Abstract

We propose a maximum a posteriori-based method that solves an important practical problem in the deconvolution of 4Pi images by simultaneously delivering an estimate of both the object and the unknown phase. The method was tested in simulations and on data from both test samples and biological specimen. It generates object estimates that are free from interference artifacts and reliably recovers arbitrary relative phases. Based on vectorial focusing theory, our theoretical analysis allowed for a simple and efficient implementation of the algorithm. Taking several 4Pi images at different relative phases of the interfering beams is shown to improve the robustness of the approach.

© 2009 Optical Society of America

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2009

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

2007

2006

2002

S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002).
[CrossRef]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002).
[CrossRef]

2001

1999

1992

1991

I. Csiszár, Ann. Stat. 19, 2032 (1991).
[CrossRef]

1987

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Baddeley, D.

Bertero, M.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

Bewersdorf, J.

S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002).
[CrossRef]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002).
[CrossRef]

Bezdek, J. C.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002).
[CrossRef]

S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002).
[CrossRef]

Boccacci, P.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

Carl, C.

Conchello, J.-A.

Cremer, C.

Csiszár, I.

I. Csiszár, Ann. Stat. 19, 2032 (1991).
[CrossRef]

Diaspro, A.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

Engelhardt, J.

Hathaway, R. J.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Hell, S.

Hell, S. W.

Howard, R. E.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Kelley, C.

C. Kelley, Iterative Method for Optimization (SIAM, 1999), Vol. 18.

Lang, M.

Markham, J.

Müller, T.

Nagorni, M.

Stelzer, E. H. K.

Vicidomini, G.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

Wilson, C. A.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Windham, M. P.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Ann. Stat.

I. Csiszár, Ann. Stat. 19, 2032 (1991).
[CrossRef]

Appl. Opt.

J. Microsc.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Optim. Theory Appl.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987).
[CrossRef]

Opt. Commun.

C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Other

C. Kelley, Iterative Method for Optimization (SIAM, 1999), Vol. 18.

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Figures (3)

Fig. 1
Fig. 1

(a) Test phantom and (b) example of a simulated image with ϕ = 7 π 8 , NA 1.46 (oil), λ exc = 635 nm , λ em = 680 nm , pinhole 0.5 AU . (c) Deconvolution with known phase and (d) PBD (phase estimated, ϕ E = 2.66 ). (e) D KL ( f , f E ) and (f) ϕ E as functions of ϕ for SI and PD mode. Both graphs are obtained as the average of five different simulated images. For all simulations we fixed the total number of photons k , n g k ( n ) , resulting in a maximum of expected photons of 50 ( SNR 7 : 1 ) and 25 ( SNR 5 : 1 ) for SI and PD images, respectively.

Fig. 2
Fig. 2

Bead images ( λ exc = 635 nm , λ em = 680 nm , SNR 6 : 1 ) and object estimate in the case of constructive [(a), (b)] and destructive [(c), (d)] phase. (e) Robust estimation of arbitrary phases ϕ using the SI or PD ( N ϕ = 2 , δ 2 = π ) .

Fig. 3
Fig. 3

(a) Axial ( x z ) slice of a 3D stack of a Golgi apparatus ( λ exc = 568 nm , λ em = 605 nm , SNR 7 : 1 ). (b) Deconvolved data without phase estimation (ϕ is set to zero) and (c) with simultaneous phase estimation (estimated phase ϕ E = 5.88 ). Intensity profiles at site indicated by arrows (d). The residual sidelobes in the deconvolved data without phase estimation are removed by PBD. Scale bar, 0.5 μ m .

Equations (14)

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P ( f , ϕ | g ) = P ( g | f , ϕ ) P ( f ) P ( ϕ ) P ( g ) .
P ( g | f , ϕ ) = k , n poi [ g k ( n ) | ( H k f ) ( n ) ] ,
J ( f , ϕ | g ) = k , n { ( H k f ) ( n ) g k ( n ) ln [ ( H k f ) ( n ) ] } + μ f 2 2 ,
f l + 1 = arg min f 0 J ( f , ϕ l | g ) ,
ϕ l + 1 = arg min ϕ J ( f l + 1 , ϕ | g ) .
f l , i + 1 = f l , i 1 + μ f l , i k ( H k T g k H k f l , i ) , i = 1 , , N SGM ,
ϕ l , i + 1 = ϕ l , i τ l , i ρ ( ϕ l , i ) , i = 1 , , N SDM ,
ρ ( ϕ l , i ) = ϕ [ J ( f l + 1 , ϕ | g ) ] ( ϕ l , i ) = k , n { ( H k f l + 1 ) ( n ) ( 1 g k ( n ) ( H k f l + 1 ) ( n ) ) } ,
h eff ( r , φ k ) = | E 1 ( r ) + exp ( i φ k ) E 2 ( r ) | 2 h CEF ( r ) ,
H k f = A 0 f + cos φ k A 1 f sin φ k A 2 f ,
H k f = sin φ k A 1 f cos φ k A 2 f ,
A 0 f = [ ( | E 1 | 2 + | E 2 | 2 ) h CEF ] f ,
A 1 f = [ 2 R ( E 1 * E 2 ) h CEF ] f ,
A 2 f = [ 2 I ( E 1 * E 2 ) h CEF ] f

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