Abstract

We apply a nonlocal adaptive spectral transform for sparse modeling of phase and amplitude of a coherent wave field. The reconstruction of this wave field from complex-valued Gaussian noisy observations is considered. The problem is formulated as a multiobjective constrained optimization. The developed iterative algorithm decouples the inversion of the forward propagation operator and the filtering of phase and amplitude of the wave field. It is demonstrated by simulations that the performance of the algorithm, both visually and numerically, is the current state of the art.

© 2012 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  2. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).
  3. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).
    [PubMed]
  4. J. L. Starck, F. Murtagh, and J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University, 2010).
    [CrossRef]
  5. 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 August 2010) .
  6. V. Katkovnik, A. Danielyan, and K. Egiazarian, “Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2011).
  7. A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. (to be published).
    [PubMed]
  8. 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]
  9. 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]
  10. P. Chatterjee and P. Milanfar, “Is denoising dead?” IEEE Trans. Image Process. 19, 895–911 (2010).
    [CrossRef]
  11. V. Katkovnik and J. Astola, “Beyond the diffraction limits: inverse numerical imaging based on transform-domain sparse modeling for phase and amplitude,” presented at the 10th Euro-American Workshop on Information Optics (WIO-2011) Benicàssim, Spain (19–24 June 2011).
  12. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef] [PubMed]
  13. V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
    [CrossRef] [PubMed]
  14. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt. 47, 3481–3493 (2008).
    [CrossRef] [PubMed]
  15. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP,, 1998).
    [CrossRef]
  16. 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]
  17. 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]
  18. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  19. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
    [CrossRef]
  20. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [CrossRef] [PubMed]
  21. 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]
  22. 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]
  23. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6, 506–509 (2010).
    [CrossRef]
  24. Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646(2010).
    [CrossRef]
  25. D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates (Student Mathematical Library, 2007).
  26. D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).
  27. M. J. Osborne, An Introduction to Game Theory (Oxford University, 2003).
  28. F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Ann. Oper. Res. , 5, 173–210 (2010).
    [CrossRef]
  29. 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.

2010

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]

P. Chatterjee and P. Milanfar, “Is denoising dead?” IEEE Trans. Image Process. 19, 895–911 (2010).
[CrossRef]

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

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Ann. Oper. Res. , 5, 173–210 (2010).
[CrossRef]

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]

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646(2010).
[CrossRef]

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]

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

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef] [PubMed]

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]

Astola, J.

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, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef] [PubMed]

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

V. Katkovnik and J. Astola, “Beyond the diffraction limits: inverse numerical imaging based on transform-domain sparse modeling for phase and amplitude,” presented at the 10th Euro-American Workshop on Information Optics (WIO-2011) Benicàssim, Spain (19–24 June 2011).

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP,, 1998).
[CrossRef]

Bertsekas, D. P.

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

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP,, 1998).
[CrossRef]

Brady, D. J.

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.

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]

Chatterjee, P.

P. Chatterjee and P. Milanfar, “Is denoising dead?” IEEE Trans. Image Process. 19, 895–911 (2010).
[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, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. (to be published).
[PubMed]

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

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 August 2010) .

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 wavefield 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]

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. (to be published).
[PubMed]

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

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 August 2010) .

Elad, M.

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

Eldar, Y. C.

Facchinei, F.

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Ann. Oper. Res. , 5, 173–210 (2010).
[CrossRef]

Fadili, J.

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

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]

Gazit, S.

Goodman, J. W.

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

Hahn, J.

Han, D.

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

Horisaki, R.

Javidi, B.

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

Kanzow, C.

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Ann. Oper. Res. , 5, 173–210 (2010).
[CrossRef]

Katkovnik, V.

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, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield 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]

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. (to be published).
[PubMed]

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

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 August 2010) .

V. Katkovnik and J. Astola, “Beyond the diffraction limits: inverse numerical imaging based on transform-domain sparse modeling for phase and amplitude,” presented at the 10th Euro-American Workshop on Information Optics (WIO-2011) Benicàssim, Spain (19–24 June 2011).

Kornelson, K.

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

Kreis, T.

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

Lam, E. Y.

Larson, D.

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

Lim, S.

Mait, J. N.

Marks, D. L.

Mattheiss, M.

Migukin, A.

Milanfar, P.

P. Chatterjee and P. Milanfar, “Is denoising dead?” IEEE Trans. Image Process. 19, 895–911 (2010).
[CrossRef]

Murtagh, F.

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

Osborne, M. J.

M. J. Osborne, An Introduction to Game Theory (Oxford University, 2003).

Rivenson, Y.

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

Schulz, T. J.

Segev, M.

Shen, F.

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 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]

Wang, A.

Weber, E.

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

Wikner, D. A.

Xu, Z.

Ann. Oper. Res.

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Ann. Oper. Res. , 5, 173–210 (2010).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Process.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. (to be published).
[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]

P. Chatterjee and P. Milanfar, “Is denoising dead?” IEEE Trans. Image Process. 19, 895–911 (2010).
[CrossRef]

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. Opt. Soc. Am. A

Opt. Express

Other

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

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

M. J. Osborne, An Introduction to Game Theory (Oxford University, 2003).

V. Katkovnik and J. Astola, “Beyond the diffraction limits: inverse numerical imaging based on transform-domain sparse modeling for phase and amplitude,” presented at the 10th Euro-American Workshop on Information Optics (WIO-2011) Benicàssim, Spain (19–24 June 2011).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP,, 1998).
[CrossRef]

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

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

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]

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 August 2010) .

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

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.

