Abstract

A variational algorithm to object wavefront reconstruction from noisy intensity observations is developed for the off-axis holography scenario with imaging in the acquisition plane. The algorithm is based on the local least square technique proposed in paper [J. Opt. Soc. Am. A 21, 367 (2004)]. First, multiple reconstructions of the wavefront are produced for various size and various directional windows applied for localization of estimation. At the second stage, a special statistical rule is applied in order to select the best window size estimate for each pixel of the image and for each of the directional windows. At the third final stage the estimates of the different directions obtained for each pixel are aggregated in the final one. Simulation experiments and real data processing prove that the developed algorithm demonstrate the performance of the extraordinary quality and accuracy for both the phase and amplitude of the object wavefront.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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2016 (2)

A. V. Belashov, N. V. Petrov, and I. V. Semenova, “Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors,” Appl. Opt. 55, 81–88 (2016).
[Crossref] [PubMed]

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016).

2015 (5)

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

T. Y. Nikolaeva and N. V. Petrov, “Characterization of particles suspended in a volume of optical medium at high concentrations by coherent image processing,” Opt. Eng. 54, 083101 (2015).
[Crossref]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015).
[Crossref] [PubMed]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015).
[Crossref]

2014 (3)

2013 (4)

2012 (1)

2011 (2)

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
[Crossref]

P. Langehanenberg, G. von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11 (2011).
[Crossref]

2010 (2)

M. K. Kim, “Principles and techniques of digital holographic microscopy,” J. Photon. Energy 1, 018005 (2010).
[Crossref]

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

2007 (1)

2005 (1)

D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005).
[Crossref]

2004 (1)

2002 (1)

V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002).
[Crossref]

2000 (1)

1999 (1)

V. Katkovnik, “A new method for varying adaptive bandwidth selection,” IEEE Trans Sig. Process. 47, 2567–2571 (1999).
[Crossref]

1997 (2)

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Springer Ser. Opt. Sci. 22, 1268–1270 (1997).

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[Crossref]

1996 (1)

1946 (1)

N. Levinson, “The Wiener (root mean square) error criterion in filter design and prediction,” J. Math. Phys. 24, 261–278 (1946).
[Crossref]

Ali, P. T. S.

Astola, J.

V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002).
[Crossref]

V. Katkovnik, K. Egiazarian, and J. Astola, “Local approximation techniques in signal and image processing,” (SPIE, 2006).
[Crossref]

Belashov, A.

Belashov, A. V.

Blu, T.

Bouman, C. A.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016).

Caprio, G. D.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Chang, X.

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

Chegal, W.

Cheng, C.-J.

Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013).
[Crossref]

Coppola, G.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Cuche, E.

Depeursinge, C.

Depeursinge., C.

Doblas, A.

Egiazarian, K.

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015).
[Crossref] [PubMed]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015).
[Crossref]

V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002).
[Crossref]

V. Katkovnik, K. Egiazarian, and J. Astola, “Local approximation techniques in signal and image processing,” (SPIE, 2006).
[Crossref]

Ekinci, K. L.

D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005).
[Crossref]

Faridian, A.

Ferraro, P.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Finke, G.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Fu, Y.

Gao, P.

Garcia-Sucerquia, J.

Hennelly, B. M.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Hu, X.

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

Huntley, J. M.

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[Crossref]

Jin, M.

Joseph, J.

Karabacak, D.

D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005).
[Crossref]

Katkovnik, V.

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015).
[Crossref] [PubMed]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015).
[Crossref]

V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002).
[Crossref]

V. Katkovnik, “A new method for varying adaptive bandwidth selection,” IEEE Trans Sig. Process. 47, 2567–2571 (1999).
[Crossref]

V. Katkovnik, K. Egiazarian, and J. Astola, “Local approximation techniques in signal and image processing,” (SPIE, 2006).
[Crossref]

Kelly, D. P.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Kemper, B.

P. Langehanenberg, G. von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11 (2011).
[Crossref]

Khare, K.

Kim, D.

Kim, M. K.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” J. Photon. Energy 1, 018005 (2010).
[Crossref]

Körner, K.

Kouh, T.

D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005).
[Crossref]

Kozacki, T.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Kujawinska, M.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Langehanenberg, P.

P. Langehanenberg, G. von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11 (2011).
[Crossref]

Lee, D. J.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016).

Lee, J.

Lee, Y.-L.

Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013).
[Crossref]

Lei, H.

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

Levinson, N.

N. Levinson, “The Wiener (root mean square) error criterion in filter design and prediction,” J. Math. Phys. 24, 261–278 (1946).
[Crossref]

Liebling, M.

