Abstract

Two image reconstruction methods for 3D lens-based intensity diffraction tomography (I-DT) are developed that account for the effects of a focusing lens placed between the object and the detector. One reconstruction method is a generalization of the original I-DT method and requires measurement of the transmitted wavefield intensity on two detector planes behind the object at each tomographic view angle. The second method employs a data-acquisition strategy in which the two intensity measurements are acquired, in turn, on a fixed detector plane, corresponding to distinct forms of lens aberration. Preliminary computer-simulation studies are conducted to demonstrate the numerical implementation of both methods and corroborate their mathematical correctness.

© 2009 Optical Society of America

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    [CrossRef]
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  22. D. Shi and M. A. Anastasio, “Off-axis holographic tomography for diffracting scalar wave fields,” Phys. Rev. E 73, 016612 (2006).
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    [CrossRef]
  35. B. Chen and J. J. Stamnes, “Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders,” Appl. Opt. 37, 1999-2010 (1998).
    [CrossRef]
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    [CrossRef]
  37. X. Pan, “A unified reconstruction theory for diffraction tomography with considerations of noise control,” J. Opt. Soc. Am. A 15, 2312-2326 (1998).
    [CrossRef]

2007 (3)

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626-642 (2007).
[CrossRef]

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. (Bellingham) 46, 107003 (2007).
[CrossRef]

T. C. Peterson and V. J. Keast, “Astigmatic intensity equation for electron microscopy based phase retrival,” Ultramicroscopy 107, 635-643 (2007).
[CrossRef]

2006 (3)

2005 (4)

2004 (1)

M. A. Anastasio and D. Shi, “On the relationship between intensity diffraction tomography and phase-contrast tomography,” in Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

2003 (4)

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

T. Beetz, C. Jacobsen, and A. Stein, “Soft x-ray diffraction tomography: simulations and first experimerimental results,” J. Phys. I 104, 31-34 (2003).
[CrossRef]

O. R. Halse, J. J. Stamnes, and A. J. Devaney, “Three-dimensional diffraction tomography by two-dimensional sectioning,” Opt. Commun. 224, 185-195 (2003).
[CrossRef]

2002 (2)

2001 (2)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165-176 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. 87, 123902 (2001).
[CrossRef] [PubMed]

2000 (2)

1998 (2)

1996 (1)

T. C. Wedberg and J. J. Stamnes, “Recent results in optical diffraction microtomography,” Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

1995 (2)

T. C. Wedberg and J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28-31 (1995).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

1994 (1)

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. (Bellingham) 33, 3243-3253 (1994), http://link.aip.org/link/?JOE/33/3243/1.
[CrossRef]

1993 (1)

1992 (1)

A. J. Devaney and A. Schatzberg, “The coherent optical tomographic microscope,” Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

1986 (1)

A. J. Devaney, “Reconstructive tomography with diffracting wave fields,” Inverse Probl. 2, 161-183 (1986).
[CrossRef]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1979 (1)

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567-587 (1979).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Allen, L. J.

W. McBride, N. L. O'Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. 61, 321-324 (2005).
[CrossRef]

L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. 87, 123902 (2001).
[CrossRef] [PubMed]

Anastasio, M. A.

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Beetz, T.

T. Beetz, C. Jacobsen, and A. Stein, “Soft x-ray diffraction tomography: simulations and first experimerimental results,” J. Phys. I 104, 31-34 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Charrire, F.

Chen, B.

Colomb, T.

Davis, T.

Depeursinge, C.

Devaney, A. J.

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338-2347 (2005).
[CrossRef]

O. R. Halse, J. J. Stamnes, and A. J. Devaney, “Three-dimensional diffraction tomography by two-dimensional sectioning,” Opt. Commun. 224, 185-195 (2003).
[CrossRef]

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. (Bellingham) 33, 3243-3253 (1994), http://link.aip.org/link/?JOE/33/3243/1.
[CrossRef]

M. Maleki and A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

A. J. Devaney and A. Schatzberg, “The coherent optical tomographic microscope,” Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

A. J. Devaney, “Reconstructive tomography with diffracting wave fields,” Inverse Probl. 2, 161-183 (1986).
[CrossRef]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Gbur, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Guo, P.