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Figures (7)

Fig. 1
Fig. 1

Phase modulation object with the binary phase (chessboard test image), noiseless data and reconstruction by the ASD algorithm. 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: wiggles, waves, etc., are clearly seen.

Fig. 2
Fig. 2

Phase modulation object with the binary phase (chessboard test image), noiseless data and reconstruction by the proposed SPAR algorithm. The visual quality of the reconstruction is perfect: wiggles, waves, and other artifacts are completely wiped out. In the cross sections, the difference between the true signal and the reconstructions is not seen.

Fig. 3
Fig. 3

Amplitude modulation by the binary chessboard test image and the phase modulation by the gray-scale spatially varying cameraman test image (noiseless data) and reconstruction by the ASD algorithm. In the cross sections, thick (red) and thin (blue) curves show the true signal and the reconstructions, respectively. The visual quality of the reconstructions is very poor: artifacts as well as the chessboard squares (from amplitude modulation) are clearly seen in the phase reconstruction.

Fig. 4
Fig. 4

Amplitude modulation by the binary chessboard test image and the phase modulation by the gray-scale cameraman test image (noiseless data) and reconstruction by the proposed SPAR algorithm. The visual quality of the reconstructions is almost perfect. In the cross sections, a difference between the true signal and the reconstructions is slightly seen only in the phase reconstruction.

Fig. 5
Fig. 5

Phase modulation by the gray-scale cameraman test image (noisy data) and reconstruction by the ASD algorithm. In the cross sections, thick (red) and thin (blue) curves show the true signal and the reconstructions, respectively. The visual quality of the reconstructions is poor: artifacts and noise are seen.

Fig. 6
Fig. 6

Phase modulation by the gray-scale cameraman test image (noisy data) and reconstruction by the proposed SPAR algorithm. RMSE values for the phase and the amplitude are shown at the top. The visual quality of the reconstruction is quite good: wiggles, waves, other artifacts, and noise are nearly wiped out. The noise level in the reconstructions is quite low in comparison with what is achieved in Fig. 5.

Fig. 7
Fig. 7

RMSE for the amplitude and phase reconstructions by the proposed SPAR algorithm versus the number of iterations and phase modulation object with the binary phase (chessboard test image; noiseless data).

Tables (2)

Tables Icon

Table 1 RMSE Values for the Phase Reconstruction a

Tables Icon

Table 2 RMSE Values for the Phase Reconstruction a

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

u z = A z u 0 ,
y = A z u 0 + σ ε ,
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 ) ,
1 μ y A z · u 0 2 2 + τ a · θ a p + τ φ · θ φ p ,
u ^ 0 = arg min u 0 1 μ y A z · u 0 2 2 ,
subject to   u 0 Ψ a θ ^ a exp ( j Ψ φ θ ^ φ ) 2 2 ε 1 ,
( θ ^ a , θ ^ φ ) = arg min θ a , θ φ τ a · θ a l p + τ φ · θ φ l p ,
subject to   θ a Φ a · abs ( u ^ 0 ) 2 2 ε 2 , θ φ Φ φ · angle ( u ^ 0 ) 2 2 ε 3 .
L 1 ( u 0 , θ a , θ φ ) = 1 μ y A z · u 0 2 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 l p + τ φ · θ φ l p +
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 = arg min u 0 L 1 ( u 0 , θ ^ a , θ ^ φ ) , ( θ ^ a , θ ^ φ ) = arg min θ a , θ φ L 2 ( u ^ 0 , θ a , θ φ ) .
u 0 t + 1 = arg min u 0 L 1 ( u 0 , θ a t , θ φ t ) ,
( θ a t + 1 , θ φ t + 1 ) = arg min θ a , θ φ L 2 ( u 0 t + 1 , θ a , θ φ ) , t = 0 , 1 ,
θ a = T h τ a γ a { Φ a · abs ( u 0 ) } , θ φ = T h τ φ γ φ { Φ φ · angle ( u 0 ) } ,
u 0 = ( 1 μ A z H A z + 1 γ 0 I n × n ) - 1 ( 1 μ A z H y + 1 γ 0 v 0 ) ,
v 0 = Ψ a θ a exp ( j Ψ φ θ φ ) ,
U ˜ z ( f ) = A ˜ z ( f ) U ˜ 0 ( f ) , u ˜ z = FFT - 1 ( U ˜ z ( f ) ) , u z = u ˜ z ( center ) ,
U ˜ 0 ( f ) = ( 1 μ A ˜ z * ( f ) Y ˜ ( f ) + 1 γ 0 V ˜ 0 ) / ( 1 μ | A ˜ z ( f ) | 2 + 1 γ 0 ) , u ˜ 0 = FFT 1 ( U ˜ 0 ( f ) ) , u 0 = u ˜ 0 ( center ) ,
U ˜ RI ( f ) = 1 μ A ˜ * ( f ) Y ˜ ( f ) / ( 1 μ | A ˜ ( f ) | 2 + 1 γ ) , u ˜ 0 RI = FFT 1 ( U ˜ RI ( f ) ) , u 0 RI = u ˜ 0 RI ( center ) ,
Input y , A ˜ ( f ) , u 0 RI .
Using   u 0 RI   construct transforms   Φ   and   Ψ .
Set   t = 0 , u 0 0 = u 0 RI , u z 0 = y , v 0 0 = y .
1 μ A z H ( A z u 0 y ) + 1 γ 0 ( u 0 v 0 ) = 0
u ^ 0 = ( 1 μ A z H A z + 1 γ 0 I n × n ) - 1 × [ 1 μ A z H y + 1 γ 0 v 0 ] .
min θ τ · θ p + 1 2 θ - B 2 2 ,
θ i = arg min θ i τ · θ i 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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