Abstract

Phase retrieval based on the Transport of Intensity Equation (TIE) has shown to be a powerful tool to obtain the phase of complex fields. Recently, it has been proven that the performance of TIE techniques can be improved when using unequally spaced measurement planes. In this paper, an algorithm is presented that recovers accurately the phase of a complex objects from a set of intensity measurements obtained at unequal plane separations. This technique employs multiple band-pass filters in the frequency domain of the axial derivative and uses these specific frequency bands for the calculation of the final phase. This provides highest accuracy for TIE based phase recovery giving minimal phase error for a given set of measurement planes. Moreover, because each of these band-pass filters has a distinct sensitivity to noise, a new plane selection strategy is derived that equalizes the error contribution of all frequency bands. It is shown that this new separation strategy allows controlling the final error of the retrieved phase without using a priori information of the object. This is an advantage compared to previous optimum phase retrieval techniques. In order to show the stability and robustness of this new technique, we present the numerical simulations.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [PubMed]
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    [Crossref] [PubMed]

2014 (11)

R. Porras-Aguilar, M. Kujawinska, and W. Zaperty, “Capture and display mismatch compensation for real-time digital holographic interferometry,” Appl. Opt. 53(13), 2870–2880 (2014).
[Crossref] [PubMed]

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” Proc. SPIE 9271, 927128 (2014).
[Crossref]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39(1), 30–33 (2014).
[Crossref] [PubMed]

M. H. Jenkins, J. M. Long, and T. K. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53(16), D29–D39 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum plane selection for transport-of-intensity-equation-based solvers,” Appl. Opt. 53(30), 7050–7058 (2014).
[Crossref] [PubMed]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22(9), 10661–10674 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[Crossref] [PubMed]

C. T. Koch, “Towards full-resolution inline electron holography,” Micron 63, 69–75 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22(14), 17172–17186 (2014).
[Crossref] [PubMed]

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53(34), J1–J6 (2014).
[PubMed]

2013 (7)

2012 (2)

2011 (3)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

P. Langehanenberg, G. Von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Res. 01, 1–11 (2011).

K. A. Nugent, “The measurement of phase through the propagation of intensity: an introduction,” Contemp. Phys. 52(1), 55–69 (2011).
[Crossref]

2010 (3)

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010).

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref] [PubMed]

2009 (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. 11, 54008A (2009).

2008 (1)

2004 (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

2001 (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

1998 (1)

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

1992 (1)

1990 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1983 (1)

1982 (1)

Asundi, A.

Balbekin, N. S.

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” Proc. SPIE 9271, 927128 (2014).
[Crossref]

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Bie, R.

Börrnert, F.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[Crossref] [PubMed]

Candès, E. J.

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66(8), 1241–1274 (2013).
[Crossref]

Chen, Q.

Claus, R. A.

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

Connolly, B.

Dauwels, J.

Falaggis, K.

Fienup, J. R.

Gaylord, T. K.

Gureyev, T. E.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Guzzinati, G.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[Crossref] [PubMed]

Huang, L.

Ivanov, V. Y.

Jenkins, M. H.

Jingshan, Z.

Jozwik, M.

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

Kemper, B.

P. Langehanenberg, G. Von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Res. 01, 1–11 (2011).

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47(4), A52–A61 (2008).
[Crossref] [PubMed]

Kim, M. K.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010).

Koch, C. T.

C. T. Koch, “Towards full-resolution inline electron holography,” Micron 63, 69–75 (2014).
[Crossref] [PubMed]

Kozacki, T.

Kujawinska, M.

Langehanenberg, P.

P. Langehanenberg, G. Von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Res. 01, 1–11 (2011).

Langer, M.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

Long, J. M.

Lubk, A.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[Crossref] [PubMed]

Martinez-Carranza, J.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Neureuther, A.

Nugent, K.

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Nugent, K. A.

K. A. Nugent, “The measurement of phase through the propagation of intensity: an introduction,” Contemp. Phys. 52(1), 55–69 (2011).
[Crossref]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Paganin, D. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

Pavlov, K. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Petrov, N. V.

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” Proc. SPIE 9271, 927128 (2014).
[Crossref]

Petruccelli, J. C.

