Ghost Imaging. Josselin Garnier (Université Paris Diderot)

Transcription

1 Grenoble December, 014 Ghost Imaging Josselin Garnier Université Paris Diderot General topic: correlation-based imaging with noise sources. Particular application: Ghost imaging.

2 Grenoble December, 014 Scalar wave equation and Green s function In this talk, we consider the scalar wave model in R d, d =,3: 1 c x u t xu = nt, x nt, x: source. c x: propagation speed parameter of the medium, assumed to be constant outside a domain with compact support. In the Fourier domain: ûω, x = ut, xe iωt dt we have ûω, x = Ĝω, x, yˆnω, yd y where the time-harmonic Green s function Ĝω, x, y is the solution of the Helmholtz equation x Ĝ+ ω c xĝ = δ x y, with the Sommerfeld radiation condition c x = c 0 at infinity: lim x d 1 x x x x i ω Ĝω, x, y = 0 c 0

3 Grenoble December, 014 Green s function estimation with ambient noise sources 1/3 1 c x u t xu = nt, x Sources nt, x: random process, stationary in time, with mean zero and covariance nt1, y 1 nt, y = Ft t 1 K y 1 δ y 1 y : statistical average. The function ˆF is the power spectral density of the sources. The function K characterizes the spatial support of the sources. The empirical cross correlation: C T τ, x 1, x = 1 T T 0 ut, x 1 ut+τ, x dt converges in probability as T to the statistical cross correlation C 1 given by C 1 τ, x 1, x = u0, x 1 uτ, x = 1 d y π dωĝω, x 1, yĝω, x, yk yˆfωe iωτ

4 Grenoble December, 014 Green s function estimation with ambient noise sources /3 B0,L x 1 x Cross correlation with noise sources distributed on a closed surface B0, L: C 1 τ, x 1, x = 1 dω π dσ yĝω, x 1, yĝω, x, yˆfωe iωτ By Helmholtz-Kirchhoff identity, we have B0,L Ĝω, x 1, x Ĝω, x 1, x = iω c 0 C 1 τ, x 1, x = c 0 4π B0,L dσ yĝω, x 1, yĝω, x, y ˆFω ω Im Ĝω, x 1, x e iωτ dω

5 Grenoble December, 014 Green s function estimation with ambient noise sources 3/3 B0,L x 1 x τ C 1 τ, x 1, x = ic 0 4π = c 0 ˆFωIm Ĝω, x 1, x e iωτ dω F τ Gτ, x 1, x F τ G τ, x 1, x The cross correlation is related to the Green s function. the passive sensors can be transformed into virtual sources. If the medium is homogeneous then Ĝω, x 1, x = Ĉ 1 ω, x 1, x = c 0 1 4π x 1 x exp i ω x 1 x c 0 ˆFω ω ω sinc x 1 x c 0

6 [1] J. Garnier and G. Papanicolaou, SIAM J. Imaging Sciences, Reflector imaging with a passive receiver array Ambient noise sources emit stationary random signals. The signals ut, x r r=1,...,nr are recorded by the receivers x r r=1,...,nr. The reflector is imaged by migration of the cross correlation matrix [1]: I y S = N r r,r =1 xr y S C T + x r y S, x r, x r c 0 c 0 with C T τ, x r, x r = 1 T T 0 ut+τ, x r ut, x r dt x 50 0 x 5 x z signal recorded at x t signal recorded at x t coda correlation x 1 x τ Good image provided the ambient noise illumination is long in time and diversified in angle [1].

