Abstract

We propose and demonstrate a computational method for complex-field imaging from many noisy intensity images with varying defocus, using an extended complex Kalman filter. The technique offers dynamic smoothing of noisy measurements and is recursive rather than iterative, so is suitable for adaptive measurements. The Kalman filter provides near-optimal results in very low-light situations and may be adapted to propagation through turbulent, scattering, or nonlinear media.

© 2011 Optical Society of America

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2010

2008

S. Pavani, and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16(5), 3484–3489 (2008).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008).
[CrossRef] [PubMed]

2007

2005

2004

H. Campbell, S. Zhang, A. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. 29(23), 2707–2709 (2004).
[CrossRef] [PubMed]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21(5), 828–831 (2004).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

2003

2002

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

2001

L. Allen, and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

R. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. 26(10), 684–685 (2001).
[CrossRef]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

1999

1998

1997

1996

1994

1993

1992

1988

1987

1986

1983

M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[CrossRef]

1982

R. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

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

1980

J. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 291–305 (1980).

1978

J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

A. Devaney, and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A 68, 1352–1354 (1978).
[CrossRef]

1977

1976

R. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976).
[CrossRef]

1973

D. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972

R. Gerchberg, and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

1960

R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82(1), 35–45 (1960).
[CrossRef]

1951

W. Bragg, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190–191 (1951).
[CrossRef] [PubMed]

Acosta, E.

Allen, L.

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

L. Allen, and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

Bilmont, M.

Boistel, R.

Bowers, C.

Bragg, W.

W. Bragg, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190–191 (1951).
[CrossRef] [PubMed]

Campbell, H.

Chapman, H.

Childlaw, R.

A. Devaney, and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A 68, 1352–1354 (1978).
[CrossRef]

Cloetens, P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008).
[CrossRef] [PubMed]

J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[CrossRef] [PubMed]

Crosby, E.

Davis, T.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

Dean, B.

Devaney, A.

A. Devaney, and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A 68, 1352–1354 (1978).
[CrossRef]

Dong, B.

Dorsch, R.

Ersoy, O.

Faulkner, H.

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

Fienup, J.

Gao, D.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

George, N.

Gerchberg, R.

R. Gerchberg, and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Gerke, T. D.

T. D. Gerke, and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4(3), 188–193 (2010).
[CrossRef]

Gonsalves, R.

R. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. 26(10), 684–685 (2001).
[CrossRef]

R. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A 4, 166–170 (1987).
[CrossRef]

R. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

R. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976).
[CrossRef]

Greenaway, A.

Gu, B.

Guigay, J.

Guigay, J. P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008).
[CrossRef] [PubMed]

Gureyev, T.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

Horstmeyer, R.

Kalman, R.

R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82(1), 35–45 (1960).
[CrossRef]

Kou, S. S.

Langer, M.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008).
[CrossRef] [PubMed]

J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[CrossRef] [PubMed]

Lee, D.

Luo, Y.

Mayo, S.

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

Mendlovic, D.

Miao, J.

Miller, P.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

Misell, D.

D. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Nugent, K.

D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

Oh, S.

Osten, W.

Oxley, M.

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

Oxley, M. P.

L. Allen, and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

Paganin, D.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef]

Parry, D.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

Pavani, S.

Paxman, R.

Pedrini, G.

Peyrin, F.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008).
[CrossRef] [PubMed]

Piestun, R.

Pogany, A.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

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Supplementary Material (1)

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Figures (7)

Fig. 1
Fig. 1

Experimental setup using laser illumination and 4f imaging system, with camera on a motion stage for obtaining multiple images in sequence. The stack of intensity images demonstrates the bilinearity of the problem - amplitude contrast is symmetric about the focal plane, while phase contrast is anti-symmetric.

Fig. 2
Fig. 2

Kalman filter schematic diagram (wn is “process noise,” assumed to be negligible).

Fig. 3
Fig. 3

Simulated phase and amplitude retrieval. (a) Noisy intensity images, (b) actual amplitude at focus, (c) actual phase, (d) recovered amplitude, (e) recovered phase (radians).

Fig. 4
Fig. 4

(a) Progress of Kalman estimator: actual intensity as field propagates, evolution of intensity estimate, actual phase (radians) as field propagates, and evolution of phase estimate (radians). (b) Actual and noise-corrupted measurements of axial intensity for a single pixel, (c) Average RMS error convergence as more images are added.

Fig. 5
Fig. 5

Noisy dataset phase retrieval comparison of techniques. (a) Actual phase at object plane, (b) traditional TIE (error=0.0150), (c) higher order TIE (error=0.0013), (d) standard iterative method after 100 iterations (error=0.0061), (e) modified iterative technique (error=0.00091), and (f) Kalman estimator (error=0.00017). All scale bars in radians.

Fig. 6
Fig. 6

Simulated results of Kalman estimation with Fourier compression. (a) Some of the measured noisy images, (b) actual amplitude at focus, (c) actual phase at focus, (d) recovered amplitude, (e) recovered phase, (f) amplitude error map (average error is 0.0099), (g) phase error map (average error is 0.0259 radians).

Fig. 7
Fig. 7

( Media 1) Experimental results of Kalman estimator. (a) Some captured images, (b) recovered amplitude, (c) recovered phase, displayed as inverse height. (d) Surface profile measurement by AFM (inverse height) and (e) its cross-section along the white line.

Equations (16)

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A ( x , y , z ) z = i λ 4 π 2 A ( x , y , z ) ,
P [ I ( x , y , z n ) | A ( x , y , z n ) ] = e γ | A ( x , y , z n ) | 2 γ | A ( x , y , z n ) | 2 I ( x , y , z n ) I ( x , y , z n ) ! ,
a ( z ) ( A ( x 1 , y 1 , z ) A ( x M , y 1 , z ) A ( x 1 , y 2 , z ) A ( x M , y M , z ) ) , η n ( I ( x 1 , y 1 , z n ) I ( x M , y 1 , z n ) I ( x 1 , y 2 , z n ) I ( x M , y M , z n ) ) ,
η n γ | a ( z n ) | 2 + υ n ,
a ^ ( z 0 ) a ( z 0 ) ,
Q ( z 0 ) [ a ( z 0 ) a ^ ( z 0 ) ] [ a * ( z 0 ) a ^ * ( z 0 ) ] T ,
P ( z 0 ) [ a ( z 0 ) a ^ ( z 0 ) ] [ a ( z 0 ) a ^ ( z 0 ) ] T .
d a ^ d z = L a ^ ,
d Q d z = L Q + Q L * T ,
d P d z = L P + P L T .
a ^ ( z n + ) = a ^ ( z n ) + K n [ η n γ | a ^ ( z n ) | 2 ] ,
Q ( z n + ) = [ I γ K n A n * ] Q ( z n ) γ K n A n * P * ( z n ) ,
P ( z n + ) = [ I γ K n A n * ] P ( z n ) γ K n A n * Q * ( z n ) ,
A n diag a ^ ( z n ) ,
K n = γ [ Q ( z n ) A n + P ( z n ) A n * ] D n 1 ,
D n γ 2 [ A n * Q ( z n ) A n + A n * P ( z n ) A n * + c . c . ] + R n ,

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