Peyrin, F.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

Porras-Aguilar, R.

Qu, W.

Reed Teague, M.

Roddier, F.

Schmalz, J. A.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Sczyrba, M.

Shanker, A.

Shevkunov, I. A.

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” Proc. SPIE 9271, 927128 (2014).
[Crossref]

Sivokon, V. P.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

Strohmer, T.

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66(8), 1241–1274 (2013).
[Crossref]

Tian, L.

Towers, C. E.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. 11, 54008A (2009).

Towers, D. P.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. 11, 54008A (2009).

Verbeeck, J.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[Crossref] [PubMed]

Von Bally, G.

P. Langehanenberg, G. Von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Res. 01, 1–11 (2011).

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47(4), A52–A61 (2008).
[Crossref] [PubMed]

Voroninski, V.

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66(8), 1241–1274 (2013).
[Crossref]

Vorontsov, M. A.

Waller, L.

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

Yu, Y.

Yuan, X.-H.

Zaperty, W.

Zhang, L.

Zhao, M.

Zuo, C.

3D Res. (1)

P. Langehanenberg, G. Von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Res. 01, 1–11 (2011).

Appl. Opt. (7)

Commun. Pure Appl. Math. (1)

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66(8), 1241–1274 (2013).
[Crossref]

Contemp. Phys. (1)

K. A. Nugent, “The measurement of phase through the propagation of intensity: an introduction,” Contemp. Phys. 52(1), 55–69 (2011).
[Crossref]

IEEE Trans. Image Process. (1)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

J. Opt. (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. 11, 54008A (2009).

J. Opt. Soc. Am. (1)

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

Micron (1)

C. T. Koch, “Towards full-resolution inline electron holography,” Micron 63, 69–75 (2014).
[Crossref] [PubMed]

Opt. Commun. (2)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[Crossref]

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[Crossref]

Opt. Express (7)

C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22(14), 17172–17186 (2014).
[Crossref] [PubMed]

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C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013).
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[Crossref] [PubMed]

C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter--theory and applications,” Opt. Express 21(5), 5346–5362 (2013).
[Crossref] [PubMed]

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Opt. Lett. (5)

Phys. Rev. A (1)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

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[Crossref]

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[Crossref]

Proc. SPIE (2)

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” Proc. SPIE 9271, 927128 (2014).
[Crossref]

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M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010).

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Figures (8)

Fig. 1
Fig. 1 a) IPTFTIE for three different defocus distances and b) RMSE frequency response of the retrieved phases. The vertical lines are placed in its corresponding COF. The lines labeled with the legend RMSES correspond to the RMSE of the retrieved phase through simulations, and the theoretical RMSE (RMSET) are denoted with the solid lines. Moreover, the analytical results of Eq. (11) and the position of the COFs are plotted with the horizontal and the vertical dashed lines respectively. The RMSES presented in this figure are the average of 25 simulations.
Fig. 2
Fig. 2 Flow chart of the MF-TIE based solver.
Fig. 3
Fig. 3 RMSET vs ε using several phase difference amplitudes and different levels of noise. The red, black and green circles correspond to the phase difference amplitudes of π/4, π/8 and π/16 respectively. a) SNR = 30dB. b) SNR = 40dB c) SNR = 50dB. The solid line is the noise boundary limit given by Eq. (26). The circles are the actual RMSET when employing the MF-TIE. The triangle, pentagon, star and square markers indicate the number of planes necessary to retrieve the phase with a constant value of the RMSET (dash line).
Fig. 4
Fig. 4 RMSE vs for various grating frequencies when using different plane selection strategies for the GP-TIE and MF-TIE for several levels of noise. a) Plane separation strategy defined in Eq. (17). b) Separation strategy defined in Eq. (25). The vertical lines are placed on its corresponding COF. The final results of these plots are the averages of 15 simulations.
Fig. 5
Fig. 5 RMSE of the retrieved phase for various levels of noise. Three different TIE solvers have been employed: the Classical TIE solver (CL-TIE) [27] with optimal equally spaced planes (triangles) [22], the GP-TIE solver [23] with exponential spacing (cross) and optimized exponential spacing (asterisk), and the MF-TIE with equal noise sensitivity strategy of Eq. (25) (circle). These simulations were carried out with eleven planes. Each marker is the average of 20 simulations.
Fig. 6
Fig. 6 Phase retrieval for three different solvers and employing the unequal phase separation defined by Eq. (25)
Fig. 7
Fig. 7 Phase retrieval in case of absorption for SNR = 50dB. a) Absorption distribution μ. b) Retrieved phase with MF-TIE. c) Retrieved phase with GP-TIE.
Fig. 8
Fig. 8 RMSE of the retrieved phase for various values of μ. a) RMSE of the retrieved phase when the in-focus intensity is not constant. b) RMSE of the retrieved phase after employing the iterative TIE algorithm showed of reference [36]. These simulations were made with SNR = 60dB, n = 5. Each marker is the average of 25 simulations.