Lin, Y.-C.

Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013).
[Crossref]

Magnusson, R.

Marquet, P.

Martínez-Corral, M.

Memmolo, P.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Miccio, L.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Michalkiewicz, A.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Molin, N.-E.

Monaghan, D. S.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Naik, D.

Netti, P. A.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Nikolaeva, T. Y.

T. Y. Nikolaeva and N. V. Petrov, “Characterization of particles suspended in a volume of optical medium at high concentrations by coherent image processing,” Opt. Eng. 54, 083101 (2015).
[Crossref]

Osten, W.

Pandey, N.

D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010).
[Crossref]

Paturzo, M.

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015).
[Crossref]

Pavillon, N.

Pedrini, G.

Petrov, N.

Petrov, N. V.

A. V. Belashov, N. V. Petrov, and I. V. Semenova, “Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors,” Appl. Opt. 55, 81–88 (2016).
[Crossref] [PubMed]

T. Y. Nikolaeva and N. V. Petrov, “Characterization of particles suspended in a volume of optical medium at high concentrations by coherent image processing,” Opt. Eng. 54, 083101 (2015).
[Crossref]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015).
[Crossref] [PubMed]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015).
[Crossref]

Saavedra, G.

Saldner, H. O.

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[Crossref]

H. O. Saldner, N.-E. Molin, and K. A. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased tv holograms,” Appl. Opt. 35, 332–336 (1996).
[Crossref] [PubMed]

Sánchez-Ortiga, E.

Seelamantula, C. S.

Semenova, I.

Semenova, I. V.

Shevkunov, I. A.

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015).
[Crossref]

V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015).
[Crossref] [PubMed]

Singh, A. K.

Stetson, K. A.

Takeda, M.

Tu, H.-Y.

Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013).
[Crossref]

Unser, M.

von Bally, G.

P. Langehanenberg, G. von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11 (2011).
[Crossref]

Weiner, A. M.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016).

Wilke, M.

Yamaguchi, I.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Springer Ser. Opt. Sci. 22, 1268–1270 (1997).

Zeng, Y.

Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015).
[Crossref] [PubMed]

Zhang, T.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Springer Ser. Opt. Sci. 22, 1268–1270 (1997).

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[Crossref]

Adv. Opt. Photonics (1)

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[Crossref]

Appl. Opt. (6)

Electronic Imaging (1)

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016).

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[Crossref]

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[Crossref]

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D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005).
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J. Math. Imaging Vision (1)

V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002).
[Crossref]

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Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1

Bias-variance balance with respect to the window size h.

Fig. 2
Fig. 2

Block-diagram of the ICI-LLS algorithm.

Fig. 3
Fig. 3

Adaptive windows.

Fig. 4
Fig. 4

Noisy holograms of different SNR simulated for the step-wise phase object.

Fig. 5
Fig. 5

Noiseless step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

Fig. 6
Fig. 6

Noisy (low noise level, SNR=40) step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

Fig. 7
Fig. 7

Noisy (high noise level, SNR=5 dB) step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

Fig. 8
Fig. 8

Adaptive varying window sizes obtained by the ICI algorithm for the cameraman phase object. These images of pixel-wise optimal window sizes correspond to the directional sectorial windows of the directions: first row left-to-right 0, π/3, 2π/3, second row left-to-right π, 4π/3, 5π/3.

Fig. 9
Fig. 9

Cameraman phase object reconstruction results:images (first row) and horizontal middle line cross-sections (second row).The cross-section of the initial phase is omitted because the difference with ICI-LLS is too small.

Fig. 10
Fig. 10

LLS phase reconstructions for cameraman phase object with different sizes of the processing windows.

Fig. 11
Fig. 11

Comparative reconstructions of the step-wise phase from very noisy hologram (SNR=5): FT, ICI-LLS and SPAR algorithms. The advantage of ICI-LLS is obvious.

Fig. 12
Fig. 12

Comparison of PSDH, LLS and ICI-LLS reconstructions for real data experiment: images (first row) and horizontal middle line cross-sections (second row).

Tables (3)

Tables Icon

Table 1 Window parameters for ICI-LLS.

Tables Icon

Table 2 Root Mean Square Errors (RMSE) for phase step object.

Tables Icon

Table 3 Root Mean Square Errors (RMSE) for cameraman phase object.

