Abstract

Image reconstruction of partially coherent light is interpreted as quantum-state reconstruction. An efficient method based on the maximum-likelihood estimation is proposed for acquiring information from blurred intensity measurements affected by noise. Connections with incoherent-image restoration are pointed out. The feasibility of the method is demonstrated numerically. Spatial and correlation details significantly below the diffraction limit are revealed in the reconstructed pattern.

© 2004 Optical Society of America

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2003 (4)

M. J. Bastiaans, K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
[CrossRef]

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

M. Ježek, J. Fiurášek, Z. Hradil, “Quantum inference of states and processes,” Phys. Rev. A 68, 012305–1-7 (2003).
[CrossRef]

S. A. Babichev, B. Brezger, A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903–1-4 (2003).
[CrossRef]

2002 (7)

R. W. Gerchberg, “A new approach to phase retrieval of a wave front,” J. Mod. Opt. 49, 1185–1196 (2002).
[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[CrossRef]

M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef]

E. H. K. Stelzer, “Beyond the diffraction limit?” Nature (London) 417, 806–807 (2002).
[CrossRef]

M. A. Man’ko, “Electromagnetic signal processing and noncommutative tomography,” J. Russ. Laser Res. 23, 433–448 (2002).
[CrossRef]

J. Řeháček, Z. Hradil, M. Zawisky, W. Treimer, M. Strobl, “Maximum likelihood absorption tomography,” Europhys. Lett. 59, 694–700 (2002).
[CrossRef]

J. Řeháček, Z. Hradil, “Invariant information and quantum state estimation,” Phys. Rev. Lett. 88, 130401–1-4 (2002).
[CrossRef] [PubMed]

2001 (5)

J. Řeháček, Z. Hradil, M. Ježek, “Iterative algorithm for reconstruction of entangled states,” Phys. Rev. A 63, 040303–1-4 (2001).
[CrossRef]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1-4 (2001).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Göckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

W. Kim, “Two-dimensional phase retrieval using a window function,” Opt. Lett. 26, 134–136 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

2000 (3)

Z. Hradil, J. Summhammer, “Quantum theory of incompatible observations,” J. Phys. A Math. Gen. 33, 7607–7612 (2000).
[CrossRef]

K. Banaszek, G. M. D’Ariano, M. G. A. Paris, M. F. Sacchi, “Maximum-likelihood estimation of the density matrix,” Phys. Rev. A 61, 010304–1-4 (2000).
[CrossRef]

Z. Hradil, J. Summhammer, G. Badurek, H. Rauch, “Reconstruction of the spin state,” Phys. Rev. A 62, 014101 (2000).
[CrossRef]

1999 (1)

Z. Hradil, J. Summhammer, H. Rauch, “Quantum tomography as normalization of incompatible observation,” Phys. Lett. A 261, 20–24 (1999).
[CrossRef]

1998 (2)

B. R. Frieden, D. J. Graser, “Closed-form maximum entropy image restoration,” Opt. Commun. 146, 79–84 (1998).
[CrossRef]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

1997 (4)

Z. Hradil, “Quantum-state estimation,” Phys. Rev. A 55, R1561–R1564 (1997).
[CrossRef]

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[CrossRef]

T. Opatrný, D. G. Welsch, W. Vogel, “Least-squares inversion for density-matrix reconstruction,” Phys. Rev. A 56, 1788–1799 (1997).
[CrossRef]

1996 (4)

G. M. D’Ariano, C. Macchiavello, M. G. A. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996).
[CrossRef]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

V. Bužek, G. Adam, G. Drobný, “Quantum state reconstruction and detection of quantum coherences on different observation levels,” Phys. Rev. A 54, 804–820 (1996).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays, Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (2)

D. F. V. James, G. S. Agarwal, “Generalized Radon transform for tomographic measurement of short pulses,” J. Opt. Soc. Am. B 12, 704–708 (1995).
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Wigner function and probability distribution for shifted and squeezed quadratures,” Quantum Semiclassic. Opt. 7, 615–623 (1995).
[CrossRef]

1994 (2)

G. M. D’Ariano, C. Macchiavello, M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (2)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Y. Vardi, D. Lee, “From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).

1989 (1)

K. Vogel, H. Risken, “Determination of quasiprobability distribution in terms of probability distribution for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

1988 (1)

B. R. Frieden, “Applications to optics and wave mechanics of the criterion of maximum Cramer–Rao bound,” J. Mod. Opt. 35, 1297–1316 (1988).
[CrossRef]

1987 (2)

D. L. Snyder, M. I. Miller, J. L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

J. Bertrand, P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

1983 (1)

1982 (4)

M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phases,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

1978 (1)

S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

1977 (3)

A. Rockmore, A. Macovski, “A maximum likelihood approach to transmission image reconstruction from projections,” IEEE Trans. Nucl. Sci. 24, 1929–1935 (1977).
[CrossRef]

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximumlikelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

J. Peřina, V. Peřinová, Z. Braunerová, “Super-resolution in linear systems with noise,” Opt. Appl. VII, 79–83 (1977).

1976 (2)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

A. Rockmore, A. Macovski, “A maximum likelihood approach to emission image reconstruction from projections,” IEEE Trans. Nucl. Sci. 23, 1428–1432 (1976).
[CrossRef]

1974 (2)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,” J. Phys. D Appl. Phys. 7, L65–L68 (1974).
[CrossRef]

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1973 (1)

A. S. Holevo, “Statistical decision theory for quantum systems,” J. Multivar. Anal. 3, 337–394 (1973).
[CrossRef]

1972 (3)

1970 (2)

G. E. Backus, F. Gilbert, “Uniqueness in the inversion of inaccurate gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

1969 (4)

B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
[CrossRef] [PubMed]

J. Peřina, V. Peřinová, “Optical imaging with partially coherent non-thermal light. II. Reconstruction of object from its image and similarity between object and its image,” Opt. Acta 16, 309–320 (1969).
[CrossRef]

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).