Gureyev, T.

Halse, O. R.

O. R. Halse, J. J. Stamnes, and A. J. Devaney, “Three-dimensional diffraction tomography by two-dimensional sectioning,” Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Heger, T. J.

Huang, Y.

Jacobsen, C.

T. Beetz, C. Jacobsen, and A. Stein, “Soft x-ray diffraction tomography: simulations and first experimerimental results,” J. Phys. I 104, 31-34 (2003).
[CrossRef]

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Kaveh, M.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Keast, V. J.

T. C. Peterson and V. J. Keast, “Astigmatic intensity equation for electron microscopy based phase retrival,” Ultramicroscopy 107, 635-643 (2007).
[CrossRef]

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165-176 (2001).
[CrossRef]

Maleki, M.

Maleki, M. H.

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. (Bellingham) 33, 3243-3253 (1994), http://link.aip.org/link/?JOE/33/3243/1.
[CrossRef]

Marquet, P.

Mayo, S.

McBride, W.

W. McBride, N. L. O'Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. 61, 321-324 (2005).
[CrossRef]

McMahon, P.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Miller, P.

Mitchell, E. A D.

Mueller, R.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Nugent, K. A.

W. McBride, N. L. O'Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. 61, 321-324 (2005).
[CrossRef]

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

O'Leary, N. L.

W. McBride, N. L. O'Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. 61, 321-324 (2005).
[CrossRef]

Oxley, M. P.

L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. 87, 123902 (2001).
[CrossRef] [PubMed]

Paganin, D.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. 87, 123902 (2001).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Pan, X.

Paterson, D.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Pavillon, N.

Peele, A.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Peterson, T. C.

T. C. Peterson and V. J. Keast, “Astigmatic intensity equation for electron microscopy based phase retrival,” Ultramicroscopy 107, 635-643 (2007).
[CrossRef]

Pogany, A.

Rappaz, B.

Rau, C.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Schatzberg, A.

A. J. Devaney and A. Schatzberg, “The coherent optical tomographic microscope,” Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

Shi, D.

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. (Bellingham) 46, 107003 (2007).
[CrossRef]

M. A. Anastasio, D. Shi, and G. Gbur, “Multispectral intensity diffraction tomography reconstruction theory: quasi-nondispersive objects,” J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

D. Shi and M. A. Anastasio, “Off-axis holographic tomography for diffracting scalar wave fields,” Phys. Rev. E 73, 016612 (2006).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, “Spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

M. A. Anastasio and D. Shi, “On the relationship between intensity diffraction tomography and phase-contrast tomography,” in Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Snigirev, A.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Stamnes, J. J.

O. R. Halse, J. J. Stamnes, and A. J. Devaney, “Three-dimensional diffraction tomography by two-dimensional sectioning,” Opt. Commun. 224, 185-195 (2003).
[CrossRef]

B. Chen and J. J. Stamnes, “Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders,” Appl. Opt. 37, 1999-2010 (1998).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Recent results in optical diffraction microtomography,” Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28-31 (1995).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

Stein, A.

T. Beetz, C. Jacobsen, and A. Stein, “Soft x-ray diffraction tomography: simulations and first experimerimental results,” J. Phys. I 104, 31-34 (2003).
[CrossRef]

Stevenson, A.

Wade, G.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Wedberg, T. C.

T. C. Wedberg and J. J. Stamnes, “Recent results in optical diffraction microtomography,” Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28-31 (1995).
[CrossRef]

Weitkamp, T.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Wilkins, S.

Wolf, E.

G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890-1892 (2002).
[CrossRef]

G. Gbur and E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194-2202 (2002).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A.Consortini, ed. (Academic, 1996), pp. 83-110.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Acta Crystallogr. (1)

W. McBride, N. L. O'Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. 61, 321-324 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Inverse Probl. (1)

A. J. Devaney, “Reconstructive tomography with diffracting wave fields,” Inverse Probl. 2, 161-183 (1986).
[CrossRef]

J. Microsc. (1)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165-176 (2001).
[CrossRef]

J. Opt. Soc. Am. A (9)

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, “Spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur and E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194-2202 (2002).
[CrossRef]