7 [1] J. Garnier and G. Papanicolaou, Inverse Problems 8, ; SIIMS 7, A successful application: seismic exploration below an overburden Data: {ut, x r ; x s,t R,r = 1,...,N r,s = 1,...,N s } Correlations: {Ct, x r, x r,t R,r,r = 1,...,N r } Ct, x r, x r = N s s=1 ut +t, x r ; x s ut, x r ; x s dt Imaging by migration of correlations [1] From Bakulin and Calvert 009 geometric set-up data migration correlation migration

8 [1] A. Valencia et al., PRL 94, ; [] J. H. Shapiro et al., Quantum Inf. Process Ghost imaging Noise source. without object in path 1; a high-resolution detector measures the spatially-resolved and time-resolved intensity I 1 t,x. with object mask in path ; a single-pixel detector measures the spatially-integrated and time-resolved intensity I t. Experimental results: the correlation of I 1 and I is an image of the object [1,]!

9 Grenoble December, 014 Ghost imaging Wave equation in paths 1 and : 1 c j x u j t xu j = nt,xδz, x = x,z R R, j = 1, Noise source: nt,xnt,x = Ft t Kxδx x with ˆFω centered on ω 0. Observations: I 1 t,x = u 1 t,x,l I t = I t,x dx = R R u t,x,l+l 0 dx Correlation: C T x = 1 T T 0 I 1 t,xi tdt 1 T T 0 1 I 1 t,xdt T T 0 I tdt Let us first carry out the analysis in homogeneous media.

10 Grenoble December, 014 Wave representation in path 1: û 1 ω,x = Ĝ 1 ω,x,l,y,0 ˆnω,ydy R with Ĝ1 ω,x,l,y,0 = Ĝ 0 ω,x,l,y,0 no object. Wave representation in path : û ω,x = Ĝ ω,x,l+l0,y,0 ˆnω,ydy R with Ĝ ω,x,l+l0,y,0 = R Ĝ0 ω,x,l+l0,r,l T rĝ0 ω,r,l,y,0 dr mask T r in the plane z = L.

11 Grenoble December, 014 Wave representation in path 1: û 1 ω,x = Ĝ 1 ω,x,l,y,0 ˆnω,ydy R with Ĝ1 ω,x,l,y,0 = Ĝ 0 ω,x,l,y,0 no object. Wave representation in path : û ω,x = Ĝ ω,x,l+l0,y,0 ˆnω,ydy R with Ĝ ω,x,l+l0,y,0 = R Ĝ0 ω,x,l+l0,r,l T rĝ0 ω,r,l,y,0 dr mask T r in the plane z = L. Result with nt, x Gaussian distributed: C T x T C 1 x with C 1 x = 1 dωdω dy dy π R KyKy ˆFωˆFω R Ĝ1 ω,x,l,y,0 Ĝ1 ω,x,l,y,0 dx Ĝ ω,x,l+l 0,y,0 Ĝ ω,x,l+l 0,y,0 R

12 Grenoble December, 014 Wave representation in path 1: û 1 ω,x = Ĝ 1 ω,x,l,y,0 ˆnω,ydy R with Ĝ1 ω,x,l,y,0 = Ĝ 0 ω,x,l,y,0 no object. Wave representation in path : û ω,x = Ĝ ω,x,l+l0,y,0 ˆnω,ydy R with Ĝ ω,x,l+l0,y,0 = R Ĝ0 ω,x,l+l0,r,l T rĝ0 ω,r,l,y,0 dr mask T r in the plane z = L. Result with nt, x Gaussian distributed: C T x T C 1 x with quasi-monochromatic approximation ˆF concentrated around ω 0 : C 1 x = dy dy KyKy R R Ĝ1 ω0,x,l,y,0 Ĝ 1 ω0,x,l,y,0 dx Ĝ ω0,x,l+l 0,y,0 Ĝ ω0,x,l+l 0,y,0 R

