Abstract

The 4f optical setup is considered with a wave field modulation by a spatial light modulator located in the focal plane of the first lens. Phase as well as amplitude of the wave field are reconstructed from noisy multiple-intensity observations. The reconstruction is optimal due to a constrained maximum likelihood formulation of the problem. The proposed algorithm is iterative with decoupling of the inverse of the forward propagation of the wave field and the filtering of phase and amplitude. The sparse modeling of phase and amplitude enables the advanced high-accuracy filtering and sharp imaging of the complex-valued wave field. Artifacts typical for the conventional algorithms (wiggles, ringing, waves, etc.) and attributed to optical diffraction can be suppressed by the proposed algorithm.

© 2012 Optical Society of America

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  36. 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 (IEEE, 2011), pp. 3514–3517.
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    [CrossRef] [PubMed]
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2012

2011

2010

2009

2008

2007

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

2006

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[CrossRef] [PubMed]

2005

2004

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797(2004).
[CrossRef]

1998

1997

1994

1993

1992

1991

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

1982

1980

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

1973

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations, J. Phys. D 6, 2200–2216 (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.

Almoro, P.

Astola, J.

Bergmann, R. B.

Bertsekas, D. P.

D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

Brady, D. J.

Brady, G. R.

Buckheit, J. B.

J. B. Buckheit and D. L. Donoho, “WaveLab and reproducible research,” Tech. Rep. 474 (Stanford University, 1995), http://www-stat.stanford.edu/~wavelab/Wavelab_850/wavelab.pdf.

Camacho, L.

Candes, E. J.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Choi, K.

Cull, C. F.

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Danielyan, A.

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

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 (IEEE, 2011), pp. 3514–3517.

Dong, B.

Donoho, D. L.

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

J. B. Buckheit and D. L. Donoho, “WaveLab and reproducible research,” Tech. Rep. 474 (Stanford University, 1995), http://www-stat.stanford.edu/~wavelab/Wavelab_850/wavelab.pdf.

Egiazarian, K.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481–3493 (2008).
[CrossRef] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

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 (IEEE, 2011), pp. 3514–3517.

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

Elad, M.

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

Eldar, Y. C.

Ersoy, O. K.

Fadili, J.

J.-L. Starck, F. Murtagh, and J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University, 2010).
[CrossRef]

Falldorf, C.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797(2004).
[CrossRef]

Fienup, J. R.

Foi, A.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Garcha, J.

Gazit, S.

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).

Glückstad, J.

J. Glückstad and D. Palima, Generalised Phase Contrast: Applications in Optics and Photonics, Springer Series in Optical Sciences (Springer, 2009), Vol.  146.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Gu, B.

Guizar-Sicairos, M.

Hahn, J.

Han, D.

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

Hanson, S.

Horisaki, R.

Irwan, R.

Ivanov, V. Y.

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6, 506–509 (2010).
[CrossRef]

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36, 1054–1058(1997).
[CrossRef] [PubMed]

Katkovnik, V.

V. Katkovnik and J. Astola, “High-accuracy wave field reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54(2012).
[CrossRef]

A. Migukin, V. Katkovnik, 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. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481–3493 (2008).
[CrossRef] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

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 (IEEE, 2011), pp. 3514–3517.

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

Kohler, C.

Kopylow, C. V.

Kornelson, K.

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

Lam, E. Y.

Lane, R. G.

R. Irwan and R. G. Lane, “Phase retrieval with prior information,” J. Opt. Soc. Am. A 15, 2302–2311 (1998).
[CrossRef]

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

Larson, D.

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

Li, J.

Lim, S.

Maallo, A. M. S.

Mait, J. N.

Marks, D. L.

Mattheiss, M.

Micy, V.

Migukin, A.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations, J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

Murtagh, F.

J.-L. Starck, F. Murtagh, and J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University, 2010).
[CrossRef]

Osten, W.

Palima, D.

J. Glückstad and D. Palima, Generalised Phase Contrast: Applications in Optics and Photonics, Springer Series in Optical Sciences (Springer, 2009), Vol.  146.

Pedrini, G.

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6, 506–509 (2010).
[CrossRef]

Rodenburg, J. M.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797(2004).
[CrossRef]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Saxton, W. O.

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

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).

Schulz, T. J.

Segev, M.

Sivokon, V. P.

Starck, J.-L.

J.-L. Starck, F. Murtagh, and J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University, 2010).
[CrossRef]

Stern, A.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6, 506–509 (2010).
[CrossRef]

Szameit, A.

