − k 3 ) , . . . , ( k n − 1 − k n )) = gcd(( k 1 − k n ) , . . . , ( k n − 1 − k n )) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Combining the MEA Theorem with the Lemmas above gives the theoretical underpinnings of the Modified Euclidean Algorithm. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Combining the MEA Theorem with the Lemmas above gives the theoretical underpinnings of the Modified Euclidean Algorithm. Corollary Let n ≥ 2 . Given a randomly chosen n-tuple of positive integers ( k 1 , . . . , k n ) ∈ { 1 , . . . , ℓ } n , gcd( k 1 τ, . . . , k n τ ) − → τ , with probability [ ζ ( n )] − 1 as ℓ − → ∞ . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d Combining the MEA Theorem with the Lemmas above gives the theoretical underpinnings of the Modified Euclidean Algorithm. Corollary Let n ≥ 2 . Given a randomly chosen n-tuple of positive integers ( k 1 , . . . , k n ) ∈ { 1 , . . . , ℓ } n , gcd( k 1 τ, . . . , k n τ ) − → τ , with probability [ ζ ( n )] − 1 as ℓ − → ∞ . Moreover, the estimate (1 − 2 1 − ω ) − 1 ≤ [ ζ ( ω )] − 1 ≤ 1 shows that the algorithm very likely produces this value in the noise-free case or with minimal noise with as few as 10 data elements. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Modified Euclidean Algorithm (MEA), Cont’d S.D. Casey and B.M. Sadler, “Pi, the primes, periodicities and probability,” The American Mathematical Monthly , Vol. 120, No. 7, pp. 594–608 (2013). S.D. Casey, “Sampling issues in Fourier analytic vs. number theoretic methods in parameter estimation,” 31 st Annual Asilomar Conference on Signals, Systems and Computers , Vol. 1, pp, 453–457 (1998). S.D. Casey and B.M. Sadler, “Modifications of the Euclidean algorithm for isolating periodicities from a sparse set of noisy measurements,” IEEE Transactions on Signal Processing , Vol. 44, No. 9, pp. 2260–2272 (1996) . B.M. Sadler and S.D. Casey, “On pulse interval analysis with outliers and missing observations,” IEEE Transactions on Signal Processing , Vol. 46, No. 11, pp. 2990–3003 (1998). The MEA can work with very sparse data sets ( > 95% missing observations). Trade-off – low noise – use MEA vs. higher noise – combine spectral analysis with MEA theory. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Mathematical Models – Multiple Periods Our data model is the union of M copies of S = { s i , j } n i j =1 with s j = k j τ + ϕ + η j , , each with different periods or “generators” Γ = { τ i } , k ij ’s and phases. Let τ M = max i { τ i } and τ m = min i { τ i } . Then our data is � � n i � M S = ϕ i + k ij τ i + η ij , i =1 j =1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Mathematical Models – Multiple Periods Our data model is the union of M copies of S = { s i , j } n i j =1 with s j = k j τ + ϕ + η j , , each with different periods or “generators” Γ = { τ i } , k ij ’s and phases. Let τ M = max i { τ i } and τ m = min i { τ i } . Then our data is � � n i � M S = ϕ i + k ij τ i + η ij , i =1 j =1 where n i is the number of elements from the i th generator the different periods or “generators” are Γ = { τ i } { k ij } is a linearly increasing sequence of natural numbers with missing observations ϕ i (the phases) are random variables uniformly distributed in [0 , τ i ) η ij ’s are zero-mean iid Gaussian with standard deviation 3 σ ij < τ/ 2 We think of the data as events from M periodic processes, and l =1 , where N = � represent it, after reindexing, as S = { α l } N i n i . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Structure of Randomness over [0 , T ) Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Structure of Randomness over [0 , T ) Theorem (Weyl’s Equidistribution Theorem) Let φ be an irrational number, j ∈ N . Let � j φ � = j φ − ⌊ j φ ⌋ . Then given a , b, 0 ≤ a < b < 1 , � � 1 n card 1 ≤ j ≤ n : � j φ � ∈ [ a , b ] − → ( b − a ) as n − → ∞ . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Structure of Randomness over [0 , T ) Assuming only minimal knowledge of the range of { τ i } , namely bounds T L , T U such that 0 < T L ≤ τ i ≤ T U , we phase wrap the data by the mapping � α l � � α l � = α l Φ ρ ( α l ) = ρ − , ρ ρ where ρ ∈ [ T L , T U ], and ⌊·⌋ is the floor function. Thus �·� is the fractional part, and so Φ ρ ( α l ) ∈ [0 , 1). Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The Structure of Randomness over [0 , T ) Assuming only minimal knowledge of the range of { τ i } , namely bounds T L , T U such that 0 < T L ≤ τ i ≤ T U , we phase wrap the data by the mapping � α l � � α l � = α l Φ ρ ( α l ) = ρ − , ρ ρ where ρ ∈ [ T L , T U ], and ⌊·⌋ is the floor function. Thus �·� is the fractional part, and so Φ ρ ( α l ) ∈ [0 , 1). Definition A sequence of real random variables { x j } ⊂ [0 , 1) is essentially uniformly distributed in the sense of Weyl if given a , b , 0 ≤ a < b < 1, � � 1 1 ≤ j ≤ n : x j ∈ [ a , b ] − → ( b − a ) n card as n − → ∞ almost surely. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Applying Weyl’s Theorem We assume that for each i , { k ij } is a linearly increasing infinite sequence of natural numbers with missing observations such that k ij − → ∞ as j − → ∞ . Weyl’s Theorem applies asymptotically. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Applying Weyl’s Theorem We assume that for each i , { k ij } is a linearly increasing infinite sequence of natural numbers with missing observations such that k ij − → ∞ as j − → ∞ . Weyl’s Theorem applies asymptotically. Theorem (C (2014)) For almost every choice of ρ (in the sense of Lebesgue measure) Φ ρ ( α l ) is essentially uniformly distributed in the sense of Weyl. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Applying Weyl’s Theorem, Cont’d Moreover, the set of ρ ’s for which this is not true are rational multiples of { τ i } . Therefore, except for those values, Φ ρ ( α ij ) is essentially uniformly distributed in [ T L , T U ) . The values at which Φ ρ ( α ij ) = 0 almost surely are ρ ∈ { τ i / n : n ∈ N } . These values of ρ cluster at zero, but spread out for lower values of n . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Applying Weyl’s Theorem, Cont’d Moreover, the set of ρ ’s for which this is not true are rational multiples of { τ i } . Therefore, except for those values, Φ ρ ( α ij ) is essentially uniformly distributed in [ T L , T U ) . The values at which Φ ρ ( α ij ) = 0 almost surely are ρ ∈ { τ i / n : n ∈ N } . These values of ρ cluster at zero, but spread out for lower values of n . We phase wrap the data by computing modulus of the spectrum, i.e., compute � � N � � � � e (2 π is ( j ) /τ ) � | S iter ( τ ) | = � . � j =1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Applying Weyl’s Theorem, Cont’d Moreover, the set of ρ ’s for which this is not true are rational multiples of { τ i } . Therefore, except for those values, Φ ρ ( α ij ) is essentially uniformly distributed in [ T L , T U ) . The values at which Φ ρ ( α ij ) = 0 almost surely are ρ ∈ { τ i / n : n ∈ N } . These values of ρ cluster at zero, but spread out for lower values of n . We phase wrap the data by computing modulus of the spectrum, i.e., compute � � N � � � � e (2 π is ( j ) /τ ) � | S iter ( τ ) | = � . � j =1 The values of | S iter ( τ ) | will have peaks at the periods τ j and their harmonics ( τ j ) / k . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period The EQUIMEA Algorithm – One Period S = { s j } n j =1 , with s j = k j τ + ϕ + η j Initialize: Sort the elements of S in descending order. Form the new set with elements ( s j − s j +1 ). Set s j ← − ( s j − s j +1 ). (Note, this eliminates the phase ϕ .) Let � τ denote the value the algorithm gives for τ , and let “ ← − ” denote replacement . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period The EQUIMEA Algorithm – One Period 1.) [Adjoin 0 after first iteration.] S iter ← − S ∪ { 0 } . 2.) [Sort.] Sort the elements of S iter in descending order. 3.) [Compute all differences.] Set S iter = � ( s j − s k ) for all j , k with s j > s k . 4.) [Eliminate zero(s).] If s j = 0, then S iter ← − S iter \ { s j } . 5.) [Adjoin previous iteration.] Form S iter ← − S iter ∪ S iter − 1 . 6.) [Compute spectrum.] Compute � � N � � � � e (2 π is ( j ) /τ ) � | S iter ( τ ) | = � . � j =1 7.) [Threshold.] Choose the largest peak. Label it as τ iter 8.) The algorithm terminates if | τ iter − τ iter − 1 | < Error . Declare τ = τ iter . If not, iter ← − ( iter + 1 ). Go to 1.) . � Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period, Cont’d Figure: EQUIMEA One Period Tau – Original Data Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period, Cont’d Figure: EQUIMEA One Period Tau – One Iteration Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period, Cont’d Figure: EQUIMEA One Period Tau – One Iteration – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period, Cont’d Figure: EQUIMEA One Period Tau – Third Iteration Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – One Period, Cont’d Figure: EQUIMEA One Period Tau – Third Iteration – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Deinterleaving Multiple Signals Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Multiple Periods The EQUIMEA Algorithm – Multiple Periods Our data model is the union of M copies of S = { s i , j } n i j =1 with s j = k j τ + ϕ + η j , , each with different periods or “generators” Γ = { τ i } , k ij ’s and phases. Let τ M = max i { τ i } and τ m = min i { τ i } . Then our data is � � n i � M S = ϕ i + k ij τ i + η ij , i =1 j =1 Let � τ denote the value the algorithm gives for τ , and let “ ← − ” denote replacement . l =1 , where N = � After reindexing, S = { α l } N i n i . Initialize: Sort the elements of S in descending order. Form the new set with elements ( s l − s l +1 ). Set s l ← − ( s l − s l +1 ). (Note, this eliminates the phase ϕ .) Set iter = 1 , i = 1, and Error . Go to 1.) Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Multiple Periods 1.) [Adjoin 0 after first iteration.] S iter ← − S ∪ { 0 } . 2.) [Sort.] Sort the elements of S iter in descending order. 3.) [Compute all differences.] Set S iter = � ( s j − s k ) with s j > s k . 4.) [Eliminate zero(s).] If s j = 0, then S iter ← − S iter \ { s j } . 5.) [Adjoin previous iteration.] Form S iter ← − S iter ∪ S iter − 1 . � � � � � N � � j =1 e (2 π is ( j ) /τ ) 6.) [Compute spectrum.] Compute | S iter ( τ ) | = � . � 7.) [Threshold.] Choose the largest peak. Label it as τ iter 8.) If | τ iter − τ iter − 1 | < Error . Declare � τ i = τ iter . If not, iter ← − ( iter + 1 ). Go to 1.) . 9.) Given τ i , frequency notch | S iter ( τ ) | for � τ i / m , m ∈ N . Let i ← − i + 1. � � � � � N � j =1 e (2 π is ( j ) /τ ) � 10.) [Compute spectrum.] Compute | S iter ( τ ) | = � . � 11.) [Threshold.] Choose the largest peak. Label it as τ i +1 . Algorithm terminates when there are no peaks. Else, go to 9.) . Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods Figure: Two Periods – OriginalData Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods, Cont’d Figure: Spectrum of Two Period Data Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods, Cont’d Figure: EQUIMEA – Two Periods – Iter1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods, Cont’d Figure: EQUIMEA – Two Periods – Iter1 – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods, Cont’d Figure: EQUIMEA – Two Periods – Iter2 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Two Periods, Cont’d Figure: EQUIMEA – Two Periods – Iter2 – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods Figure: Three Periods – OriginalData Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods, Cont’d Figure: Spectrum of Three Period Data Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods, Cont’d Figure: EQUIMEA – Three Periods – Iter1 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods, Cont’d Figure: EQUIMEA – Three Periods – Iter1 – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods, Cont’d Figure: EQUIMEA – Three Periods – Iter2 Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function The EQUIMEA Algorithm – Three Periods, Cont’d Figure: EQUIMEA – Three Periods – Iter2 – Spectrum Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Estimating the ϕ i ’s τ i of τ i , exp(2 π i k j τ i For a good estimate � τ i ) ≈ 1. For η j ≪ τ i ≈ � τ i , b η j / � τ i ≪ 1, and so exp(2 π i η j / � τ i ) ≈ exp(0) = 1. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Estimating the ϕ i ’s τ i of τ i , exp(2 π i k j τ i For a good estimate � τ i ) ≈ 1. For η j ≪ τ i ≈ � τ i , b η j / � τ i ≪ 1, and so exp(2 π i η j / � τ i ) ≈ exp(0) = 1. Therefore, � n � � τ i � exp(2 π i s ij 2 π arg ) τ i � j =1 � n � � τ i � exp(2 π i k ij τ i ) exp(2 π i η ij ) exp(2 π i ϕ = 2 π arg ) τ i τ i τ i � � � j =1 � n � � � � τ i exp(2 π i ϕ i τ i n · exp(2 π i ϕ i � = � ≈ 2 π arg ) 2 π arg ) τ i τ i � � j =1 � � � � �� � � τ i exp(2 π i ϕ i τ i exp(2 π i ϕ i � = � = arg n + arg ) 2 π arg ) 2 π τ i � τ i � τ i 2 πϕ i � = = ϕ i 2 π τ i � Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Estimating the k ij ’s’s We present two methods of getting an estimate on the set of k ij ’s. Let round( · ) denotes rounding to the nearest integer. Given a good estimate � ϕ i , the first is to form the set σ = { k ij τ i + ϕ i + η ij − � φ } n j =1 . Given the estimate � τ i , estimate k ij by � k ij τ i + ϕ i + η ij − � � ϕ i � k ij = round . τ i � Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Estimating the k ij ’s’s We present two methods of getting an estimate on the set of k ij ’s. Let round( · ) denotes rounding to the nearest integer. Given a good estimate � ϕ i , the first is to form the set σ = { k ij τ i + ϕ i + η ij − � φ } n j =1 . Given the estimate � τ i , estimate k ij by � k ij τ i + ϕ i + η ij − � � ϕ i � k ij = round . τ i � Let σ ′ = { K ij τ − i + η ′ ij } n − 1 j =1 ∪ { k ( i , n i ) τ i + ϕ i + η in i − � ϕ i } , where K ij = k ij − k ( i , j +1) and η ′ ij = η ij − η ( i , j +1) . Given the estimate � τ i , � k ( i , ni ) τ i + ϕ i + η ( i , ni ) − b � ϕ i estimate k ( i , n i ) by � k ( i , n i ) = round and K ij by b τ i � K ij τ i + η ′ � ij � K ij = round . τ i � Then, � k ( i , n i − 1) = � K ( i , n i − 1) + � k ( i , n i ) , � k ( i , n i − 2) = � K ( i , n i − 2) + � k ( i , n i − 1) , and so on. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Epilogue: The Riemann Zeta Function A nice “byproduct” of the MEA work is a novel way to compute values of the Riemann Zeta Function. Stephen Casey The Analysis of Periodic Point Processes
Motivation: Signal and Image Signatures π , the Primes, and Probability The Modified Euclidean Algorithm (MEA) Deinterleaving Multiple Signals (EQUIMEA) Epilogue: The Riemann Zeta Function Epilogue: The Riemann Zeta Function A nice “byproduct” of the MEA work is a novel way to compute values of the Riemann Zeta Function. Definition Riemann Zeta Function : For { z ∈ C : z = x + iy , x > 1 } , ∞ � n − z . ζ ( z ) = n =1 Stephen Casey The Analysis of Periodic Point Processes
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