Equations (27)

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u s = B o exp ( j φ o ) + A r exp ( j φ r ) ,
φ r = 2 π ( x sin α x + y sin α y ) / λ .
I = | B o exp ( j φ o ) + A r exp ( j φ r ) | 2 = B o 2 + A r 2 + B o A r ( exp ( j ( φ o φ r ) ) + exp ( j ( φ o φ r ) ) ) .
U = | B o | 2 + | A r | 2 and Z = u o A r .
I = U + ( V * Z + Z * V ) ,
A r = ( U ± U 2 4 | Z | 2 2 ) 1 / 2 ,
B o = | Z | / A r .
Y = I + σ ε ,
J m = t X m w ( t ) [ Y ( t ) ( U + ( V * ( t ) Z + Z * V ( t ) ) ) ] 2 .
[ 1 α α * α * 1 β * α β 1 ] [ U ^ Z ^ Z ^ * ] = [ c 1 c 2 c 3 ] ,
α = t X m w ( t ) V * ( t ) β = t X m w ( t ) ( V * ( t ) ) 2 , c 1 = t X m w ( t ) Y ( t ) c 2 = t X m w ( t ) V ( t ) Y ( t ) c 3 = t X m w ( t ) V * ( t ) Y ( t ) .
A b ^ = c , A = [ 1 α α * α * 1 β * α β 1 ] , b ^ = [ U ^ Z ^ Z ^ * ] , c = [ c 1 c 2 c 3 ]
E { Δ b ^ } = b E { b ^ } = b A 1 E { c } E { c } = [ t X m w ( t ) I ( t ) , t X m w ( t ) V ( t ) I ( t ) , t X m w ( t ) V * ( t ) I ( t ) ] T ,
A e 0 = Δ c ,
Δ c = ( Δ c 1 , Δ c 2 , Δ c 3 ) T Δ c 1 = t X m w ( t ) ε ( t ) , Δ c 2 = t X m w ( t ) V ( t ) ε ( t ) , Δ c 3 = t X m w ( t ) V * ( t ) ε ( t ) .
cov e 0 = E { e 0 e 0 H } = A 1 E { Δ c Δ c H } A 1 = A 1 cov Δ c A 1 .
cov Δ c = σ 2 [ q X m w 2 ( t ) q X m w 2 ( t ) V * ( t ) q X m w 2 ( t ) V ( t ) q X m w 2 ( t ) V ( t ) q X m w 2 ( t ) q X m w 2 ( t ) V 2 ( t ) q X m w 2 ( t ) V * ( t ) q X m w 2 ( t ) ( V * ( t ) ) 2 q X m w 2 ( t ) ] .
cov e 0 ( 1 , 1 ) = σ U 2 , cov e 0 ( 2 , 2 ) = σ Z 2 , cov e 0 ( 3 , 3 ) = σ Z * 2 = σ Z 2 .
[ Δ U Δ Z Δ Z * ] = Δ A [ Δ φ o Δ B o Δ A r ] ,
Δ A = [ 0 2 B o 2 A r j B o exp ( j φ o ) A r exp ( j φ o ) A r B o exp ( j φ o ) j B o exp ( j φ o ) A r exp ( j φ o ) A r B o exp ( j φ o ) ] .
cov φ , B o , A r = Δ A 1 cov e 0 Δ A H
σ φ o 2 ( x ) = σ Z 2 ( x ) 2 B o 2 ( x ) A r 2 ( x ) , σ B o 2 ( x ) = 2 A r 2 ( x ) σ Z 2 ( x ) + B o 2 ( x ) σ U 2 ( x ) 4 ( A r 2 ( x ) B o 2 ( x ) ) 2 , σ A r 2 ( x ) = A r 2 ( x ) σ U 2 ( x ) + 2 B o 2 ( x ) σ Z 2 ( x ) 4 ( A r 2 ( x ) B o 2 ( x ) ) 2 .
σ φ o 2 ( x ) = σ Z 2 ( x ) B o 2 ( x ) A r 2 ( x ) + σ U 2 ( x ) 4 B o 4 ( x ) , σ B o 2 ( x ) = σ Z 2 ( x ) 4 B o 2 ( x )
h + = arg min h E { ( φ ^ o , h φ o ) 2 } .
[ h + ( x ) , φ ^ o , h + ( x ) , σ φ ^ o , h + 2 ( x , ) ] = ICI ( φ ^ o , h ( x ) , σ φ ^ o , h 2 ( x ) , h H ) ,
J m = t X m w h , d ( t ) [ Y ( t ) ( U + ( V * ( t ) Z + Z * V ( t ) ) ) ] 2 ,
φ ^ o ( x ) = d D φ ^ o , h d + , d ( x ) 1 σ φ o , h d + , d 2 ( x ) d D 1 σ φ o , h d + , d 2 ( x ) .

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