M. Maleki and A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338-2347 (2005).
[CrossRef]

X. Pan, “A unified reconstruction theory for diffraction tomography with considerations of noise control,” J. Opt. Soc. Am. A 15, 2312-2326 (1998).
[CrossRef]

M. A. Anastasio and X. Pan, “Computationally efficient and statistically robust image rexonstruction in 3D diffraction tomography,” J. Opt. Soc. Am. A 17, 391-400 (2000).
[CrossRef]

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626-642 (2007).
[CrossRef]

M. A. Anastasio, D. Shi, and G. Gbur, “Multispectral intensity diffraction tomography reconstruction theory: quasi-nondispersive objects,” J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

J. Phys. I (1)

T. Beetz, C. Jacobsen, and A. Stein, “Soft x-ray diffraction tomography: simulations and first experimerimental results,” J. Phys. I 104, 31-34 (2003).
[CrossRef]

Meas. Sci. Technol. (1)

T. C. Wedberg and J. J. Stamnes, “Recent results in optical diffraction microtomography,” Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

Opt. Commun. (3)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

O. R. Halse, J. J. Stamnes, and A. J. Devaney, “Three-dimensional diffraction tomography by two-dimensional sectioning,” Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Opt. Eng. (Bellingham) (2)

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. (Bellingham) 46, 107003 (2007).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Conventional measurement geometry of I-DT. A plane wave illuminates a weakly scattering object, and at each tomographic view angle the intensity of the transmitted wave field is recorded on two parallel detector planes.

Fig. 2
Fig. 2

Schematic of lens-based I-DT.

Fig. 3
Fig. 3

Measurement geometry for two-detector lens-based I-DT.

Fig. 4
Fig. 4

(a) Re { n ( r ) } 1 and (b) Im { n ( r ) } for phantom #2.

Fig. 5
Fig. 5

Images that depict Re { n ( r ) } 1 for phantom #1 reconstructed by use of the reconstruction formula in Subsection 4A from (a) noiseless and (b) noisy data with noise level 0.1%.

Fig. 6
Fig. 6

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 5a, 5b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 7
Fig. 7

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } of phantom #2 reconstructed by use of the reconstruction formula in Subsection 4A from noiseless data.

Fig. 8
Fig. 8

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 7a, 7b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 9
Fig. 9

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } of phantom #2 reconstructed by use of the reconstruction formula in Subsection 4A from noisy data with noise level 0.01%.

Fig. 10
Fig. 10

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 9a, 9b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 11
Fig. 11

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } of phantom #2 reconstructed by use of the reconstruction formula in Subsection 4B from the noiseless simulation data with a lens focusing error corresponding to c = 0.0001 m .

Fig. 12
Fig. 12

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 11a, 11b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 13
Fig. 13

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } for phantom #2 reconstructed by use of the reconstruction formula in Subsection 4B from data with noise level 0.01% and a lens focusing error corresponding to c = 0.0001 m .

Fig. 14
Fig. 14

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 13a, 13b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 15
Fig. 15

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } for phantom #2 reconstructed from the noiseless aberrational simulation data by use of one-detector lens-based I-DT with lens focusing errors of c 1 = 0.0001 m and c 2 = 0.0002 m , respectively.

Fig. 16
Fig. 16

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 15a, 15b, respectively. The profiles of the true phantom are displayed in dashed curves.

Fig. 17
Fig. 17

Images that depict (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } for phantom #2 reconstructed from the noisy aberrational simulation data by use of one-detector lens-based I-DT with lens focusing errors of c 1 = 0.0001 m and c 2 = 0.0002 m and noise level 0.000001%.

Fig. 18
Fig. 18

Solid curves in (a) and (b) display the profiles through the central rows of Figs. 17a, 17b, respectively. The profiles of the true phantom are displayed in dashed curves.

