Abstract

An iterative phase-estimation method for the calculation of a specimen’s phase function or optical-path-length (OPL) distribution from differential-interference-contrast (DIC) microscopy images is presented. The method minimizes the least-squares discrepancy measure by use of the conjugate-gradient technique to estimate the phase function from multiple DIC images acquired at different specimen rotations. The estimate is regularized with a quadratic smoothness penalty. Results from testing the method with simulations and measured DIC images show improvement in the estimated phase when at least two rotationally diverse DIC images instead of a single DIC image are used for the estimation. The OPL of a cell that is estimated from two DIC images was found to be much more reliable than the OPL computed from single DIC images (which had a coefficient of variation equal to 15.8%).

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).
  2. R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
    [PubMed]
  3. C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” D.Sc. thesis (Washington University, Sever Institute of Technology, St. Louis, Mo., 1998).
  4. C. Preza, D. L. Snyder, J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential interference contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
    [CrossRef]
  5. T. J. Holmes, W. J. Levy, “Signal-processing characteristics of differential-interference-contrast microscopy,” Appl. Opt. 26, 3929–3939 (1987).
    [CrossRef] [PubMed]
  6. K. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” M.Sc. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).
  7. E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
    [CrossRef]
  8. E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
    [CrossRef] [PubMed]
  9. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  10. N. Baba, H. Tomita, N. Miura, “Iterative reconstruction method in phase-diversity imaging,” Appl. Opt. 33, 4428–4433 (1994).
    [CrossRef] [PubMed]
  11. E. Lantz, “Retrieval of a phase-and-amplitude submicrometric object from images obtained in partially coherent microscopy,” J. Opt. Soc. Am. A 8, 791–800 (1991).
    [CrossRef]
  12. V. Y. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
    [CrossRef]
  13. C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
    [CrossRef]
  14. C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).
  15. I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
    [CrossRef]
  16. D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  17. D. L. Snyder, Random Point Processes (Wiley, New York, 1975).
  18. R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. (UK) 7, 149–154 (1964).
    [CrossRef]
  19. M. Aoki, Introduction to Optimization Techniques (Macmillan, New York, 1971).
  20. E. Polak, Computational Methods in Optimization (Academic, New York, 1971).
  21. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).
  22. T. M. Apostol, Calculus, 2nd ed. (Wiley, New York, 1969), Vol. 2.
  23. M. Pluta, Advanced Light Microscopy: Principles and Basic Properties (PWN-Polish Scientific, Warsaw, 1988).
  24. E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
    [CrossRef] [PubMed]

1999

C. Preza, D. L. Snyder, J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential interference contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

1998

E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
[CrossRef] [PubMed]

1997

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

1994

1993

1992

1991

E. Lantz, “Retrieval of a phase-and-amplitude submicrometric object from images obtained in partially coherent microscopy,” J. Opt. Soc. Am. A 8, 791–800 (1991).
[CrossRef]

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

1987

1969

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

1964

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. (UK) 7, 149–154 (1964).
[CrossRef]

Allen, R. D.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Aoki, M.

M. Aoki, Introduction to Optimization Techniques (Macmillan, New York, 1971).

Apostol, T. M.

T. M. Apostol, Calculus, 2nd ed. (Wiley, New York, 1969), Vol. 2.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).

Aten, A. A.

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

Aten, J. A.

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
[CrossRef] [PubMed]

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Baba, N.

Conchello, J.-A.

C. Preza, D. L. Snyder, J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential interference contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Csiszár, I.

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

Dana, K.

K. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” M.Sc. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

David, G. B.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Fienup, J. R.

Fletcher, R.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. (UK) 7, 149–154 (1964).
[CrossRef]

Hammoud, A. M.

Hoebe, R. A.

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

Holmes, T. J.

Ivanov, V. Y.

Lang, W.

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

Lantz, E.

Levy, W. J.

Markham, J.

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Meerman, G. J. T.

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

Miura, N.

Nomarski, G.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Paxman, R. G.

Pluta, M.

M. Pluta, Advanced Light Microscopy: Principles and Basic Properties (PWN-Polish Scientific, Warsaw, 1988).

Polak, E.

E. Polak, Computational Methods in Optimization (Academic, New York, 1971).

Preza, C.

C. Preza, D. L. Snyder, J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential interference contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” D.Sc. thesis (Washington University, Sever Institute of Technology, St. Louis, Mo., 1998).