13 Grenoble December, 014 C 1 x = dy dy KyKy R R Ĝ1 ω0,x,l,y,0 Ĝ 1 ω0,x,l,y,0 dx Ĝ ω0,x,l+l 0,y,0 Ĝ ω0,x,l+l 0,y,0 R With Ĝ1 ω0,x,l,y,0 = Ĝ0 ω0,x,l,y,0 and Ĝ ω0,x,l+l 0,y,0 = R Ĝ0 ω0,x,l+l 0,r,L T rĝ0 ω0,r,l,y,0 dr we get C 1 x = dr dr T rt r R R dykyĝ0 ω0,x,l,y,0 Ĝ 0 ω0,r,l,y,0 R dy Ky Ĝ0 ω0,x,l,y,0 Ĝ 0 ω0,r,l,y,0 R dx Ĝ 0 ω0,x,l+l 0,r,L Ĝ 0 ω0,x,l+l 0,r,L R

14 Grenoble December, 014 C 1 x = dr dr T rt r R R dykyĝ0 ω0,x,l,y,0 Ĝ 0 ω0,r,l,y,0 R dy Ky Ĝ0 ω0,x,l,y,0 Ĝ 0 ω0,r,l,y,0 R dx Ĝ 0 ω0,x,l+l 0,r,L Ĝ 0 ω0,x,l+l 0,r,L R Use reciprocity: C 1 x = dr dr T rt r R R dykyĝ0 ω0,x,l,y,0 Ĝ 0 ω0,y,0,r,l R dy Ky Ĝ0 ω0,r,l,y,0 Ĝ 0 ω0,y,0,x,l R dx Ĝ 0 ω0,r,l,x,l+l 0 Ĝ 0 ω0,x,l+l 0,r,L R

15 Grenoble December, 014 C 1 x = dr dr T rt r R R dykyĝ0 ω0,x,l,y,0 Ĝ 0 ω0,y,0,r,l R dy Ky Ĝ0 ω0,r,l,y,0 Ĝ 0 ω0,y,0,x,l R dx Ĝ 0 ω0,r,l,x,l+l 0 Ĝ 0 ω0,x,l+l 0,r,L R Quick guess: C 1 x dr dr T rt r R R h K r x h K r x h 0 r r C T x

16 Grenoble December, 014 Resolution analysis: nt,xnt,x = Ft t Kxδx x, Kx = I 0 exp x r 0 Paraxial Green s function: Result: with Ĝ 0 ω0,x,l,y,0 = 1 4πL exp i ω 0 L+i ω 0 x y c 0 c 0 L C 1 x = R hx r T r dr hx = I 0r0 4 7 π L exp, x ρ 4ρ gi0 = c 0L gi0 ω0 r 0 Resolution: ρ gi0 λ 0 L/r 0, λ 0 = πc 0 /ω 0 Rayleigh resolution formula.

17 Grenoble December, 014 Extension for partially coherent source Gauss-Schell model: nt,xnt,x = Ft t I 0 exp x+x 4r 0 x x 4ρ 0 Result: with C 1 x = Hx,r = I 0r 0ρ 0c 0 3ω 4 0 ρ gi1 R gi1 R Hx,r T r dr exp x r 4ρ gi1 x+r 4R gi1, ρ gi1 = c 0L + ρ 0 ω0 r 0 4 = ρ gi0 + ρ 0 4, R gi1 = c 0L + r 0 ω0 ρ Loss of resolution due to the partial coherence of the source: ρ gi1 > ρ gi0. - Fully incoherent case ρ 0 0: cf previous case ρ gi1 = ρ gi0. - Fully coherent case r 0 = ρ 0 : the kernel is Hx,r = I 0 ρ4 0 c 0 3ω0 4ρ gi1 which means there is no resolution at all. exp x ρ gi1 r, ρ gi1

18 Grenoble December, 014 Wave propagation in a random medium Random medium model: 1 c x = 1 1+µ x c 0 c 0 is a reference speed, µ x is a zero-mean random process. The background Green s function deterministic and known: x Ĝ 0 ω, x, y+ ω c 0 Ĝ 0 ω, x, y = δ x y The physical Green s function random and unknown: Note: x Ĝω, x, y+ ω c 0 1+µ x Ĝω, x, y = δ x y - the statistics of Ĝω, x, y x is a nonlinear function of the statistics of µ x x. - a detailed stochastic analysis is possible in different regimes of separation of scales small wavelength, large propagation distance, large bandwidth,...