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Vorontsov, M. A.

Weber, E.

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

Wikner, D. A.

Xu, Z.

Yang, G.

Zalevsky, Z.

Zhang, F.

Zhang, G.

Zhang, Y.

Zhuang, J.

Appl. Opt.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218(1994).
[CrossRef] [PubMed]

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36, 1054–1058(1997).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481–3493 (2008).
[CrossRef] [PubMed]

P. Almoro, A. M. S. Maallo, and S. Hanson, “Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing,” Appl. Opt. 48, 1485–1493 (2009).
[CrossRef] [PubMed]

C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. 48, 4003–4008 (2009).
[CrossRef] [PubMed]

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] [PubMed]

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010).
[CrossRef] [PubMed]

K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49, H1–H10 (2010).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797(2004).
[CrossRef]

IEEE Trans. Image Process.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

Int. J. Comput. Vis.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

J. Disp. Technol.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6, 506–509 (2010).
[CrossRef]

J. Microsc. Spectrosc. Electron.

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

J. Mod. Opt.

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations, J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

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).

Other

D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

J. B. Buckheit and D. L. Donoho, “WaveLab and reproducible research,” Tech. Rep. 474 (Stanford University, 1995), http://www-stat.stanford.edu/~wavelab/Wavelab_850/wavelab.pdf.

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

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 (IEEE, 2011), pp. 3514–3517.

J. Glückstad and D. Palima, Generalised Phase Contrast: Applications in Optics and Photonics, Springer Series in Optical Sciences (Springer, 2009), Vol.  146.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

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

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

J.-L. Starck, F. Murtagh, and J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University, 2010).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

4 f optical system: object, Fourier, and image planes are in focus of two lenses, optical mask (SLM) in Fourier focus plane of first lens.

Fig. 2
Fig. 2

Amplitude modulation object with the binary phase (chessboard test image). The reconstruction by the no filtering algorithm without phase/amplitude filtering. In the cross sections, thick (red) and thin (blue) curves show the true signal and the reconstructions, respectively. RMSE values for the phase and amplitude are shown at the top. The visual quality of the reconstruction is quite poor: noise, wiggles, waves, etc., are clearly seen.

Fig. 3
Fig. 3

Phase modulation object with the Lena test image. The reconstruction by the no filtering algorithm without phase/amplitude filtering. The visual quality of the reconstruction is quite poor: noise, wiggles, waves, etc., are clearly seen.

Fig. 4
Fig. 4

Amplitude modulation object with the binary phase (chessboard test image). Reconstruction by the 4 f -SPAR algorithm. The visual quality of the reconstruction is very good; noise and diffraction artifacts are wiped out. Compare with Fig. 2, where the filtering is not used.

Fig. 5
Fig. 5

Phase modulation object with the Lena test image. Reconstruction by the 4 f -SPAR algorithm. The visual quality of the reconstruction is very good, in particular versus Fig. 3, where the filtering is not used.

Fig. 6
Fig. 6

Cross sections of abs 2 ( m ˜ r [ k 1 , k 2 ] ) ) for ϰ = 1 , 0.99, 0.95, 0.5.

Tables (2)

Tables Icon

Table 1 RMSE Values for Amplitude and Phase Reconstructions a

Tables Icon

Table 2 RMSE Values for Amplitude and Phase Reconstructions a

Equations (53)

