Abstract

An iterative Gerchberg–Saxton-type algorithm with a support constraint for twin-image removal from reconstructed Gabor inline holograms of single plane objects is described. It is applied to simulated holograms and to holograms of ice crystals recorded in the laboratory and in atmospheric clouds in situ. The algorithm is characterized by a distinction between object and background region and an iterative adaption of the object mask. Applying the algorithm to recorded inline holograms of atmospheric objects, the twin-image artifacts are removed successfully, for the first time allowing for a proper access to the in situ phase information on atmospheric ice crystals. It is also demonstrated that, after application of the algorithm, previously indiscernible internal object features can become visible for large Fresnel numbers.

© 2009 Optical Society of America

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2008 (3)

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

M. Gross, M. Atlan, and E. Absil, “Noise and aliases in off-axis and phase-shifting holography,” Appl. Opt. 47, 1757-1766(2008).
[CrossRef] [PubMed]

J. M. Desse, P. Picart, and P. Tankam, “Digital three-color holographic interferometry for flow analysis,” Opt. Express 16, 5471-5480 (2008).
[CrossRef] [PubMed]

2007 (5)

2006 (2)

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

M. K. Kim, Y. Lingfeng, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A Pure Appl. Opt. 8, S518-S523 (2006).
[CrossRef]

2004 (2)

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

J. P. Fugal, R. A. Shaw, E. W. Saw, and A. V. Sergeyev, “Airborne digital holographic system for cloud particle measurements,” Appl. Opt. 43, 5987-5995 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771-778 (2002).
[CrossRef]

2001 (2)

P. Korecki, G. Materlik, and J. Korecki, “Complex γ-ray hologram: solution to twin images problem in atomic resolution imaging,” Phys. Rev. Lett. 86, 1534-1537 (2001).
[CrossRef] [PubMed]

M. Tegze and G. Faigel, “X-ray holography: theory and experiment,” J. Phys. Condens. Matter 13, 10613-10623(2001).
[CrossRef]

2000 (1)

1998 (1)

1997 (1)

1993 (1)

1991 (1)

1987 (3)

1982 (1)

1978 (1)

1972 (1)

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

1968 (1)

1966 (2)

D. Gabor and W. P. Goss, “Interference microscope with total wavefront reconstruction,” J. Opt. Soc. Am. 56, 849-858(1966).
[CrossRef]

C. Magono and C. W. Lee, “Meteorological classification of natural snow crystals,” J. Fac. Sci. Hokkaido Univ. 7, 321-335(1966).

1963 (1)

1962 (1)

1956 (1)

A. Lohmann, “Optische Einseitenbandübertragung angewandt auf das Gabor-Mikroskop,” J. Mod. Opt. 3:2, 97-99(1956).
[CrossRef]

1951 (1)

L. W. Bragg and G. L. Rogers, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190-191 (1951).
[CrossRef] [PubMed]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778(1948).
[CrossRef] [PubMed]

Absil, E.

Atlan, M.

Borrmann, S.

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

Bragg, L. W.

L. W. Bragg and G. L. Rogers, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190-191 (1951).
[CrossRef] [PubMed]

Bryngdahl, O.

Burkeeva, I.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Cedola, A.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Chapman, H. N.

Crimmins, T. R.

Cuche, E.

Curtius, J.

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

De Caro, L.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Depeursinge, C.

Desse, J. M.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), p. 501.

Faigel, G.

M. Tegze and G. Faigel, “X-ray holography: theory and experiment,” J. Phys. Condens. Matter 13, 10613-10623(2001).
[CrossRef]

Fienup, J. R.

Fink, H. W.

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Formanek, P.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Fugal, J. P.

Gabor, D.

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 (Jena) 35, 237-246 (1972).

Giannini, C.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Fourier Optics (Roberts, 2005).

Goss, W. P.

Gross, M.

Guagliardi, A.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Joyeux, D.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography (Springer-Verlag, 2005).

Kim, M. K.

M. K. Kim, Y. Lingfeng, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A Pure Appl. Opt. 8, S518-S523 (2006).
[CrossRef]

Klett, J. D.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer, 1997), p. 54.

