Super-resolution meets algebraic geometry

Transcription

1 Super-resolution meets algebraic geometry Stefan Kunis joint work with H. Michael Möller, Thomas Peter, Tim Römer, and Ulrich von der Ohe

2 Outline Univariate trigonometric moment problem Multivariate trigonometric moment problem Moment problem on the unit sphere

3 Univariate trigonometric moment problem model problem d = 1 ˆf 1,..., ˆf M C \ {}, t 1,..., t M [, 1), t i t j finitely supported complex measure µ = ˆf j δ tj its trigonometric moments f : Z C f (k) := 1 e 2πikt dµ(t) = ˆf j e 2πikt j task: compute t j and ˆf j from f (k), k n

4 Univariate trigonometric moment problem Super-resolution imaging d = 1, M = 3, n = 16 (33 samples) 3 point sources optical system digital camera measure µ on [, 1) low pass S n µ samples S n µ(j/n) (S nµ(t) = k n f (k)e2πikt )

5 Univariate trigonometric moment problem Super-resolution imaging d = 1, M = 3, n = 4 (9 samples) 3 point sources optical system digital camera measure µ on [, 1) low pass S n µ samples S n µ(j/n) (S nµ(t) = k n f (k)e2πikt )

6 Univariate trigonometric moment problem t j [, 1), z j = e 2πit j T := {z C : z = 1} Prony polynomial p : C C M p(z) := (z z j ) = main point ˆp l z l l= ˆp l f (l m) = l= = ˆf j e 2πit ( ) j m ˆp l e 2πit j l l= ˆf j z m j p(z j ) = (where we used f (k) := 1 e2πikt dµ(t) = M ˆf j e 2πikt j )

7 Univariate trigonometric moment problem Prony s method, n = M Input: moments f (k), k = M,..., M 1 Set up the rank-m Toeplitz matrix T := (f (k l)) l=,...,m C M+1 M+1 k=,...,m 2 Compute ker T = span{ˆp} 3 Compute zeros z j of p(z) = M k= ˆp kz k 4 Compute coefficients ˆf j Output: parameters z j = e 2πit j, ˆf j, j = 1,..., M

8 Multivariate trigonometric moment problem d = 1 ˆf 1,..., ˆf M C \ {}, t 1,..., t M [, 1), t i t j finitely supported complex measure µ = ˆf j δ tj its trigonometric moments f : Z C f (k) := [,1) e 2πikt dµ(t) = ˆf j e 2πikt j task: compute t j and ˆf j from f (k), k n

9 Multivariate trigonometric moment problem d > 1 ˆf 1,..., ˆf M C \ {}, t 1,..., t M [, 1) d, t i t j finitely supported complex measure µ = ˆf j δ tj its trigonometric moments f : Z d C f (k) := e 2πikt dµ(t) = [,1) d ˆf j e 2πikt j task: compute t j and ˆf j from f (k), k n

10 Multivariate trigonometric moment problem d > 1, M parameters Ω := {z j = e 2πit j : j = 1,..., M} T d C d [n] = {,..., n} d, N := (n + 1) d, identify C N and polynomials Π n := span{z k : k [n]} C[Z 1,..., Z d ] evaluation map Π n p Ap = (p(z)) z Ω A := ( zj k multilevel Toeplitz matrix ),...,M;k [n] CM N C N N T := (f (k l)) k,l [n] = A diag(ˆf )A

11 Multivariate trigonometric moment problem Question Under which condition does hold true? Ω = V (ker T ) := {z C d : p(z) = for all p ker T }

12 Multivariate trigonometric moment problem Theorem... z j = e 2πit j T d, q := min i j t j t i, then (n 1)q > log d implies V (ker T ) = Ω. Proof. Algebraic techniques (Curtow/Fialkow,...) and full rank of A n 1 if (n 1)q > 2d Kunis/Potts (n 1)q > d Ingham, Komornik/Loreti, Potts/Tasche (n 1)q > log d (n 1)q > c d (n 1)q > 1 still open, d = 1 Potts/Tasche, Moitra d c d

13 Multivariate trigonometric moment problem Corollary (Dual certificate, sum of squares) p(t) = 1 p l (e 2πit ) 2, p l ONB for ker(t ), N l=1 is a trigonometric polynomial of degree n and fulfills p(t) 1 for all t [, 1) d p(t) = 1 if and only if t = t j Convex optimization approach (Candès/Fernandez-Granda,...) min µ TV s.t. f µ (k) = f (k) p... semidefinite program

14 Multivariate trigonometric moment problem, d = 1 M = 3, n = i i sum of squares 1 p zeros in C

15 Multivariate trigonometric moment problem, d = 2 M = n = 2, t 1 = (, ), t 2 = ( 1 2, 1 2 ), f (k) = (1, 1)k + ( 1, 1) k T 1 T 2 T T = T 2 T 1 T 2, T 1 = 2, T 2 = 2 2 T 1 T 2 T ker T : p 1 = 1 + Z 2 1, p 2 = Z 1 + Z 2, sum of squares 1 p zeros in [, 1) 2

16 Multivariate trigonometric moment problem, d = 3 M = 2, n = 1 zeros of p1 in [, 1)3 zeros of p1 and p2 in [, 1)3

17 Moment problem on the unit sphere coefficients ˆf j R, and parameter x j S d 1, j = 1,..., M, µ : P(S d 1 ) R, µ = ˆf j δ xj moment sequence is the spherical harmonic sum f : {(k, l) : k N, l = 1,..., N k } R (k, l) Yk l (x) dµ(x) = ˆf j Yk l (x j ) S d 1 nonequispaced spherical Fourier matrix ( ) Y n := Yk l (x j ),...,M k N,k n,l=1,...,n k R M N

18 Moment problem on the unit sphere Theorem... x j S d 1, (n 1)q > 2.5πd, q := min j l arccos and the entries of V (ker H n ) = Ω (x j x l ), then H n = Y n diag(ˆf 1,..., ˆf M )Y n are computed from the moments f (k, l), k 2n, l = 1,..., N k.

19 Moment problem on the unit sphere M = 3, n = 2 M = 5, n = 3 Stefan Kunis, Thomas Peter, Tim Römer, and Ulrich von der Ohe. A multivariate generalization of Prony s method. Linear Algebra Appl., pages 31 47, 216. Stefan Kunis, H. Michael Möller, and Ulrich von der Ohe. Prony s method on the sphere. arxiv:163.22