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c ( v ) = 1 j λ f F x { u 0 ( x ) } ( v λ f ) = 1 j λ f U 0 ( v λ f ) .
u r ( x ) = 1 j λ f F v { c ( v ) · M r ( v ) } ( x λ f ) .
U r ( v ) = U 0 ( v ) M r ( λ f · v ) .
U 0 [ l 1 , l 2 ] = FFT { u 0 [ k 1 , k 2 ] } k 1 = 0 N 1 1 k 2 = 0 N 2 1 u 0 [ k 1 , k 2 ] exp ( j 2 π ( k 1 l 1 / N 1 + k 1 l 2 / N 2 ) ) ,
u 0 [ k 1 , k 2 ] = FFT 1 { U 0 [ l 1 , l 2 ] } 1 N 1 N 2 l 1 = 0 N 1 1 l 2 = 0 N 2 1 U 0 [ l 1 , l 2 ] exp ( j 2 π ( k 1 l 1 / N 1 + k 1 l 2 / N 2 ) ) ,
Δ x 1 Δ v 1 λ f = 1 N 1 , Δ x 2 Δ v 2 λ f = 1 N 2 .
U r [ l 1 , l 2 ] = U 0 [ l 1 + N 1 , l 2 + N 2 ] · M r [ l 1 + N 1 , l 2 + N 2 ] , 0 l i N i 1 , i = 1 , 2.
U ˜ 0 [ l 1 , l 2 ] = U 0 [ l 1 + N 1 , l 2 + N 2 ] , M ˜ r [ l 1 , l 2 ] = M r [ l 1 + N 1 , l 2 + N 2 ] ,
U r [ l 1 , l 2 ] = U ˜ 0 [ l 1 , l 2 ] · M ˜ r [ l 1 , l 2 ] , 0 l i N i 1 , i = 1 , 2.
o r [ k 1 , k 2 ] = | u r [ k 1 , k 2 ] | 2 + σ r ε r [ k 1 , k 2 ] , r = 1 , ... , K ,
Y = Ψ θ ,
θ = Φ Y .
abs ( u 0 ) = Ψ a θ a , angle ( u 0 ) = Ψ φ θ φ ,
θ a = Φ a · abs ( u 0 ) , θ φ = Φ φ · angle ( u 0 ) ,
u 0 = Ψ a θ a exp ( j Ψ φ θ φ ) .
J = r = 1 K 1 2 σ r 2 k ( o r [ k ] | u r [ k ] | 2 ) 2 + τ φ · θ φ l p + τ a · θ a l p ,
u r [ k ] = FFT 1 { U r [ l ] } = FFT 1 { U ˜ 0 [ l ] · M ˜ r [ l ] } .
L 1 ( u ˜ 0 , { u r } , θ a , θ φ ) = r = 1 K 1 2 σ r 2 k ( o r [ k ] | u r [ k ] | 2 ) 2 +
r = 1 K 1 γ r k | u r [ k ] FFT 1 { U ˜ 0 [ l ] · M ˜ r [ l ] } | 2 +
1 γ 0 u ˜ 0 Ψ a θ a exp ( j Ψ φ θ φ ) 2 2 ,
L 2 ( u ˜ 0 , θ a , θ φ ) = τ a · θ a 0 + τ φ · θ φ 0 +
1 2 γ a θ a Φ a mod ( u ˜ 0 ) 2 2 + 1 2 γ φ θ φ Φ φ angle ( u ˜ 0 ) 2 2 ,
u ˜ 0 t + 1 = argmin u ˜ 0 , { u r } L 1 ( u ˜ 0 , { u r } , θ a t , θ φ t ) ,
( θ a t + 1 , θ φ t + 1 ) = argmin θ a , θ φ L 2 ( u ˜ 0 t + 1 , θ a , θ φ ) .
u ˜ 0 * = argmin u ˜ 0 , { u r } L 1 ( u ˜ 0 , { u r } , θ a * , θ φ * ) , ( θ a * , θ φ * ) = argmin θ a , θ φ L 2 ( u ˜ 0 * , θ a , θ φ ) .
u r t + 1 [ k ] = u r t + 1 / 2 [ k ] | u r t + 1 / 2 [ k ] | o r [ k ] .
M ˜ r [ l 1 , l 2 ] = exp ( 2 π j · z r 1 Δ v 2 ( l 1 2 + l 2 2 ) / f 2 / λ ) ,
M ˜ r [ l 1 , l 2 ] = exp ( 2 π j ψ r [ l 1 , l 2 ] ) , r = 1 , ... , K ,
u r = u ˜ 0 m ˜ r
1 N 1 k 1 = 0 N 1 1 e j 2 π k 1 l 1 N 1 = δ l 1 + N 1 η 1 ,
