Abstract

Generally, wave field reconstructions obtained by phase-retrieval algorithms are noisy, blurred, and corrupted by various artifacts such as irregular waves, spots, etc. These distortions, arising due to many factors, such as nonidealities of the optical system (misalignment, focusing errors), dust on optical elements, reflections, and vibration, are hard to localize and specify. It is assumed that there is a cumulative disturbance called “background,” which describes mentioned distortions in the coherent imaging system manifested at the sensor plane. Here we propose a novel iterative phase-retrieval algorithm compensating for these distortions in the optical system. An estimate of this background is obtained via special calibration experiments, and then it is used for the object reconstruction. The algorithm is based on the maximum likelihood approach targeting on the optimal object reconstruction from noisy data and imaging enhancement using a priori information on the object amplitude. In this work we demonstrate the compensation of the distortions of the optical trace for a complex-valued object with a binary amplitude. The developed algorithm results in state-of-the-art filtering, and sharp reconstruction imaging of the object amplitude can be achieved.

© 2012 Optical Society of America

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References

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  32. A. Danielyan, V. Katkovnik, and K. Egiazarian, “Image deblurring by augmented Lagrangian with BM3D frame prior,” presented at Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland, 16–18 August 2010.
  33. A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
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    [CrossRef]
  35. N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybernet. 9, 62–66 (1979).
    [CrossRef]
  36. V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
    [CrossRef]
  37. M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
    [CrossRef]
  38. M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
    [CrossRef]

2012

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).

V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012).
[CrossRef]

S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20, 452–461 (2012).
[CrossRef]

2011

2010

M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef]

2006

2005

2003

2001

G. Pedrini, S. Schedin, and H. J. Tiziani, “Aberration compensation in digital holographic reconstruction of microscopic objects,” J. Mod. Opt. 48, 1035–1041 (2001).

S. Grilli, P. Ferraro, S. D. Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express 9, 294–302 (2001).
[CrossRef]

2000

1999

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
[CrossRef]

1996

1994

1992

1982

1981

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522(1981).

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

1979

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybernet. 9, 62–66 (1979).
[CrossRef]

1978

1976

1973

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972

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

Agour, M.

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “Automated compensation of misalignment in phase retrieval based on a spatial light modulator,” Appl. Opt. 50, 4779–4787 (2011).
[CrossRef]

M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef]

M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
[CrossRef]

Almoro, P.

Arrizon, V.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
[CrossRef]

Astola, J.

V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).

A. Migukin, V. Katkovnik, and J. Astola, “Wave field reconstruction from multiple plane intensity-only data: augmented Lagrangian algorithm,” J. Opt. Soc. Am. A 28, 993–1002 (2011).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude,” arXiv:1108.3251v1.

Bergmann, R. B.

Bregmenn, R. B.

M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
[CrossRef]

Carreon, E.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
[CrossRef]

Coppola, G.

Cuche, E.

Danielyan, A.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).

V. Katkovnik, A. Danielyan, and K. Egiazarian, “Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique,” in Proceedings of the International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 3514–3517.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “Image deblurring by augmented Lagrangian with BM3D frame prior,” presented at Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland, 16–18 August 2010.

Depeursinge, C.

Ding, J.

Dong, B.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

Dorsch, R.

Egiazarian, K.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).

V. Katkovnik, A. Danielyan, and K. Egiazarian, “Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique,” in Proceedings of the International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 3514–3517.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “Image deblurring by augmented Lagrangian with BM3D frame prior,” presented at Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland, 16–18 August 2010.

Elad, M.

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

Ersoy, O. K.

Falldorf, C.

M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
[CrossRef]

M. Agour, C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “Automated compensation of misalignment in phase retrieval based on a spatial light modulator,” Appl. Opt. 50, 4779–4787 (2011).
[CrossRef]

C. Falldorf, M. Agour, C. v. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49, 1826–1830 (2010).
[CrossRef]

M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
[CrossRef]

Ferraro, P.

Fienup, J. R.

Finizio, A.

Gao, J.

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

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 35, 237–246 (1972).

Gong, L. Y.

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Grilli, S.

Gu, B.

Gureyev, T. E.

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

Han, D.

D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, AMS, 2007).

Huang, J.

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

Ivanov, V. Yu.

Katkovnik, V.

V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).

A. Migukin, V. Katkovnik, and J. Astola, “Wave field reconstruction from multiple plane intensity-only data: augmented Lagrangian algorithm,” J. Opt. Soc. Am. A 28, 993–1002 (2011).
[CrossRef]

V. Katkovnik, A. Danielyan, and K. Egiazarian, “Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique,” in Proceedings of the International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 3514–3517.

A. Migukin, V. Katkovnik, and J. Astola, “Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude,” arXiv:1108.3251v1.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “Image deblurring by augmented Lagrangian with BM3D frame prior,” presented at Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland, 16–18 August 2010.

Kopylow, C. v.

Kornelson, K.

D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, AMS, 2007).

Kreis, Th.

Th. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Larson, D.