Equations (65)

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f ( r ) = k 2 4 π [ n 2 ( r ) 1 ] ,
U ( x , y ; z ) U i ( x , y ; z ) exp [ ψ ( x , y ; z ) ] ,
ψ ̂ ( u , v ; d ) = ( 2 π ) 2 i w F ̂ [ K = u s ̂ 1 + v s ̂ 2 + ( w k ) s ̂ 0 ] exp [ i ( w k ) d ] ,
w k 2 u 2 v 2
ψ ̂ ( u , v ; z ) = 1 ( 2 π ) 2 R 2 d x d y ψ ( x , y , z ) exp [ i ( u x + v y ) ]
F ̂ ( K ) = 1 ( 2 π ) 3 V d r f ( r ) exp [ i K r ]
F ̂ [ u , v ] F ̂ [ K = u s ̂ 1 + v s ̂ 2 + ( w k ) s ̂ 0 ] for u 2 + v 2 k 2 .
D I ( x , y ; z ) log [ I ( x , y ; z ) I i ( x , y ; z ) ] = ψ ( x , y ; z ) + ψ * ( x , y ; z ) ,
D ̂ I ( u , v ; z ) = 1 ( 2 π ) 2 R 2 d x d y D I ( x , y , z ) exp [ i ( u x + v y ) ] .
D ̂ Δ ( u , v ) D ̂ I ( u , v ; d ) D ̂ I ( u , v ; d + Δ ) exp [ i ( w k ) Δ ] .
F ̂ [ u , v ] = w i ( 2 π ) 2 exp [ i ( w k ) d ] 1 exp [ 2 i ( w k ) Δ ] D ̂ Δ ( u , v ) .
1 d i + 1 d o 1 f = 0 .
U m ( x m , y m ; d ) = 1 M U 1 ( x m M , y m M ) 2 h ( x m , y m ) ,
M d i d o .
h ( x m , y m ) = R 2 d x ̃ d y ̃ A ( λ d i x ̃ , λ d i y ̃ ) exp [ i 2 π ( x m x ̃ + y m y ̃ ) ] ,
A ( x l , y l ) = { 1 ( x l , y l ) inside the lens aperture 0 otherwise } .
U 1 ( x , y ) = U o ( x , y ) 2 P d 1 ( x , y ) ,
U 1 ( x m M , y m M ) = 1 M 2 U o ( x m M , y m M ) 2 P d 1 ( x m M , y m M ) .
U m ( x m , y m ; d ) = 1 M 3 U o ( x m M , y m M ) 2 G ( x m , y m ) ,
G ( x m , y m ) P d 1 ( x m M , y m M ) 2 h ( x m , y m ) .
I ( x m , y m ; d ) U m ( x m , y m ; d ) 2 .
ψ o ( x , y ) 1 , x , y R .
U o ( x , y ) U o * ( x , y ) = exp [ ψ o ( x , y ) + ψ o * ( x , y ) ] 1 + ψ o ( x , y ) + ψ o * ( x , y ) .
I ( x m , y m ; d ) 1 M 6 [ D 2 + D * ψ o ( x m M , y m M ) 2 G ( x m , y m ) + D ψ o * ( x m M , y m M ) 2 G * ( x m , y m ) ] ,
D d x m d y m G ( x m , y m ) .
I ̂ ( u m , v m ; d ) = 1 M 6 { D 2 δ ( u m , v m ) + D * [ ( 2 π ) 2 M 2 ψ ̂ o ( M u m , M v m ) G ̂ ( u m , v m ) ] + D [ ( 2 π ) 2 M 2 ψ ̂ o * ( M u m , M v m ) G ̂ * ( u m , v m ) ] } ,
ψ ̂ o ( u , v ) = ( 2 π ) 2 i k 2 u 2 v 2 F ̂ [ u , v ] ,
I ̂ ( u m , v m ; d ) 1 M 6 D 2 δ ( u m , v m ) = 1 M 4 D * ( 2 π ) 4 i k 2 M 2 ( u m 2 + v m 2 ) F ̂ [ M u m , M v m ] G ̂ ( u m , v m ) 1 M 4 D ( 2 π ) 4 i k 2 M 2 ( u m 2 + v m 2 ) F ̂ * [ M u m , M v m ] G ̂ * ( u m , v m ) .