19 [1] J. Garnier and K. Sølna, Ann. Appl. Probab Wave propagation in a random medium: the paraxial regime Consider the time-harmonic form of the scalar wave equation x = x,z z + û+ ω c 0 1+µx,z û = 0. Consider the paraxial regime λ l c L. More precisely, in the scaled regime the function ˆφ ε defined by ω ω ε 4, µx,z ε3 µ x ε, z ε, û ε ω,x,z = e i ωz ε 4 c 0 ˆφε ω ε 4, x ε,z satisfies ε 4 z ˆφ ε + i ωc0 zˆφε + ˆφε + ω c 0 1 ε µ x, z ˆφε ε = 0.

20 [1] J. Garnier and K. Sølna, Ann. Appl. Probab Wave propagation in a random medium: the paraxial regime Consider the time-harmonic form of the scalar wave equation x = x,z z + û+ ω c 0 1+µx,z û = 0. Consider the paraxial regime λ l c L. More precisely, in the scaled regime the function ˆφ ε defined by ω ω ε 4, µx,z ε3 µ x ε, z ε, û ε ω,x,z = e i ωz ε 4 c 0 ˆφε ω ε 4, x ε,z satisfies ε 4 z ˆφ ε + i ωc0 zˆφε + ˆφε + ω c 0 1 ε µ x, z ˆφε ε = 0. In the regime ε 1, the forward-scattering approximation in direction z is valid and ˆφ = lim ε 0 ˆφε satisfies the Itô-Schrödinger equation [1] i ω ω zˆφ+ ˆφ+ Ḃx,zˆφ = 0 c 0 with Bx,z Brownian field E[Bx,zBx,z ] = γx x z z, γx = E[µ0,0µx,z]dz. c 0

21 [1] J. Garnier and K. Sølna, Ann. Appl. Probab Wave propagation in a random medium: the paraxial regime Consider the time-harmonic form of the scalar wave equation x = x,z z + û+ ω c 0 1+µx,z û = 0. Consider the paraxial regime λ l c L. More precisely, in the scaled regime the function ˆφ ε defined by ω ω ε 4, µx,z ε3 µ x ε, z ε, û ε ω,x,z = e i ωz ε 4 c 0 ˆφε ω ε 4, x ε,z satisfies ε 4 z ˆφ ε + i ωc0 zˆφε + ˆφε + ω c 0 1 ε µ x, z ˆφε ε = 0. In the regime ε 1, the forward-scattering approximation in direction z is valid and ˆφ = lim ε 0 ˆφε satisfies the Itô-Schrödinger equation [1] dˆφ = ic 0 ω ˆφdz + iω c 0 ˆφ dbx,z with Bx,z Brownian field E[Bx,zBx,z ] = γx x z z, γx = E[µ0,0µx,z]dz.

22 [1] J. Garnier and K. Sølna, Ann. Appl. Probab We introduce the fundamental solution ˆφ ω,x,z,x 0,z 0 : dˆφ = ic 0 ω ˆφdz + iω c 0 ˆφ dbx,z starting from ˆφ ω,x,z = z 0,x 0,z 0 = δx x 0. In a homogeneous medium µ 0, B 0 the fundamental solution is ˆφ 0 ω,x,z,x0,z 0 exp iω x x0 c 0 z z 0 =. z z iπc 0 0 ω In a random medium: E [ˆφ ω,x,z,x0,z 0 ] = ˆφ 0 ω,x,z,x0,z 0 exp γ0ω z z 0 8c 0 where γx = E[µ0,0µx,z]dz = Strong damping of the coherent wave, with the scattering mean free path l sca = 8c 0/γ0ω., E [ˆφ ω,x,z,x0,z 0 ˆφ ω,x,z,x 0,z 0 ] = ˆφ 0 ω,x,z,x0,z 0 ˆφ0 ω,x,z,x 0,z 0 exp where γ x = 1 0 γ0 γxsds. γ x x ω z z 0 4c 0,

23 Grenoble December, 014 Ghost imaging in the random paraxial regime The medium in paths 1 and is heterogeneous typically, turbulent atmosphere. They are two independent realizations of the same distribution.

