Abstract

A methodology for retrieving the unknown object distribution and point-spread functions (PSFs) from a set of images acquired in the presence of temporal phase aberrations is presented in this paper. The method works by finding optimal complimentary linear filters for multi-frame deconvolution. The algorithm uses undemanding computational operations and few a priori, making it simple, fast and robust even at low signal-to-noise ratios. Results of numerical simulations and experimental tests are given as empirical proof, alongside comparisons with other algorithms found in the literature.

© 2017 Optical Society of America

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References

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  1. 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).
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  3. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
    [Crossref]
  4. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [Crossref]
  5. G. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [Crossref]
  6. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [Crossref]
  7. L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157 (1994).
    [Crossref] [PubMed]
  8. T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
    [Crossref]
  9. C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
    [Crossref]
  10. D. Turaga and T. E. Holy, “Image-based calibration of a deformable mirror in wide-field microscopy,” Appl. Opt. 49, 2030–2040 (2010).
    [Crossref] [PubMed]
  11. P. M. Johnson, M. E. Goda, and V. L. Gamiz, “Multiframe phase-diversity algorithm for active imaging,” J. Opt. Soc. Am. A 24, 1894–1900 (2007).
    [Crossref]
  12. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series vol. 7 (MIT, 1949).
  13. P. Jansson, Deconvolution of Images and Spectra (Academic, 1997).
  14. J. Idier, ed., Bayesian Approach to Inverse Problems (Wiley, 2008).
    [Crossref]
  15. V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.
  16. F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Imag. Process. 21, 1687–1700 (2012).
    [Crossref]
  17. S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution methods: A review,” in “Blind Image Deconvolution,” (Springer, 2014), pp. 37–60.
  18. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [Crossref]
  19. D. R. Luke, “Finding best approximation pairs relative to a convex and prox-regular set in a hilbert space,” SIAM J. Optim. 19, 714–739 (2008).
    [Crossref]
  20. G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
    [Crossref]
  21. T. F. Chan and C.-K. Wong, “Convergence of the alternating minimization algorithm for blind deconvolution,” Linear Algebra Appl. 316, 259–285 (2000).
    [Crossref]
  22. A. Pakhomov and K. Losin, “Processing of short sets of bright speckle images distorted by the turbulent earth’s athmosphere,” Opt. Commun. 125, 5–12 (1996).
    [Crossref]
  23. A. Pakhomov, “Fast digital processing of images of artificial earth satellites,” J. Commun. Technol. Electron. 52, 1114–1124 (2007).
    [Crossref]
  24. M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.
  25. M. Loktev, O. Soloviev, S. Savenko, and G. Vdovin, “Speckle imaging through turbulent atmosphere based on adaptable pupil segmentation,” Opt. Lett. 36, 2656–2658 (2011).
    [Crossref] [PubMed]
  26. D. Wilding, O. Soloviev, P. Pozzi, G. Vdovin, and M. Verhaegen, “Tangential Iterative Projections (TIP) Source Code,” figshare (2017) [retrieved 7 August 2017], https://doi.org/10.6084/m9.figshare.5281087 .
  27. J. Christou and K. Hege, “Iterative Deconvolution Algorithm in C (IDAC)” (Arizona Board of Regents, 2000). http://cfao.ucolick.org/software/idac/

2015 (1)

G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
[Crossref]

2012 (1)

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Imag. Process. 21, 1687–1700 (2012).
[Crossref]

2011 (1)

2010 (1)

2008 (1)

D. R. Luke, “Finding best approximation pairs relative to a convex and prox-regular set in a hilbert space,” SIAM J. Optim. 19, 714–739 (2008).
[Crossref]

2007 (2)

A. Pakhomov, “Fast digital processing of images of artificial earth satellites,” J. Commun. Technol. Electron. 52, 1114–1124 (2007).
[Crossref]

P. M. Johnson, M. E. Goda, and V. L. Gamiz, “Multiframe phase-diversity algorithm for active imaging,” J. Opt. Soc. Am. A 24, 1894–1900 (2007).
[Crossref]

2005 (1)

T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
[Crossref]

2002 (1)

2000 (1)

T. F. Chan and C.-K. Wong, “Convergence of the alternating minimization algorithm for blind deconvolution,” Linear Algebra Appl. 316, 259–285 (2000).
[Crossref]

1998 (1)

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
[Crossref]

1996 (1)

A. Pakhomov and K. Losin, “Processing of short sets of bright speckle images distorted by the turbulent earth’s athmosphere,” Opt. Commun. 125, 5–12 (1996).
[Crossref]

1994 (1)

1993 (1)

1992 (1)

1988 (1)

1982 (2)

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

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

1972 (1)

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).