I ̂ ( u M , v M ; d ) δ ( u M , v M ) M 2 = ( 2 π ) 2 i w { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] A * ( λ d o 2 π u , λ d o 2 π v ) F ̂ * [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] } ,
1 d i + Δ + 1 d o 1 f = 0 ,
Δ d o d o = Δ f 2 ( d i f ) ( d i + Δ f ) .
I ̂ ( u M , v M ; d + Δ ) δ ( u M , v M ) ( M ) 2 = ( 2 π ) 2 i w { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k ( d 1 + Δ ) ] A * ( λ d o 2 π u , λ d o 2 π v ) F ̂ * [ u , v ] exp [ i u 2 + v 2 2 k ( d 1 + Δ ) ] } ,
M d i d o .
F ̂ [ u , v ] = w i ( 2 π ) 2 D ̂ Δ ( u , v ) exp [ i u 2 + v 2 2 k d 1 ] 1 exp [ i u 2 + v 2 k Δ ] ,
D ̂ Δ ( u , v ) = D ̂ I ( u , v ; d ) D ̂ I ( u , v ; d + Δ ) exp [ i u 2 + v 2 2 k Δ ] ,
D ̂ I ( u , v ; d ) = I ̂ ( u M , v M ; d ) δ ( u M , v M ) M 2 ,
D ̂ I ( u , v ; d + Δ ) = I ̂ ( u M , v M ; d + Δ ) δ ( u M , v M ) ( M ) 2 .
( u 2 + v 2 ) Δ = 2 k n π ,
A ̃ ( x , y ) = A ( x , y ) exp [ i k ϵ ( x , y ) ] ,
h a ( x m , y m ) = R 2 d x ̃ d y ̃ A ̃ ( λ d i x ̃ , λ d i y ̃ ) exp [ i 2 π ( x m x ̃ + y m y ̃ ) ]
h ̂ a ( u m , v m ) = 1 ( 2 π ) 2 A ( λ d i 2 π u m , λ d i 2 π v m ) exp [ i k ϵ ( λ d i 2 π u m , λ d i 2 π v m ) ] ,
G ̂ ( u m , v m ) = M 2 ( 2 π ) 2 exp [ i k d 1 ] exp [ i M 2 ( u m 2 + v m 2 ) 2 k d 1 ] exp [ i k ϵ ( λ d i 2 π u m , λ d i 2 π v m ) ] A ( λ d i 2 π u m , λ d i 2 π v m ) ,
D = M 2 exp [ i k d 1 ] exp [ i k ϵ ( 0 , 0 ) ] .
I ̂ ( u M , v M ; d ) δ ( u M , v M ) M 2 = ( 2 π ) 2 i w { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] exp [ i k ( ϵ ( λ d o 2 π u , λ d o 2 π v ) ϵ ( 0 , 0 ) ) ] A * ( λ d o 2 π u , λ d o 2 π v ) F ̂ * [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] exp [ i k ( ϵ ( 0 , 0 ) ϵ ( λ d o 2 π u , λ d o 2 π v ) ) ] } .
I ̂ ( u M , v M ; d + Δ ) δ ( u M , v M ) ( M ) 2 = ( 2 π ) 2 i w { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k ( d 1 + Δ ) ] exp [ i k ( ϵ ( λ d o 2 π u , λ d o 2 π v ) ϵ ( 0 , 0 ) ) ] A * ( λ d o 2 π u , λ d o 2 π v ) F ̂ * [ u , v ] exp [ i u 2 + v 2 2 k ( d 1 + Δ ) ] exp [ i k ( ϵ ( 0 , 0 ) ϵ ( λ d o 2 π u , λ d o 2 π v ) ) ] } .
F ̂ [ u , v ] = w i ( 2 π ) 2 D ̂ Δ ( u , v ) exp [ i u 2 + v 2 2 k d 1 + i k ϵ ( 0 , 0 ) ] exp [ i k ϵ ( λ d o 2 π u , λ d o 2 π v ) ] exp [ i u 2 + v 2 k Δ ] E ( u , v )
E ( u , v ) exp [ i k ( ϵ ( λ d o 2 π u , λ d o 2 π v ) + ϵ ( λ d o 2 π u , λ d o 2 π v ) ϵ ( λ d o 2 π u , λ d o 2 π v ) ) ] ,