Korecki, J.

P. Korecki, G. Materlik, and J. Korecki, “Complex γ-ray hologram: solution to twin images problem in atomic resolution imaging,” Phys. Rev. Lett. 86, 1534-1537 (2001).
[CrossRef] [PubMed]

Korecki, P.

P. Korecki, G. Materlik, and J. Korecki, “Complex γ-ray hologram: solution to twin images problem in atomic resolution imaging,” Phys. Rev. Lett. 86, 1534-1537 (2001).
[CrossRef] [PubMed]

Koren, G.

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Lagomarsino, S.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Latychevskaia, T.

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Lee, C. W.

C. Magono and C. W. Lee, “Meteorological classification of natural snow crystals,” J. Fac. Sci. Hokkaido Univ. 7, 321-335(1966).

Lehmann, M.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Leith, E. N.

Lenk, A.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Lichte, H.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Linck, M.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Lingfeng, Y.

M. K. Kim, Y. Lingfeng, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A Pure Appl. Opt. 8, S518-S523 (2006).
[CrossRef]

Liu, G.

Lohmann, A.

O. Bryngdahl and A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. 58, 620-624 (1968).
[CrossRef]

A. Lohmann, “Optische Einseitenbandübertragung angewandt auf das Gabor-Mikroskop,” J. Mod. Opt. 3:2, 97-99(1956).
[CrossRef]

Magono, C.

C. Magono and C. W. Lee, “Meteorological classification of natural snow crystals,” J. Fac. Sci. Hokkaido Univ. 7, 321-335(1966).

Mann, C. J.

M. K. Kim, Y. Lingfeng, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A Pure Appl. Opt. 8, S518-S523 (2006).
[CrossRef]

Marquet, P.

Materlik, G.

P. Korecki, G. Materlik, and J. Korecki, “Complex γ-ray hologram: solution to twin images problem in atomic resolution imaging,” Phys. Rev. Lett. 86, 1534-1537 (2001).
[CrossRef] [PubMed]

Matoba, O.

Matzeck, C.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Meng, H.

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

Metzger, T. H.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Miao, J.

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), p. 501.

Mocuta, C.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Nakamura, T.

Nitta, K.

Osten, W.

Pan, G.

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

Pedrini, G.

Pelliccia, D.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77, 081408(R) (2008).
[CrossRef]

Picart, P.

Polack, F.

Pruppacher, H. R.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer, 1997), p. 54.

Pu, Y.

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

Raupach, S. M. F.

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

Rogers, G. L.

L. W. Bragg and G. L. Rogers, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190-191 (1951).
[CrossRef] [PubMed]

Saw, E. W.

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 (Jena) 35, 237-246 (1972).

Sayre, D.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography (Springer-Verlag, 2005).

Scott, P. D.

Sergeyev, A. V.

Shaw, R. A.

Simon, P.

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Tankam, P.

Tegze, M.

M. Tegze and G. Faigel, “X-ray holography: theory and experiment,” J. Phys. Condens. Matter 13, 10613-10623(2001).
[CrossRef]

Tiziani, H. J.

Upatnieks, J.

Vössing, H. J.

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

Woodward, H.

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

Yamaguchi, I.

Zhang, F.

Zhang, T.

Zhang, Y.

Annu. Rev. Mater. Res. (1)

H. Lichte, P. Formanek, A. Lenk, M. Linck, C. Matzeck, M. Lehmann, and P. Simon, “Electron holography: application to materials questions,” Annu. Rev. Mater. Res. 37, 539-588(2007).
[CrossRef]

Appl. Opt. (6)

J. Fac. Sci. Hokkaido Univ. (1)

C. Magono and C. W. Lee, “Meteorological classification of natural snow crystals,” J. Fac. Sci. Hokkaido Univ. 7, 321-335(1966).