Astola, J.

V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.

Ayers, G.

Bauschke, H. H.

Caulfield, H. J.

Chan, T.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
[Crossref]

Chan, T. F.

T. F. Chan and C.-K. Wong, “Convergence of the alternating minimization algorithm for blind deconvolution,” Linear Algebra Appl. 316, 259–285 (2000).
[Crossref]

Chaudhuri, S.

S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution methods: A review,” in “Blind Image Deconvolution,” (Springer, 2014), pp. 37–60.

Combettes, P. L.

Dainty, J. C.

Egiazarian, K.

V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.

Fienup, J. R.

Gamiz, V. L.

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).

Goda, M. E.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Holy, T. E.

Hong, H.

T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
[Crossref]

Jansson, P.

P. Jansson, Deconvolution of Images and Spectra (Academic, 1997).

Johnson, P. M.

Katkovnik, V.

V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.

Li, G.

G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
[Crossref]

Loktev, M.

M. Loktev, O. Soloviev, S. Savenko, and G. Vdovin, “Speckle imaging through turbulent atmosphere based on adaptable pupil segmentation,” Opt. Lett. 36, 2656–2658 (2011).
[Crossref] [PubMed]

M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.

Losin, K.

A. Pakhomov and K. Losin, “Processing of short sets of bright speckle images distorted by the turbulent earth’s athmosphere,” Opt. Commun. 125, 5–12 (1996).
[Crossref]

Luke, D. R.

D. R. Luke, “Finding best approximation pairs relative to a convex and prox-regular set in a hilbert space,” SIAM J. Optim. 19, 714–739 (2008).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

Milanfar, P.

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Imag. Process. 21, 1687–1700 (2012).
[Crossref]

Pakhomov, A.

A. Pakhomov, “Fast digital processing of images of artificial earth satellites,” J. Commun. Technol. Electron. 52, 1114–1124 (2007).
[Crossref]

A. Pakhomov and K. Losin, “Processing of short sets of bright speckle images distorted by the turbulent earth’s athmosphere,” Opt. Commun. 125, 5–12 (1996).
[Crossref]

Paliy, D.

V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.

Paxman, R. G.

Plemmons, R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
[Crossref]

Rameshan, R.

S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution methods: A review,” in “Blind Image Deconvolution,” (Springer, 2014), pp. 37–60.

Savenko, S.

M. Loktev, O. Soloviev, S. Savenko, and G. Vdovin, “Speckle imaging through turbulent atmosphere based on adaptable pupil segmentation,” Opt. Lett. 36, 2656–2658 (2011).
[Crossref] [PubMed]

M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.

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).

Schulz, T. J.

Shen, J.

T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
[Crossref]

Soloviev, O.

M. Loktev, O. Soloviev, S. Savenko, and G. Vdovin, “Speckle imaging through turbulent atmosphere based on adaptable pupil segmentation,” Opt. Lett. 36, 2656–2658 (2011).
[Crossref] [PubMed]

M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.

Sroubek, F.

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Imag. Process. 21, 1687–1700 (2012).
[Crossref]

Turaga, D.

Vdovin, G.

M. Loktev, O. Soloviev, S. Savenko, and G. Vdovin, “Speckle imaging through turbulent atmosphere based on adaptable pupil segmentation,” Opt. Lett. 36, 2656–2658 (2011).
[Crossref] [PubMed]

M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.

Velmurugan, R.

S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution methods: A review,” in “Blind Image Deconvolution,” (Springer, 2014), pp. 37–60.

Vogel, C. R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
[Crossref]

Wen, C.

G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
[Crossref]

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series vol. 7 (MIT, 1949).

Wong, C.-K.

T. F. Chan and C.-K. Wong, “Convergence of the alternating minimization algorithm for blind deconvolution,” Linear Algebra Appl. 316, 259–285 (2000).
[Crossref]

Yaroslavsky, L. P.

Zhang, A.

G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
[Crossref]

Zhang, T.