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

Reeves, C. M.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. (UK) 7, 149–154 (1964).
[CrossRef]

Rosenberger, F. U.

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

Schulz, T. J.

Sivokon, V. P.

Snyder, D. L.

C. Preza, D. L. Snyder, J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential interference contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

Stap, J.

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).

Tomita, H.

van Munster, E. B.

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
[CrossRef] [PubMed]

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

van Vliet, L. J.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Vorontsov, M. A.

White, R. L.

Winter, E. K.

E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
[CrossRef] [PubMed]

Ann. Stat.

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

Appl. Opt.

Comput. J. (UK)

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. (UK) 7, 149–154 (1964).
[CrossRef]

Cytometry

E. B. van Munster, J. Stap, R. A. Hoebe, G. J. T. Meerman, J. A. Aten, “Difference in volume of X- and Y-chromosome-bearing bovine sperm heads matches difference in DNA content,” Cytometry 35, 125–128 (1999).
[CrossRef] [PubMed]

J. Microsc.

E. B. van Munster, L. J. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

E. B. van Munster, E. K. Winter, J. A. Aten, “Measurement-based evaluation of optical pathlength distributions reconstructed from simulated differential interference contrast images,” J. Microsc. 191, 170–177 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Z. Wiss. Mikrosk.

R. D. Allen, G. B. David, G. Nomarski, “The Zeiss–Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69, 193–221 (1969).
[PubMed]

Zeiss Inf.

W. Lang, “Nomarski differential interference contrast microscopy II. Formation of the interference image,” Zeiss Inf. 17, 12–16 (1969).

Other

C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” D.Sc. thesis (Washington University, Sever Institute of Technology, St. Louis, Mo., 1998).

K. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” M.Sc. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1992).

C. Preza, D. L. Snyder, F. U. Rosenberger, J. Markham, J.-A. Conchello, “Phase estimation from transmitted-light DIC images using rotational diversity,” in Image Reconstruction and Restoration II, T. Schulz, ed., Proc. SPIE3170, 97–107 (1997).
[CrossRef]

C. Preza, E. B. van Munster, A. A. Aten, D. L. Snyder, F. U. Rosenberger, “Determination of direction-independent optical path-length distribution of cells using rotational-diversity transmitted-light differential interference contrast (DIC) images,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE3261, 60–70 (1998).

M. Aoki, Introduction to Optimization Techniques (Macmillan, New York, 1971).

E. Polak, Computational Methods in Optimization (Academic, New York, 1971).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).

T. M. Apostol, Calculus, 2nd ed. (Wiley, New York, 1969), Vol. 2.

M. Pluta, Advanced Light Microscopy: Principles and Basic Properties (PWN-Polish Scientific, Warsaw, 1988).

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Synthetic rotationally diverse DIC images of a computer-generated blob with different orientation of the shear. The shear is along the horizontal axis in (a), and it is rotated every 45 deg clockwise from (a) to (d). The scale bar is 7.8 µm.

Fig. 2
Fig. 2

[(a) (top row)] True phase image of a blob and [(b)–(e) (top row)] estimated phase images obtained from a different number of synthetic rotationally diverse images, (b) from eight images with 22.5-deg shear rotation, (c) from four images with 45-deg shear rotation, (d) from two images with orthogonal shear directions, and (e) from a single image with the shear direction along the horizontal axis. The images in the bottom row are (a) an initial guess used in the estimation and (b)–(e) the absolute value of the difference between the true phase and each corresponding estimated phase image shown on the top row. The scale bar is 7.8 µm.

Fig. 3
Fig. 3

Comparison of the MSE computed over the whole image versus number of rotationally diverse images, K. The MSE of phase images of the blob estimated (a) from noisy images without regularization, (b) from noisy images with regularization using α=5×10-4, and (c) from noise-free images without regularization. The MSE is plotted with a logarithmic scale.

Fig. 4
Fig. 4

Comparison of the cost function evaluated at every iteration during the estimation of the blob phase images with an initial guess computed from the data and different sets of diversity images, K: 1, a single image (K=1); 2, two images separated by 90-deg rotation (K=2); 3, three images separated by 45-deg rotation (K=3); 4, four images separated by 45-deg rotation (K=4); and 5, eight images separated by 22.5-deg rotation (K=8). In each case the cost function was normalized by its value at the first iteration for comparison.

Fig. 5
Fig. 5

Synthetic-DIC images generated from (a) a two-point object, and (b) a single-point object with the same relative phase are very similar. Horizontal profiles from the center of the two images are compared in the bottom panel. The direction of shear is along the horizontal axis.