J. Mod. Opt. (1)

A. Lohmann, “Optische Einseitenbandübertragung angewandt auf das Gabor-Mikroskop,” J. Mod. Opt. 3:2, 97-99(1956).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

S. M. F. Raupach, H. J. Vössing, J. Curtius, and S. Borrmann, “Digital crossed-beam holography for in situ imaging of atmospheric ice particles,” J. Opt. A Pure Appl. Opt. 8, 796-806(2006).
[CrossRef]

M. K. Kim, Y. Lingfeng, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A Pure Appl. Opt. 8, S518-S523 (2006).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

J. Phys. Condens. Matter (1)

M. Tegze and G. Faigel, “X-ray holography: theory and experiment,” J. Phys. Condens. Matter 13, 10613-10623(2001).
[CrossRef]

Meas. Sci. Technol. (1)

H. Meng, G. Pan, Y. Pu, and H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. 15, 673-685 (2004).
[CrossRef]

Nature (2)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778(1948).
[CrossRef] [PubMed]

L. W. Bragg and G. L. Rogers, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190-191 (1951).
[CrossRef] [PubMed]

Opt. Eng. (1)

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Optik (Jena) (1)

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

Phys. Rev. B (1)

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[CrossRef]

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[CrossRef] [PubMed]

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Figures (14)

Fig. 1
Fig. 1

Gabor setup and the twin-image problem. (a) For recording, the object is illuminated by coherent light, here a collimated laser beam. The resulting intensity in the hologram plane is recorded (top). For ordinary reconstruction, this recorded intensity is illuminated again by a collimated laser beam to reconstruct the field in the object plane (bottom; phases not shown). Shown below are the (b) amplitude and (c) phase of an imaginary/fictitious object used to simulate a hologram (the object’s image perceived visually would be proportional to the amplitude squared, i.e., the intensity), and the (d) amplitude and (e) phase of the complex-valued wave field upon ordinary reconstruction from the simulated hologram intensity.

Fig. 2
Fig. 2

Cascaded adaptive-mask algorithm.

Fig. 3
Fig. 3

Fictitious phase object. (a) The panel shows the phase of a fictitious pure phase object (constant amplitude not shown); the field of view is 804 μ m × 804 μ m . The phase is given in radians. (b) The panel displays the central part of the amplitude of the simulated diffracted wave field at a distance of 0.072 m , the according phase distribution would be lost upon recording. The field of view is 1675 μ m × 1675 μ m .

Fig. 4
Fig. 4

Application of the cascaded adaptive-mask algorithm to a simulated hologram (305 iterations): (a) the initial guess of the object region, (b) adapted mask, (c) and (d) amplitude and phase of a reconstruction from the hologram amplitude assuming a uniform hologram phase, (e) and (f) amplitude and phase of a reconstruction of the complex-valued hologram obtained by the algorithm. The phases are given in radians. The field of view for all panels is 804 μ m × 804 μ m .

Fig. 5
Fig. 5

Violation of the respective constraints in the hologram plane and the object plane. (a)  V h , (b)  V o (see text). The error peaks have partially been truncated to allow for a better view of the minima.

Fig. 6
Fig. 6

Effect of the mask accuracy on the algorithm: (a) a mask tightly covering the object, (b) and (c) amplitude and phase upon reconstruction of the complex-valued hologram obtained after 20 iterations constantly using mask (a), d) a circular mask larger than the object, (e) and (f) amplitude and phase upon reconstruction of the complex-valued hologram obtained after 100 iterations constantly using mask (d). The phases are given in radians.

Fig. 7
Fig. 7

Effect of intensity noise. (a)–(c) Result obtained after 305 iterations, starting with the initial mask shown in Fig. 4a: (a) Adapted mask, (b) amplitude, (c) phase. (d)–(f) Result obtained after 503 iterations: (d) adapted mask, (e) amplitude, (f) phase. Amplitudes are given in arbitrary units, phases are given in radians. (g)  V h , (h)  V o .

Fig. 8
Fig. 8

Application of the algorithm to natural dendritic ice crystals ( r N q = 1.20 , F max = 10.36 , 420 iterations): (a) initial mask, (b) adapted mask, (c) and (d) amplitude and phase of the ordinary reconstruction from the hologram intensity, (e) and (f) amplitude and phase obtained upon reconstruction from the hologram amplitude, assuming a uniform hologram phase, (g) and (h) amplitude and phase obtained upon backpropagation of the complex-valued hologram obtained by the algorithm, (i) V o . Amplitudes are given in arbitrary units, phases are given in radians. The field of view is 1.6 mm × 1.6 mm .