D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, AMS, 2007).

Leach, J.

Magnus, J. R.

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed.(Wiley, 1999).

Magro, C.

Marquet, P.

Mendlovic, D.

Meucci, R.

Migukin, A.

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).

A. Migukin, V. Katkovnik, and J. Astola, “Wave field reconstruction from multiple plane intensity-only data: augmented Lagrangian algorithm,” J. Opt. Soc. Am. A 28, 993–1002 (2011).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude,” arXiv:1108.3251v1.

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Neudecker, H.

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed.(Wiley, 1999).

Nicola, S. D.

Osten, W.

Otsu, N.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybernet. 9, 62–66 (1979).
[CrossRef]

Pedrini, G.

Pierattini, G.

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 35, 237–246 (1972).

Schedin, S.

G. Pedrini, S. Schedin, and H. J. Tiziani, “Aberration compensation in digital holographic reconstruction of microscopic objects,” J. Mod. Opt. 48, 1035–1041 (2001).

Sivokon, V. P.

Testorf, M.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
[CrossRef]

Tiziani, H. J.

G. Pedrini, S. Schedin, and H. J. Tiziani, “Aberration compensation in digital holographic reconstruction of microscopic objects,” J. Mod. Opt. 48, 1035–1041 (2001).

von Kopylow, C.

M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
[CrossRef]

M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
[CrossRef]

Vorontsov, M. A.

Wang, Z.

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

Weber, E.

D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, AMS, 2007).

Xue, Q.

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

Yang, G.

Zalevsky, Z.

Zhang, Y.

Zhao, S. M.

Zheng, B. Y.

Zhuang, J.

Acta Opt. Sin.

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522(1981).

Appl. Opt.

IEEE Trans. Image Process.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).

IEEE Trans. Inf. Theory

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

IEEE Trans. Syst. Man Cybernet.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybernet. 9, 62–66 (1979).
[CrossRef]

J. Mod. Opt.

G. Pedrini, S. Schedin, and H. J. Tiziani, “Aberration compensation in digital holographic reconstruction of microscopic objects,” J. Mod. Opt. 48, 1035–1041 (2001).

J. Opt.

M. Agour, C. Falldorf, and C. von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12, 055401(2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Opt. Commun.

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 16, 207–213 (1999).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

Opt. Express

Opt. Lett.

Optik

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

Proc. SPIE

M. Agour, C. Falldorf, C. von Kopylow, and R. B. Bregmenn, “The effect of misalignment in phase retrieval based on a spatial light modulator,” Proc. SPIE 8082, 80820M (2011).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented Lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).

Q. Xue, Z. Wang, J. Huang, and J. Gao, “The elimination of the errors in the calibration image of 3D measurement with structured light,” Proc. SPIE 8430, 84300N (2012).
[CrossRef]

Other

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed.(Wiley, 1999).

Th. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, AMS, 2007).

V. Katkovnik, A. Danielyan, and K. Egiazarian, “Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique,” in Proceedings of the International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 3514–3517.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “Image deblurring by augmented Lagrangian with BM3D frame prior,” presented at Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland, 16–18 August 2010.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

A. Migukin, V. Katkovnik, and J. Astola, “Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude,” arXiv:1108.3251v1.

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Figures (11)

Fig. 1.
Fig. 1.

Experimental setup of the 4f optical system used for recording measurement data [15]. Lenses L1 and L2 in the 4f configuration provide an accurate mapping of the object wave field to the parallel observation (sensor) plane. An optical mask with the complex-valued transmittance Mr located at the Fourier plane (a phase-modulating SLM) enables linear filter operations.

Fig. 2.
Fig. 2.

Flowchart of the proposed phase-retrieval technique with the background compensation. The upper block highlighted by a dashed line represents the background calibration procedure, where the complex-valued estimate of uB is found by AL [34]. The reconstruction of the object using the background estimate is obtained by the proposed SPAR–BC algorithm.

Fig. 3.
Fig. 3.

Reconstructions of the “disturbed” object computed by (left column) AL and (right column) FA [38]. In the top row the amplitude reconstructions are presented by (a) AL and (c) FA. In the bottom row we demonstrate the phase estimates by (c) AL and (d) FA. The AL object reconstruction [u˜00=a˜00exp(i·φ˜00) with a˜00 from (a) and φ˜00 from (c)] is used for the initialization of SPAR–BC.

Fig. 4.
Fig. 4.

(a) Amplitude of the smoothed background BM3Da(abs(u^B)). (b) Initial guess for the object amplitude abs(u00) found with the smoothed background amplitude abs(u˜00)[k]/BM3Da(abs(u^B))[k].

Fig. 5.
Fig. 5.

Cross sections of the amplitude estimate of the background [thin curve, a^B=abs(u^B)] and its smoothed version computed by a BM3D filter [thick curve, BM3Da(a^B)].

Fig. 6.
Fig. 6.

