Abstract

Speckle intensity measurements utilized for phase retrieval (PR) are sequentially taken with a digital camera, which introduces quantization error that diminishes the signal quality. Influences of quantization on the speckle intensity distribution and PR are investigated numerically and experimentally in the static wavefront sensing setup. Results show that 3 to 4 bits are adequate to represent the speckle intensities and yield acceptable reconstructions at relatively fast convergence rates. Computer memory requirements may be eased down by 2.4 times if a 4 bit instead of an 8 bit camera is used. This may facilitate rapid speckle data acquisition for dynamic wavefront sensing.

© 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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2010 (3)

2009 (5)

2008 (4)

2007 (1)

2006 (4)

2005 (2)

2004 (1)

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747–5751 (2004).
[CrossRef]

2003 (1)

2002 (1)

1997 (2)

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

B. Zhao, “Effect of intensity-correlated error due to quantization and noise on phase-shifting method,” Opt. Lasers Eng. 28, 199–211 (1997).
[CrossRef]

1972 (1)

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

Agour, M.

Almoro, P.

Almoro, P. F.

Anand, A.

Araki, T.

Bao, P.

Bergmann, R. B.

Burton, D.

Falldorf, C.

Fienup, J. R.

Frauel, Y.

Fujio, M.

Gerchberg, R. W.

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

Guizar-Sicairos, M.

Gundu, P. N.

Hanson, S.

Hanson, S. G.

Jacquot, P.

P. Jacquot, “Speckle interferometry: a review of the principal methods in use for experimental mechanics applications,” Strain 44, 57–69 (2008).
[CrossRef]

Javidi, B.

Kopylow, C. v.

Lalor, M.

Lebenc, S.

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

Likara, B.

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

Lilley, F.

Maallo, A.

Mills, G.

Minami, M.

Miura, J.

Možina, M.

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

Naughton, T.

Osten, W.

Pedrini, G.

Pernuš, F.

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

Saxton, W. O.

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

Schirmer, M.

Shortt, A.

Skydan, O.

Surrel, Y.

Tajahuerce, E.

Takajo, H.

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747–5751 (2004).
[CrossRef]

Takeda, M.

Tomaževica, D.

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

Wang, W.

Yakota, M.

Yamaguchi, I.

Yamamoto, K.

Yang, S.

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747–5751 (2004).
[CrossRef]

Yasui, T.

Zhang, F.

Zhang, Y.

Zhao, B.

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

B. Zhao, “Effect of intensity-correlated error due to quantization and noise on phase-shifting method,” Opt. Lasers Eng. 28, 199–211 (1997).
[CrossRef]

Appl. Opt. (10)

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for three-dimensional object reconstruction and recognition,” Appl. Opt. 41, 4124–4132 (2002).
[CrossRef] [PubMed]

O. Skydan, F. Lilley, M. Lalor, and D. Burton, “Quantization error of CCD cameras and their influence on phase calculations in fringe pattern analysis,” Appl. Opt. 42, 5302–5307(2003).
[CrossRef] [PubMed]

G. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
[CrossRef] [PubMed]

I. Yamaguchi, K. Yamamoto, G. Mills, and M. Yakota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45, 975–983 (2006).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[CrossRef] [PubMed]

P. F. Almoro and S. G. Hanson, “Random phase plate for wavefront sensing via phase retrieval and a volume speckle field,” Appl. Opt. 47, 2979–2987 (2008).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Angular displacement and deformation analyses using a speckle-based wavefront sensor,” Appl. Opt. 48, 932–940 (2009).
[CrossRef] [PubMed]

P. Almoro, A. Maallo, and S. Hanson, “Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing,” Appl. Opt. 48, 1485–1493 (2009).
[CrossRef] [PubMed]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef] [PubMed]

Biomed. Opt. Express (1)

Euro. J. Pharm. Sci. (1)

M. Možina, D. Tomaževiča, S. Lebenc, F. Pernuš, and B. Likara, “Digital imaging as a process analytical technology tool for fluid-bed pellet coating process,” Euro. J. Pharm. Sci. 41, 156–162 (2010).
[CrossRef]

J. Euro. Opt. Soc. Rapid Publ. (1)

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Euro. Opt. Soc. Rapid Publ. 4, 09002 (2009).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747–5751 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

B. Zhao, “Effect of intensity-correlated error due to quantization and noise on phase-shifting method,” Opt. Lasers Eng. 28, 199–211 (1997).
[CrossRef]

Opt. Lett. (4)

Optik (Stuttg.) (1)

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

Strain (1)

P. Jacquot, “Speckle interferometry: a review of the principal methods in use for experimental mechanics applications,” Strain 44, 57–69 (2008).
[CrossRef]

Other (2)

P. F. Almoro, and S. G. Hanson are preparing a manuscript to be called “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser.”

P. F. Almoro and S. G. Hanson are preparing a manuscript to be called “Single-plane multiple speckle pattern phase retrieval using a deformable mirror.”

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Figures (6)

Fig. 1
Fig. 1

Quantization process. A continuous signal ( I ( x , y ) ) is cast into discrete values ( I Q ( x , y ) ) when passed through a digitizing device (Q). This is then saved as an image ( I s ( i , j ) ) in computer I with finite gray levels based on the bit depth of Q.

Fig. 2
Fig. 2

Numerical simulations of speckle patterns and reconstructions for different bit depths. (a) is the input smooth test wavefront. (b)–(f) are sampled speckle images at different bit depths. (g) is the pixel density histogram for bit depths N = 2 , 4, and 8. (h)–(l) are phase reconstructions for different bit depths, and (m) is the normalized correlation plot, with the reconstruction from N = 8 as the standard.

Fig. 3
Fig. 3

(a)–(c) Axial and (d)–(f) transverse intensity scans of a VSF and the corresponding correlations for different bit depths. Axial and transverse line scans obtained at bit depth N = 8 are used as standards.

Fig. 4
Fig. 4

Memory requirements for 20 speckle patterns at different bit depths.

Fig. 5
Fig. 5

Experimental results. (a)–(d) and (b)–(h) are the speckle intensity line scans, in the transverse and axial directions, respectively, for bit depths N = 1 , 2, 3, and 8 (left to right). (i)–(l) are the corresponding reconstructed phase maps.

Fig. 6
Fig. 6

Influences of bit depth and number of iterations on the rate of convergence. (a) and (b) show portions of the phase difference maps for increasing numbers of iterations, i, (left to right) for bit depths N = 3 and 4, respectively. Phase map obtained for N = 8 and i = 10 [Fig. 5l] is used as standard. (c) Graphs depicting the rates of decrease, for N = 3 and 4, in the RMSD of the phase difference maps.

Equations (7)

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

I Q ( x , y ) = int [ I ( x , y ) B I max ] × I max B .
B = 2 N 1.
I S ( i , j ) = int [ I Q ( x , y ) max ( I Q ) × 255 ] .
z 0 = Λ trans D λ .
Λ trans = 2 Δ x .
Δ z = Λ long = 8 λ ( z o D ) 2 .
A ( i , j ) = I S ( i , j ) .

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