Abstract

Reconstruction of an image (or shape or wavefront) from measurements of the derivatives of the image in two orthogonal directions is a common problem. We demonstrate how a particular reconstructor, commonly referred to as the Fried algorithm, can be used with megapixel derivative images to recover the original image. Large datasets are handled by breaking the derivative images into smaller tiles, applying the Fried algorithm and stitching the tiles back together. The performance of the algorithm is demonstrated using differential interference contrast microscopy on a known test object.

© 2012 Optical Society of America

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References

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  1. S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35, 447–449 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
    [CrossRef]
  5. M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006).
    [CrossRef]
  6. K. J. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” Master’s thesis (Massachusetts Institute of Technology, 1992).
  7. D. D. Duncan, D. G. Fischer, A. Dayton, and S. A. Prahl, “Quantitative Carré differential interference contrast microscopy to assess phase and amplitude,” J. Opt. Soc. Am. A 28, 1297–1306(2011).
    [CrossRef]
  8. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  9. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  10. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
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  12. D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
    [CrossRef]
  13. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  14. R. J. Noll, “Phase estimates from slope-type wavefront sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
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  15. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  16. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
    [CrossRef]
  17. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
    [CrossRef]

2011

2010

S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35, 447–449 (2010).
[CrossRef]

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

2009

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

2006

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006).
[CrossRef]

2005

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

2001

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

1998

1983

1980

1978

1977

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

Ayers, F. R.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

Barbastathis, G.

Bevilacqua, F.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[CrossRef]

Cogswell, C. J.

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

Cuccia, D. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[CrossRef]

Dana, K. J.

K. J. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” Master’s thesis (Massachusetts Institute of Technology, 1992).

Daneshbod, M.

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

Dayton, A.

Duncan, D. D.

D. D. Duncan, D. G. Fischer, A. Dayton, and S. A. Prahl, “Quantitative Carré differential interference contrast microscopy to assess phase and amplitude,” J. Opt. Soc. Am. A 28, 1297–1306(2011).
[CrossRef]

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

Durkin, A. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[CrossRef]

Fischer, D. G.

D. D. Duncan, D. G. Fischer, A. Dayton, and S. A. Prahl, “Quantitative Carré differential interference contrast microscopy to assess phase and amplitude,” J. Opt. Soc. Am. A 28, 1297–1306(2011).
[CrossRef]

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

Fried, D. L.

Harris, T. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, Volume II: Devices, Measurements, & Properties, 2nd ed. (McGraw-Hill, 1995), Chap. 33, pp. 33.3–33.101.

Hudgin, R. H.

Inoué, S.

King, S. V.

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Kou, S. S.

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

Noll, R. J.

Prahl, S. A.

D. D. Duncan, D. G. Fischer, A. Dayton, and S. A. Prahl, “Quantitative Carré differential interference contrast microscopy to assess phase and amplitude,” J. Opt. Soc. Am. A 28, 1297–1306(2011).
[CrossRef]

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

Preza, C.

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Sheppard, C. J. R.

S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35, 447–449 (2010).
[CrossRef]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

Shribak, M.

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

Southwell, W. H.

Teague, M. R.

Thomas, M. E.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, Volume II: Devices, Measurements, & Properties, 2nd ed. (McGraw-Hill, 1995), Chap. 33, pp. 33.3–33.101.

Tromberg, B. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[CrossRef]

Tropf, W. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, Volume II: Devices, Measurements, & Properties, 2nd ed. (McGraw-Hill, 1995), Chap. 33, pp. 33.3–33.101.

Waller, L.

Appl. Opt.

J. Biomed. Opt.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

J. Microsc.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Opt. Lett.

Proc. SPIE

D. D. Duncan, D. G. Fischer, M. Daneshbod, and S. A. Prahl, “Differential interference contrast microscopy for the quantitative assessment of tissue organization,” Proc. SPIE 7570, 75700C (2010).
[CrossRef]

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Other

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, Volume II: Devices, Measurements, & Properties, 2nd ed. (McGraw-Hill, 1995), Chap. 33, pp. 33.3–33.101.

K. J. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” Master’s thesis (Massachusetts Institute of Technology, 1992).

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Figures (8)

Fig. 1.
Fig. 1.

Illustration of Fried geometry. The derivative values (Dxi,j,Dyi,j) for each pixel are known and are related to the unknown values ϕi,j at each corner through Eq. (1).

Fig. 2.
Fig. 2.

Each tile Mi,j is a (m+1)×(n+1) matrix of pixels and overlaps adjacent tiles. The unknown constant of integration ci,j for each tile is chosen to minimize the differences along the tile edges.

Fig. 3.
Fig. 3.

Intrinsic reconstruction error of the Fried algorithm for a periodic phase grating.

Fig. 4.
Fig. 4.

Phase derivatives Dx and Dy obtained from DIC measurements of the test object. The white square shows the area on the bottom (uncoated) step that was averaged and subtracted to set the phase for this step to zero.

Fig. 5.
Fig. 5.

Recovered phase in radians. The square indicates the region that corresponds to the area measured with the AFM.

Fig. 6.
Fig. 6.

Phase estimates obtained using AFM (left) and DIC (right). The color bar on the right has units of radians. The vertical dashed line indicates the phases graphed in Fig. 7.

Fig. 7.
Fig. 7.

Comparison phases obtained with the DIC, AFM, and SPI (spiral phase integration) techniques along the dashed lines shown in Fig. 6.

Fig. 8.
Fig. 8.

Relationship between the phases of each pixel determined using AFM (horizontal axis) and DIC (vertical axis) methods. The least-squares fit is ϕDIC=0.045+1.08ϕAFM.

Equations (11)

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

Dxi,j=(ϕi,j+1+ϕi+1,j+1)(ϕi,j+ϕi+1,j)2,Dyi,j=(ϕi+1,j+ϕi+1,j+1)(ϕi,j+ϕi,j+1)2,
Xi,jX¯mjm+i.
[WxWy]Φ¯=[D¯xD¯y],
Φ¯=W1D¯,
ϕ(x,y)=sin[2π(x+y)f]
cu,v+1cu,v=1ni=1m+1(Mu,vi,n+1Mu,v+1i,1)=Exu,v,
cu+1,vcu,v=1nj=1n+1(Mu,vm+1,jMu+1,v1,j)=Eyu,v.
[VxVy]c¯=[E¯xE¯y],
Vc¯=E¯,
Vx=[100+1000000100+1000000100+1000000100+1000000100+1000000100+1].
ϕ(x,y)=2πh(x,y)λ[n(λ)1]

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