Object reconstruction by SPAR–BC. The reconstruction of the object amplitude abs(u^0) with (a) smoothed background BM3Da(abs(u^B)) and (b) original background (abs(u^B)). (c) Result of postfiltering of the object amplitude BM3Da(abs(u^0)) [given in (a)], τa=0.04. (d) Object phase estimate angle(u^0) with the smoothed background.

Fig. 7.
Fig. 7.

Cross sections of (solid thin curve) the initial guess a00=abs(u00), (solid thick) the reconstructed amplitude a^0=abs(u^0), and (dashed curve) the filtered amplitude estimate BM3Da(a^0). These results are related to the imaging presented in Figs. 4(b), 6(a), and 6(c), respectively. The cross sections are given along the dashed line in Fig. 6.

Fig. 8.
Fig. 8.

Cross sections of the reconstructed object phase φ^0=angle(u^0) and the filtered phase BM3Dφ(φ^0), τφ=0.08. These results are shown along the dashed lines shown in Fig. 6(d).

Fig. 9.
Fig. 9.

Fragments (384×384) of a^0 with different updates of the object amplitude in SPAR–BC: (a) flattening and BM3D filtering (original SPAR–BC), (b) BM3D filtering only (which is similar to D–AL [24] with the background compensation), and (c) binarization only. The original SPAR–BC (a) enables a relatively flat surface with no significant noise with some oversmoothing. Exclusion of the flattening in BM3D filtering (b) results in a strong degradation of the reconstructing levels. The binarization only used to estimate a^0 (c) leads to more contrast imaging but strong corruption by impulse noise.

Fig. 10.
Fig. 10.

Cross sections of the fragments of a^0 presented in Fig. 9. Here the only use of the BM3D filter for the object amplitude update is denoted by “D–AL with BC.”

Fig. 11.
Fig. 11.

Partition produced according to the flattening with BM3D filtering [Eqs. (25)–(26)]. H(a00) is the histogram for the initial estimate of the object amplitude (solid curve, a00=abs(u00)), presented in Fig. 4(b). H(a^0) is the histogram for the resulting object amplitude estimate after 25 iterations [dashed curve, a^0=abs(u^0)], illustrated in Fig. 6(a). (ρ0, β10, β00) and (ρ^, β^1, β^0) are Otsu’s threshold, the upper and lower levels of the initial and resulting object estimates, respectively.

Equations (29)

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u˜0(x)=u0(x)·uB(x),
uF(vλf)=1iλfF{u0(x)}(vλf),
ur(y)=1iλfF{uF(vλf)·Mr(vλf)}(y).
Δv1Δx1Nx=λf,Δv2Δx2Ny=λf.
Ur[l1,l2]=1NxNyF{F{U0}Mr}[l1,l2],
ur=Ar·u˜0,r=1,K,
Mr[η1,η2]=exp(i2πλzr1Δv12|η1|2f2Δv22|η2|2f2).
a0[k]=abs(u0[k])={β1,forkX1X,β0,forkX0X,
or[l]=|ur[l]|2+εr[l],r=1,K,
θa=Φa·a0,θφ=Φφ·φ0(analysis),
a0=Ψa·θa,φ0=Ψφ·θφ(synthesis).
J1({or},u˜0,{ur},{Λr},v˜0)=r=1K1σr2[12or|ur|222+1γrurAr·u˜022+2γrRe{ΛrH·(urAr·u˜0)}]+1γ0u˜0v˜022.
J2,a(θa,a0)=12θaΦa·a022+τa·θa1,
s.t.a0[k]={β1,forkX1,β1,forkX0,
J2,φ(θφ,φ0)=12θφΦφ·φ022+τφ·θφ1.
JAL=r=1K1σr2[12orB|ur|222+1γrurAr·uB22+2γrRe{ΛrH·(urAr·uB)}]+μ·uB22.
u0t[k]=u˜0t[k]/u^B[k],
θat=argminθaJ2,a(θa,abs(u0t)),
θφt=argminθφJ2,φ(θφ,angle(u0t)),
v˜0t=Ψaθatexp(i·Ψφθφt)u^B,
urt=argminurJ1(or,u˜0t,ur,Λrt,v˜0t),
Λrt+1=Λrt+αr·(urtAr·u˜0t),r=1,K,
u˜0t+1=argminu˜oJ1({or},u˜0,{urt},{Λrt},v˜0t).
a0t+1/2=BM3Da(abs(u0t)),φ0t+1/2=BM3Dφ(angle(u0t)),v˜0t=a0t+1/2exp(i·φ0t+1/2)u^B,
X0t={a0t[k]:0a[k]0tρt},X1t={a0t[k]:a0t[k]>ρt}.
β0t=mediana0t[k]X0t(a0t[k]),β1t=mediana0t[k]X1t(a0t[k]).
a0t+1/3=BM3Da(abs(u0t)β0t)+β0t,a0t+1/2=BM3Da(a0t+1/3β1t)+β1t,
u˜0t+1=r=1KBr·(urt+1+Λrt)+κ·v˜0t+1,
Br=(r=1KArHAr+κ·In×n)1·ArH,

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