Fig. 9
Fig. 9

Enlarged views of (a) Fig. 8c and (b) Fig. 8g. The field of view for both panels is 871 μ m × 871 μ m .

Fig. 10
Fig. 10

Application of the algorithm to the hologram of a small platelike crystal ( r N q = 0.56 , F max = 0.044 , 65 iterations): (a) initial mask, (b) adapted mask, (c) and (d) amplitude and phase of the ordinary reconstruction from the hologram intensity, (e) and (f) amplitude and phase obtained upon reconstruction from the hologram amplitude, assuming a uniform hologram phase, (g) and (h) amplitude and phase obtained upon backpropagation of the complex-valued hologram obtained by the algorithm, (i) V o . Amplitudes are given in arbitrary units, phases are given in radians. The field of view is 369 μ m × 369 μ m for all panels.

Fig. 11
Fig. 11

Removal of the twin image: (a) the amplitude of the wave field obtained by ordinary reconstruction from the intensity as given by a recorded hologram [see Fig. 10c] and (b) the amplitude of the backpropagated, complex-valued wave field as obtained after application of the algorithm [see Fig. 10g]. The field of view is 2.01 mm × 2.01 mm for both panels.

Fig. 12
Fig. 12

Application of the algorithm to the hologram of natural ice crystals without pronounced inner structure ( r N q = 0.89 , F max = 8.42 , 70 iterations): (a) initial mask, (b) adapted mask, (c) and (d) amplitude and phase of the ordinary reconstruction from the hologram intensity, (e) and (f) amplitude and phase obtained upon reconstruction from the hologram amplitude, assuming a uniform hologram phase, (g) and (h) amplitude and phase obtained upon backpropagation of the complex-valued hologram obtained by the algorithm, (i)  V o . Amplitudes are given in arbitrary units, phases are given in radians. The field of view is 1.943 mm × 1.943 mm .

Fig. 13
Fig. 13

Structural details: (a) and (c) enlarged details of Fig. 12g, (b) and (d) enlarged details of Fig. 12h. The field of view is 536 μ m × 536 μ m for the top row and 603 μ m × 603 μ m for the bottom row. Note that the stellar end of the bottom-right column [(a), (b)] seems to sit on tips extending from the column’s end surface (see discussion in the text).

Fig. 14
Fig. 14

Effect of detector size and pixel pitch on K - + . This figure displays the real parts of the convolution of the forward propagation kernel with the back propagation kernel [(a), (b), (d), (e)], and the amplitude images [(c), (f)] resulting from an application of the K - + to a simulated, purely ( π - ) phase-shifting circular object, illustrating the phase contrast effect. The imaginary parts of the central peaks of the K - + are of the order of - 10 - 14 (not shown). The indicated Nyquist ratios correspond to distances z = 0.072 m [(a)–(c)] and 0.300 m [(d)–(f)], respectively. The subscript x of the Nyquist ratio has been omitted as the pixels are quadratic.

Equations (8)

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U 0 bg = { 0     ( x , y ) mask ( x , y ) mask U ( x , y , 0 ) / N     ( x , y ) mask ,
U 0 obj = { U ( x , y , 0 )     ( x , y ) mask 0     ( x , y ) mask ,
V o = ( ( x , y ) A o ( x , y ) ) / N ,     ( x , y ) M ,
V h = ( ( ξ , η ) | A h ( ξ , η ) - A ^ h ( ξ , η ) | ) / N t ,     ( ξ , η ) ,
F max = r max 2 z λ ,
U - + = U K - K + = U K - + ,
r x N q = L x 2 R N q , x ,
R N q , x = - Δ x 2 + Δ x 2 4 - z 2 - Δ x 4 / λ 2 + 0.5 Δ x 2 - λ 2 / 16 1 - 4 Δ x 2 / λ 2 ,

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