FFT l 1 , l 2 { U 0 [ l 1 , l 2 ] } [ k 1 , k 2 ] = l 1 = 0 N 1 1 l 2 = 0 N 2 1 U 0 [ l 1 , l 2 ] e j 2 π ( k 1 l 1 N 1 + k 2 l 2 N 2 ) = k 1 = 0 N 1 1 k 2 = 0 N 2 1 u 0 [ k 1 , k 2 ] l 1 = 0 N 1 1 l 2 = 0 N 2 1 e j 2 π ( ( k 1 + k 1 ) l 1 N 1 + ( k 2 + k 2 ) l 2 N 2 ) = N 1 N 2 k 1 = 0 N 1 1 k 2 = 0 N 2 1 u 0 [ k 1 , k 2 ] · δ k 1 + k 1 + N 1 η 1 · δ k 2 + k 2 + N 2 η 2 ,
FFT l 1 , l 2 { U 0 [ l 1 , l 2 ] } [ k 1 , k 2 ] = u 0 [ N 1 k 1 , N 2 k 2 ] , 0 k 1 N 1 1 , 0 k 2 N 2 1 .
c ( l 1 Δ v 1 , l 2 Δ v 2 ) = 1 j λ f k 1 = 0 N 1 1 k 2 = 0 N 2 1 u 0 ( k 1 Δ x 1 , k 2 Δ x 2 ) e j 2 π ( k 1 Δ x 1 · l 1 Δ v 1 λ f + k 2 Δ x 2 · l 2 Δ v 2 λ f ) Δ x 1 Δ x 2 .
c [ l 1 , l 2 ] = Δ x 1 Δ x 2 j λ f FFT { u 0 [ k 1 , k 2 ] } = Δ x 1 Δ x 2 j λ f U 0 [ l 1 , l 2 ] .
u r [ k 1 , k 2 ] = Δ v 1 Δ v 2 j λ f FFT { c [ l 1 , l 2 ] · M r [ l 1 , l 2 ] } .
U r [ l 1 , l 2 ] = FFT k 1 , k 2 { u r [ k 1 , k 2 ] } [ l 1 , l 2 ] = N 1 N 2 Δ v 1 Δ v 2 j λ f c [ l 1 + N 1 , l 2 + N 2 ] · M r [ l + N 1 , l 2 + N 2 ] .
U r [ l 1 , l 2 ] = N 1 N 2 Δ v 1 Δ v 2 j λ f · Δ x 1 Δ x 2 j λ f × U 0 [ l 1 + N 1 , l 2 + N 2 ] · M r [ l 1 + N 1 , l 2 + N 2 ] = U 0 [ l 1 + N 1 , l 2 + N 2 ] · M r [ l 1 + N 1 , l 2 + N 2 ] .
L 1 / u r * [ k ] = 1 σ r 2 ( | u r [ k ] | 2 o r [ k ] ) · u r [ k ] + 1 γ r ( u r [ k ] z r [ k ] ) = 0 ,
u r [ k ] = z r [ k ] ( | u r [ k ] | 2 o r [ k ] ) · γ r σ r 2 + 1 .
| u r [ k ] | = | z r [ k ] | | ( | u r [ k ] | 2 o r [ k ] ) · γ r σ r 2 + 1 | .
u r [ k ] = z r [ k ] ( | u ¯ r [ k ] | 2 o r [ k ] ) · γ r σ r 2 + 1 .
u r [ k ] = G ( o r [ k ] , z r [ k ] ) .
| ( | u ¯ r [ k ] | 2 o r [ k ] ) + σ r 2 / γ r | · | u ¯ r [ k ] | = | z r [ k ] | · ( σ r 2 / γ r ) .
u r [ k ] = z r [ k ] | z r [ k ] | | z r [ k ] | σ r 2 / γ r | u ¯ r [ k ] | 2 o r [ k ] + σ r 2 / γ r .
u r [ k ] z r [ k ] | z r [ k ] | | u ¯ r [ k ] | = z r [ k ] | z r [ k ] | o r [ k ] .
u r [ k ] = z r [ k ] | z r [ k ] | o r [ k ] .
L 1 = r 1 γ r n l | U r [ l ] M ˜ r [ l ] · U ˜ 0 [ l ] | 2 + 1 γ 0 n l | U ˜ 0 [ l ] V ˜ 0 [ l ] | 2 .
L 1 / U ˜ 0 * [ l ] = r 1 γ r n ( U r [ l ] M ˜ r [ l ] · U ˜ 0 [ l ] ) M ˜ r * [ l ] + 1 γ 0 n ( U ˜ 0 [ l ] V ˜ 0 [ l ] ) = 0 ,
U ˜ 0 [ l ] = r 1 γ r U r [ l ] M ˜ r * [ l ] + 1 γ 0 V ˜ 0 [ l ] r 1 γ r + 1 γ 0 .
min θ τ · θ l p + 1 2 θ B 2 2 ,
θ i = argmin θ i τ · θ i l p + 1 2 ( θ i B i ) 2 .
θ = T h τ ( B ) ,
θ = T h τ ( B ) = { T h τ soft ( B ) = sign ( B ) max ( | B | τ , 0 ) , if     l p = l 1 , T h 2 τ hard ( B ) = B 1 ( | B | 2 τ ) , if     l p = l 0 ,

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