24 Grenoble December, 014 If the correlation function of the noise sources is nt,xnt,x = Ft t I 0 exp x r 0 δx x, if we denote the integrated correlation function of the medium fluctuations by then with Hx = I 0r π 3 L 4 γx = C 1 x = and γ x = 1 0 γ0 γxsds. R dβexp E[µ0, 0µx, z]dz R Hx y T y dy, β γ r 0 βω 0L c 0 +i ω 0r 0 x β c 0 L.

25 Grenoble December, 014 If the medium is strongly scattering, in the sense that the propagation distance is larger than the scattering mean free path L/l sca 1, with then with l sca = 8c 0 γ0ω0, Hx = I 0r 4 0ρ gi0 7 π L 4 ρ gi exp x 4ρ gi ρ gi = c 0L ω0 + 4c 0L 3 r 0 3ω0 l = ρ scalcor gi0 + 4c 0L 3 3ω0 l, scalcor and the correlation radius of the medium l cor is defined by: γx = γ0[1 x /l cor +o x /l cor]. Scattering only slightly reduces the resolution! This imaging method is robust with respect to medium noise.,

26 Grenoble December, 014 Ghost imaging in the random paraxial regime The medium in paths 1 and is heterogeneous typically, turbulent atmosphere. Imagine that they are the same realization just for the beauty of the analysis.

27 If the correlation function of the noise sources is nt,xnt,x = Ft t I 0 exp x δx x, r0 if we denote the integrated correlation function of the medium fluctuations by if L/l sca 1, with γx = E[µ0, 0µx, z]dz, l sca = 8c 0, γ0ω0 if γx = γ0[1 x /lcor +o x /lcor], then C 1 x = Hx y T y dy, R with Hx = I 0r0 4 7 π L exp x, 4 4ρ gi3 with the radius 1 ρ gi3 = 1 ρ gi0 + 16L, l sca lcor the radius of the convolution kernel is reduced by scattering and can even be smaller than the Rayleigh resolution formula: enhanced resolution compared to the homogeneous case similar phenomenon observed in time-reversal experiments! Grenoble December, 014

28 Grenoble December, 014 Virtual ghost imaging - The medium in path is randomly heterogeneous. - There is no other measurement than the integrated transmitted intensity I t. - The realization of the source is known use of a Spatial Light Modulator and the medium is taken to be homogeneous in the virtual path 1 one can compute the field and therefore its intensity I 1 t,x in the virtual output plane of path 1.

29 Grenoble December, 014 If the correlation function of the noise sources is nt,xnt,x = Ft t I 0 exp x δx x, r0 if we denote the integrated correlation function of the medium fluctuations by if L/l sca 1, with γx = E[µ0, 0µx, z]dz, l sca = 8c 0, γ0ω0 if γx = γ0[1 x /lcor +o x /lcor], then C 1 x = Hx y T y dy, R with Hx = I 0r0ρ 4 gi0 7 π L 4 ρ exp x gi4 4ρ, gi4 with the radius ρ gi4 = c 0L ω 0 r 0 + c 0L 3 3ω 0 l scal cor = ρ gi0 + c 0L 3 3ω0 l. scalcor a one-pixel camera can give a high-resolution image of the object in scattering media!

30 Grenoble December, 014 Conclusion Imaging with noise sources and/or through scattering media is possible using correlation-based techniques. First application: seismic interferometry, virtual source imaging. Second application: ghost imaging. Many other applications!