Abstract

Efforts of phase and amplitude retrieval from diffraction patterns have almost exclusively been applied for nonperiodic objects. We investigated the quality of retrieval of nonperiodic objects embedded in a sinusoidal background, using the approach of iterative hybrid input–output with oversampling. Two strategies were employed; one by filtering in the frequency domain prior to phase retrieval, and the other by filtering the phase or amplitude image after retrieval. Results obtained indicate better outcomes with the latter approach provided detector noise is not excessive.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2009 (1)

2008 (5)

2006 (3)

2005 (1)

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

2004 (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

2003 (2)

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61-80 (2003).
[CrossRef]

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

1999 (1)

J. Miao, P. Charalambous, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

1998 (1)

1982 (1)

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258-261(2006).
[CrossRef]

Chakarov, D.

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258-261(2006).
[CrossRef]

Dual, J.

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

Eurenius, L.

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Feinup, J. R.

Fienup, J. R.

Haake, A.

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

Hägglund, C.

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Han, S.

G. Ying, Q. Wei, X. Shen, and S. Han, “A two-step phase retrieval method in Fourier-transform ghost imaging,” Opt. Commun. 281, 5130-5132 (2008).
[CrossRef]

Hennelly, B.

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61-80 (2003).
[CrossRef]

Hodgson, K. O.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

Hu, X.

Ishikawa, T.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

Joiner, D.

Kasemo, B.

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Larabell, C. A.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

LeGros, M. A.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

Li, S.

Liu, H.

Miao, J.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transform of non-periodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

Nakajima, N.

Neild, A.

T. W. Ng, D. Joiner, A. Neild, and J. Whitehill, “Computer-aided analysis of optical diffraction by low-frequency liquid surface acoustic waves,” Appl. Opt. 48, C159-C164 (2009).
[CrossRef] [PubMed]

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

Ng, T. W.

Nishino, Y.

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

Nugent, K. A.

Olsson, E.

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258-261(2006).
[CrossRef]

Quiney, H. M.

Radziwill, G.

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

Rodenburg, J. M.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Sayre, D.

J. Miao, P. Charalambous, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transform of non-periodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

Shen, X.

G. Ying, Q. Wei, X. Shen, and S. Han, “A two-step phase retrieval method in Fourier-transform ghost imaging,” Opt. Commun. 281, 5130-5132 (2008).
[CrossRef]

Sheridan, J. T.

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61-80 (2003).
[CrossRef]

Wei, Q.

G. Ying, Q. Wei, X. Shen, and S. Han, “A two-step phase retrieval method in Fourier-transform ghost imaging,” Opt. Commun. 281, 5130-5132 (2008).
[CrossRef]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258-261(2006).
[CrossRef]

Whitehill, J.

Williams, G. J.

Wu, X.

Wu, Y.

Yan, A.

Ying, G.

G. Ying, Q. Wei, X. Shen, and S. Han, “A two-step phase retrieval method in Fourier-transform ghost imaging,” Opt. Commun. 281, 5130-5132 (2008).
[CrossRef]

Appl. Opt. (4)

Biotechnol. Bioeng. (1)

A. Haake, A. Neild, G. Radziwill, and J. Dual, “Positioning, displacement and localization of cells using ultrasonic forces,” Biotechnol. Bioeng. 92, 8-14 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Nat. Photonics (1)

L. Eurenius, C. Hägglund, E. Olsson, B. Kasemo, and D. Chakarov, “Grating formation by metal-nanoparticle-mediated coupling of light into waveguided modes,” Nat. Photonics 2, 360-364 (2008).
[CrossRef]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258-261(2006).
[CrossRef]

Nature (1)

J. Miao, P. Charalambous, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Opt. Commun. (2)

G. Ying, Q. Wei, X. Shen, and S. Han, “A two-step phase retrieval method in Fourier-transform ghost imaging,” Opt. Commun. 281, 5130-5132 (2008).
[CrossRef]

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61-80 (2003).
[CrossRef]

Opt. Express (3)

Phys. Rev. Lett. (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110-112 (2003).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Schematic description of diffraction pattern recording of a sample in which (inset) the object is embedded in sinusoidal substrate or medium.

Fig. 2
Fig. 2

Images (a) autumn, (b) house, and (c) crystal used in terms of phase or amplitude to evaluate the retrieval approach.

Fig. 3
Fig. 3

(a) Original phase image, (b) original phase image overlaid by a high-frequency carrier ( 1 / π ) , (c) original phase image overlaid by a low-frequency carrier ( 1 / 2 π ) . The corresponding Fourier amplitudes are depicted in (d), (e), and (f), respectively.

Fig. 4
Fig. 4

(a) Original amplitude image, (b) original amplitude image overlaid by a high-frequency carrier ( 1 / π ) , (c) original amplitude image overlaid by a low-frequency carrier ( 1 / 2 π ) . The corresponding Fourier amplitudes are depicted in (d), (e), and (f), respectively.

Fig. 5
Fig. 5

(a) Retrieved phase of spectrum image with no overlay [Fig. 3d]. The phase of spectrum images with a high-frequency ( 1 / π ) overlay [Fig. 3b] retrieved by using methods (A) and (B) are presented in (b) and (d), respectively. The phase of spectrum image with a low-frequency ( 1 / 2 π ) overlay [Fig. 3c] retrieved by using methods (A) and (B) are presented in (c) and (e), respectively. Regardless of the sinusoidal frequency, the approach of filtering in the frequency domain prior to phase retrieval produces better outcomes.

Fig. 6
Fig. 6

Reconstruction error versus iteration number from the various phase retrieval conditions. The outcome of using method (B) with the lower-frequency ( 1 / 2 π ) overlay and without an overlay converged to the smallest error values fastest in a verbatim manner. Method (B) with the higher-frequency ( 1 / π ) overlay converged to the lower values after more iterations. Method (A) converged to high error values, with the higher-frequency ( 1 / π ) overlay condition performing marginally better.

Fig. 7
Fig. 7

Reconstruction error versus iteration number from the various phase retrieval conditions with lower-frequency overlay ( 1 / 2 π ) with methods (A) and (B), using no noise and noise levels of ξ = 4.98 , 1.66 , 0.99 . The increase in error occurs with noise level for both methods and becomes method independent when ξ is equal to or below 1.66.

Fig. 8
Fig. 8

Retrieved phase of spectrum image with low-frequency ( 1 / 2 π ) overlay, using method (A) for (a)  ξ = 4.98 , (b)  ξ = 1.66 , and (c)  ξ = 0.99 , as well as with method (B) for (d)  ξ = 4.98 , (e)  ξ = 1.66 , and (f)  ξ = 0.99 .

Equations (6)

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f ( x , y ) = g ( x , y ) e i θ ( x , y )
f ( x , y ) = g ( x , y ) e i [ θ ( x , y ) + p 1 sin ( p 2 x + ϕ ) ] ,
f ( x , y ) = [ g ( x , y ) + p 1 sin ( p 2 x + ϕ ) ] e i θ ( x , y ) ,
Noise = F ( u , v ) ξ × , random
| F ( u , v ) | = { M ( u , v ) S | F ( u , v ) | ( u , v ) S ,
H ( u , v ) = { 0 u = u o 1 otherwise ,

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