Tables (1)

Tables Icon

Table 1 Defocus distances for various SNR, ε and RMSET

Equations (30)

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· [ I 0 φ ] = k I ( r , z ) I ( r , z ) 2 z ,
φ = FT 1 I ^ 0 [ FT ( A ( r , z ) ) 4 π λ f 2 z ] ,
PTF T I E = π λ f 2 z ,
PTF= sin ( π λ f 2 z ) .
PTF PTF T I E = 1 ε ,
ζ ( ε , z ) = 6 ε π λ z .
RMSE 2 = K 1 z 2 c 0 + z 4 K 2 W | F T 1 [ f 2 [ z 3 I ˜ | z = 0 ] ] | 2 d 2 r ,
I ˜ ( f , z ) = δ ( f ) + 2 M cos ( π λ f 2 z ) 2 Φ sin ( π λ f 2 z ) ,
3 z 3 I ˜ ( f , z ) | z = 0 = 2 ( π λ f 2 ) 3 Φ .
RMSE 2 = C 1 [ σ 2 ( π λ ) 2 ζ 4 6 ε + C 2 C 1 ( 6 ε ( π λ ) 2 ) 2 ( f j ζ ) 8 ] ,
RMSE 2 = C 1 [ σ 2 ( π λ ) 2 ζ 4 6 ε 1 ] .
0 < ζ ( z N ) < ζ ( z N 1 ) < < ζ ( z 2 ) f m a x ,
φ i = F T 1 [ F ( i ) ( A ( z i ) ) 4 π λ z i f 2 I 0 ] .
φ ( x , y ) = i = 1 N φ i ( x , y ) .
RMSE i 2 = ( k σ π ) 2 1 32 π 2 z i 2 1 W 2 W | FT 1 [ F ( i ) / f 2 ] | 2 d 2 r + z i 4 K 2 W | FT 1 [ [ F ( i ) × z 3 I ˜ / f 2 ] ] | 2 d 2 r .
RMSE i 2 = ( k σ π ) 2 1 32 π 2 z i 2 α i ,
z j = g 0 j 1 Δ z ,
RMSE N 2 = ( k σ π ) 2 1 32 π 2 z N 2 α N ,
α N = W 2 Δ x 2 π [ 2 Δ f 2 π λ z N ( 6 ε ) 1 / 2 ] ,
z N = 2 1 ( RMSE N 2 K 1 π ) [ b 2 6 ε + b 2 2 6 ε + 8 ( RMSE N 2 K 1 π ) ] ,
α N-1 = W 2 Δ x 2 π [ ζ N 2 ζ N 1 2 ] ,
RMSE N 2 = K 1 π z N 1 2 Δ f 2 [ ζ N 2 ζ N 1 2 ] .
z N 1 2 + Γ [ z N 1 z N ] = 0 ,
z N 1 = 2 1 ( Γ + Γ 2 + 4 z N Γ ) .
z i 1 = 2 1 ( Γ + Γ 2 + 4 z i Γ ) .
RMSE T = N RMSE N ,
ψ = k I ( r , z ) I ( r , z ) 2 z .
φ = 2 { [ I 1 ψ ] } ,
ψ = i = 1 N ψ i .
φ = i = 1 N φ i = i = 1 N 2 { [ I 1 ψ i ] }

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