Fig. 6
Fig. 6

Profiles from estimated phase images of a phantom with two points. Horizontal profiles from the true phase, and estimated phase images of the two-point object computed with a different number of diversity DIC images, K.

Fig. 7
Fig. 7

Estimated phase images of a blob [Fig. 2(a)] obtained from noisy rotationally diverse images (with SNR=18 dB) without regularization (top row) and with regularization using a penalty weight α=5×10-4 (bottom row) (a) from eight images with 22.5-deg shear rotation, (b) from four images with 45-deg shear rotation, (c) from three images with 45-deg shear rotation, (d) from two images with orthogonal shear directions, and (e) from a single image.

Fig. 8
Fig. 8

Rotationally diverse DIC images of a single bovine sperm head acquired by rotating the cell ∼45 deg counterclockwise from (a) to (d). The images were then rotated so that the orientation of the shear rotates while the cell is fixed. The shear is along the 45-deg axis in (a), and it is rotated every time ∼45 deg clockwise from (a) to (d). The scale bar is ∼8 µm.

Fig. 9
Fig. 9

Estimated phase images of a cell computed from the images shown in Fig. 8 using regularization (with a smoothing parameter α=2×10-3) and a uniform initial guess (a) from eight images with 45-deg shear rotation, (b) from four images with 45-deg shear rotation, (c) from two images with 90-deg shear rotation, and (d) from a single image. On the average, the images were estimated after 70 iterations of the method. The scale bar is ∼8 µm.

Tables (1)

Tables Icon

Table 1 Effect of Rotational Diversity on the IOPL of a Cella

Equations (24)

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

i(x)=-+α(ξ)-+f(xo)h(x-xo)hc(ξ;xo)dxo2dξ,
hc(ξ;xo)=1jλfcexp[j2π(xoξ+yoη)/λfc];
h(x)=12[exp(-jΔθ)p(x-Δx, y, z)-exp(jΔθ)p(x+Δx, y, z)],
ik(x)=ξΞα(ξ)xoχf(xo)hk(x-xo)hc(ξ;xo)2,
ik(x)=a1xoχf(xo)hk(x-xo)2,
E(ϕ)=k=0K-1xχ[dk(x)-ik(x)]2,
ϕm+1=ϕm-βmhm,m=0, 1 , ,
E(ϕ)ϕ(xo)=2k=0K-1xχik(x)ϕ(xo)[ik(x)-dk(x)].
ik(x)ϕ(xo)=2 ReξΞα(ξ)ek(ξ)ϕ(xo)ek(ξ)*,
ek(ξ)=xoχexp[-jϕ(xo)]hk(x-xo)hc(ξ;xo).
ek(x)ϕ(xo)=-j exp[-jϕ(xo)]hk(x-xo)hc(ξ;xo),
ik(x)ϕ(xo)=2 Im{exp[-jϕ(xo)]hk(x-xo)ak(x;xo)},
ak(x;xo)=xoχexp[jϕ(xo)]hk*(x-xo)js(xo;xo),
js(xo;xo)=ξΞα(ξ)hc(ξ;xo)hc*(ξ;xo),
ak(x)=a1xoχexp[jϕ(xo)]hk*(x-xo).
Er(ϕ)=k=0Kxχ[dk(x)-ik(x)]2+αS(ϕ),
S(ϕ)=R3|ϕ|2dx
S(ϕ)=i=0N-1j=0N-1([ϕ(i+i, j)-ϕ(i, j)]2+[ϕ(i, j+1)-ϕ(i, j)]2+[ϕ(i-1, j)-ϕ(i, j)]2+[ϕ(i, j-1)-ϕ(i, j)]2.
Er(ϕ)ϕ(xo)=E(ϕ)ϕ(xo)+αS(ϕ)ϕ(xo),
S(ϕ)ϕ(i, j)=16ϕ(i, j)-4[ϕ(i+1, j)+ϕ(i, j+1)+ϕ(i, j-1)+ϕ(i-1, j)].
i(x, y)=a1 sin2[0.5{ϕ(x-Δx, y)-ϕ(x+Δx, y)}+Δθ].
ϕ(x, y)=F-1{Φ(f, y)},
Φ(f, y)=0iff=0orf=n2Δxn=1, 2 ,jS(f, y)2 sin(2πfΔx)otherwise,
S(f, y)=F{2{sin-1[i(x, y)/a1]-Δθ}},

Metrics