D ̂ Δ ( u , v ) = D ̂ I ( u , v ; d ) D ̂ I ( u , v ; d + Δ ) exp [ i u 2 + v 2 2 k Δ ] exp [ i k ( ϵ ( λ d o 2 π u , λ d o 2 π v ) ϵ ( λ d o 2 π u , λ d o 2 π v ) ) ] ,
I ̂ ( u M , v M ; d , ϵ m ) δ ( u M , v M ) M 2 = ( 2 π ) 2 i w { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] exp [ i k ( ϵ m ( λ d o 2 π u , λ d o 2 π v ) ϵ m ( 0 , 0 ) ) ] A * ( λ d o 2 π u , λ d o 2 π v ) F ̂ * [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] exp [ i k ( ϵ m ( 0 , 0 ) ϵ m ( λ d o 2 π u , λ d o 2 π v ) ) ] } ,
F ̂ [ u , v ] = w i ( 2 π ) 2 D ̂ ϵ ( u , v ) exp [ i u 2 + v 2 2 k d 1 ] E 1 ( u , v ) + E 2 ( u , v )
E 1 ( u , v ) exp [ i k ( ϵ 1 ( u , v ) ϵ 1 ( 0 , 0 ) ) ] ,
E 2 ( u , v ) exp [ i k ( ϵ 1 ( 0 , 0 ) ϵ 1 ( λ d o 2 π u , λ d o 2 π v ) 2 ϵ 2 ( 0 , 0 ) + ϵ 2 ( λ d o 2 π u , λ d o 2 π v ) + ϵ 2 ( λ d o 2 π u , λ d o 2 π v ) ) ] .
D ̂ ϵ ( u , v ) = D ̂ I ( u , v ; d , ϵ 1 ) D ̂ I ( u , v ; d , ϵ 2 ) exp [ i k ( ϵ 2 ( λ d o 2 π u , λ d o 2 π v ) ϵ 2 ( 0 , 0 ) + ϵ 1 ( 0 , 0 ) ϵ 1 ( λ d o 2 π u , λ d o 2 π v ) ) ] ,
D ̂ I ( u , v ; d , ϵ 1 ) = I ̂ ( u M , v M ; d , ϵ 1 ) δ ( u M , v M ) M 2 ,
D ̂ I ( u , v ; d , ϵ 2 ) = I ̂ ( u M , v M ; d , ϵ 2 ) δ ( u M , v M ) M 2 .
1 d i + 1 d o 1 f = c ,
ϵ ( y ) = c 2 y 2 .
A ( y l ) = { 1 , if y l 0.01 0 , otherwise } .
h ̂ ( u m , v m ) = 1 ( 2 π ) 2 A ( λ d i 2 π u m , λ d i 2 π v m ) .
G ̂ ( u m , v m ) = ( 2 π ) 2 [ M 2 P ̂ d 1 ( M u m , M v m ) h ̂ ( u m , v m ) ] .
P ̂ d 1 ( M u m , M v m ) = 1 ( 2 π ) 2 exp [ i k d 1 ] exp [ i M 2 ( u m 2 + v m 2 ) 2 k d 1 ] .
G ̂ ( u m , v m ) = M 2 ( 2 π ) 2 exp [ i k d 1 ] exp [ i M 2 ( u m 2 + v m 2 ) 2 k d 1 ] A ( λ d i 2 π u m , λ d i 2 π v m ) ,
D = ( 2 π ) 2 G ̂ ( u m , v m ) u m v m = 0 = M 2 exp [ i k d 1 ] .
I ̂ m ( u m , v m ; d ) δ ( u m , v m ) M 2 = exp [ i M 2 ( u m 2 + v m 2 ) 2 k d 1 ] A ( λ d i 2 π u m , λ d i 2 π v m ) ( 2 π ) 2 i k 2 M 2 ( u m 2 + v m 2 ) F ̂ [ M u m , M v m ] exp [ i M 2 ( u m 2 + v m 2 ) 2 k d 1 ] A * ( λ d i 2 π u m , λ d i 2 π v m ) ( 2 π ) 2 i k 2 M 2 ( u m 2 + v m 2 ) F ̂ * [ M u m , M v m ] .
I ̂ m ( u M , v M ; d ) δ ( u M , v M ) M 2 = ( 2 π ) 2 i k 2 ( u 2 + v 2 ) { A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] A ( λ d o 2 π u , λ d o 2 π v ) F ̂ [ u , v ] exp [ i u 2 + v 2 2 k d 1 ] } ,

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