1968 (1)

G. E. Backus, F. Gilbert, “The resolving power of growth earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
[CrossRef]

1967 (2)

1963 (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
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K. Vogel, H. Risken, “Determination of quasiprobability distribution in terms of probability distribution for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

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A. Rockmore, A. Macovski, “A maximum likelihood approach to transmission image reconstruction from projections,” IEEE Trans. Nucl. Sci. 24, 1929–1935 (1977).
[CrossRef]

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P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Chap. 2, pp. 29–186.

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,” J. Phys. D Appl. Phys. 7, L65–L68 (1974).
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P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1958).

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J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (5)

Fig. 1
Fig. 1

Optical intensity I(q) of the true object in the input plane.

Fig. 2
Fig. 2

Simulated data fi in 64 pixels (points) affected by 20% background noise sample the optical intensity I(x) in the output plane (curve) for imaging axial arrangement, din=0.75 m, s=0.

Fig. 3
Fig. 3

Exponentially fast convergence of the square difference between two successive iterations during the extremization process.

Fig. 4
Fig. 4

Optical intensity I(q) of the reconstructed object in the input plane (points) compared with the true object (thin lines).

Fig. 5
Fig. 5

Contour lines (thin lines) of the reconstructed mutual intensity Γ(q, q). The positions of the diagonal-intensity spots as well as the positions of the off-diagonal correlations match the true ones (thick lines).

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

ρˆ=kλk|φkφk|.
ρ^+=ρˆ,Tr[ρˆ]=1,ψ|ρˆ|ψ0,|ψ,
Γ(x, x)=Tr[ρˆ|xx|]=kλkx|φkφk|x=kλkφk(x)φk*(x)=ψ*(x)ψ(x)ens,
Γ*(x, x)=Γ(x, x),dxI(x)=1,
Γ(x, x)0.
W(x, p)=1πdxexp(-i2px)Γ(x+x, x-x).
|ψout=Tˆ|ψin.
z Tˆ=LˆTˆ,
ρ^out=Tˆρ^inT^+,Tˆ=exp-i Hˆz.
x^outp^out=T^+x^inp^inTˆ.
x^outp^out=Tx^inp^in=ABCDx^in+sp^in+r,
W(x, p)=W(Dx-Bp-s, -Cx+Ap-r).
x0p0=1z2-z1x1z2-x2z1x2-x1.
Tˆ=-aadξ|ξξ|
ψout(x)=x|ψout=dx0x|Tˆ|x0x0|ψin=dx0h(x, x0)ψin(x0).
h(x, x0)=x|Tˆ|x0,
Γout(x, x)=dqdqh(x, q)h*(x, q)Γin(q, q).
Aˆ|a=a|a,a|aa|=1ˆ,a|a=δaa.
Π^b0,bΠ^b=1ˆ,
I(x)=Γout(x, x)=p(x)=Tr[ρ^out|xx|].
O^i=Δidx|xx|,
pi=Tr[ρ^outO^i]=Tr[ρˆΠ^i],Π^i=T^+O^iTˆ.
ψout=dx0h(x-x0)ψin(x0).
ψ˜in=ψ˜out+N˜h˜.
Tr[ρˆΠ^i]=fi,
i|fi-Tr[ρˆΠ^i]|2,
L(ρˆ)ipiNfi,
lnL(ρˆ)=ifiln pi=ifilnTr[ρˆΠ^i].
ρ^est=arg maxˆρlnL(ρˆ).
δ lnL(ρˆ)δρˆρ^est=0,
δ lnL(ρˆ)δφk|=ifipiΠ^i|φk
Rˆρˆ=ρˆ.
Rˆ=ifipiΠ^i,
dxR(q, x)Γ(x, q)=Γ(q, q),
R(q, x)=ifipiPi(q, x)
Pi(q, x)=Δidxh*(x, q)h(x, x)
pi=dqdqΓ(q, q)Pi(q, q).
Γ(1)(q, q)=ifiΔidxh*(x, q)h(x, q)dξΔidx|h(x, ξ)|2.
Γ(n+1)(q, q)=dxR(n)(q, x)Γ(n)(x, q).
Γ(x, x)=I(x)δ(x-x),
dqR(q, q)I(q)=I(q),
pi=dqPi(q, q)I(q).
ifidqPi(q, q)I(q)dqPi(x, q)I(q)=I(x).
h(x, x0)=x|exp-idout2k p^2-aadξ|ξξ|×exp-ik2f x^2exp-idin2k p^2×exp(-ispˆ)|x0,
h(x, x0)=h(x, x0)E(x, x0).
h(x, x0)=expik2x2dout+(x0+s)2din-θ2Δ
E(x, x0)=12erf1-i2kΔθΔ+a-12erf1-i2kΔθΔ-a
Δ=1/din+1/dout-1/f,
θ=θin+θout=x0+sdin+xdout,
erf(z)=2π0zdt exp(-t2).
R=Cλdout/a

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