T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
[Crossref]

Adaptive Optical System Technologies (1)

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies 3353, 994–1005 (1998).
[Crossref]

Appl. Opt. (3)

IEEE Trans. Imag. Process. (1)

F. Sroubek and P. Milanfar, “Robust multichannel blind deconvolution via fast alternating minimization,” IEEE Trans. Imag. Process. 21, 1687–1700 (2012).
[Crossref]

J. Commun. Technol. Electron. (1)

A. Pakhomov, “Fast digital processing of images of artificial earth satellites,” J. Commun. Technol. Electron. 52, 1114–1124 (2007).
[Crossref]

J. Opt. Soc. Am. A (4)

Linear Algebra Appl. (1)

T. F. Chan and C.-K. Wong, “Convergence of the alternating minimization algorithm for blind deconvolution,” Linear Algebra Appl. 316, 259–285 (2000).
[Crossref]

Opt. Commun. (1)

A. Pakhomov and K. Losin, “Processing of short sets of bright speckle images distorted by the turbulent earth’s athmosphere,” Opt. Commun. 125, 5–12 (1996).
[Crossref]

Opt. Eng. (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

T. Zhang, H. Hong, and J. Shen, “Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions,” Opt. Eng. 44, 017005 (2005).
[Crossref]

Opt. Lett. (2)

Optik (1)

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).

SIAM J. Optim. (1)

D. R. Luke, “Finding best approximation pairs relative to a convex and prox-regular set in a hilbert space,” SIAM J. Optim. 19, 714–739 (2008).
[Crossref]

Syst. Control Lett. (1)

G. Li, C. Wen, and A. Zhang, “Fixed point iteration in identifying bilinear models,” Syst. Control Lett. 83, 28–37 (2015).
[Crossref]

Other (8)

S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution methods: A review,” in “Blind Image Deconvolution,” (Springer, 2014), pp. 37–60.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series vol. 7 (MIT, 1949).

P. Jansson, Deconvolution of Images and Spectra (Academic, 1997).

J. Idier, ed., Bayesian Approach to Inverse Problems (Wiley, 2008).
[Crossref]

V. Katkovnik, D. Paliy, K. Egiazarian, and J. Astola, “Frequency domain blind deconvolution in multiframe imaging using anisotropic spatially-adaptive denoising,” in “Signal Processing Conference, 2006 14th European, ” (IEEE, 2006), pp. 1–5.

D. Wilding, O. Soloviev, P. Pozzi, G. Vdovin, and M. Verhaegen, “Tangential Iterative Projections (TIP) Source Code,” figshare (2017) [retrieved 7 August 2017], https://doi.org/10.6084/m9.figshare.5281087 .

J. Christou and K. Hege, “Iterative Deconvolution Algorithm in C (IDAC)” (Arizona Board of Regents, 2000). http://cfao.ucolick.org/software/idac/

M. Loktev, G. Vdovin, O. Soloviev, and S. Savenko, “Experiments on speckle imaging using projection methods,” in “SPIE Optical Engineering+ Applications,” J. J. Dolne, T. J. Karr, V. L. Gamiz, S. Rogers, and D. P. Casasent, eds. (International Society for Optics and Photonics, 2011), pp. 81650M.

Supplementary Material (1)

NameDescription
» Code 1       TIP Algorithm Source Code

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Figures (11)

Fig. 1
Fig. 1 An algorithm flowchart showing the steps of TIP. The different projection steps are shown in different colours and the explicit operations are given by the lines between the boxes that contain the data. P 1 is multi-frame linear deconvolution, P 2 is the projection to the feasible set O , P 3 is linear deconvolution for every frame, and P 4 is the projection to the feasible set . The details are shown in Table 1.
Fig. 2
Fig. 2 On the left, the full frame object is shown, the algorithm processes the image in this size. For clarity, a smaller region-of-interest (ROI) is reproduced on the right. This example shows the reconstruction of the TIP algorithm is very close to the object for this data set and provides an improvement over the input data. The PSFs for this dataset are shown in Fig. 3.
Fig. 3
Fig. 3 Comparison between the real PSFs, top row, used to generate the images seen in Fig. 2 and those retrieved by the TIP algorithm, bottom row. The size of the support is shown on the image, a circle with radius 11px.
Fig. 4
Fig. 4 Comparison between the object (top), an acquired frame (middle), and the TIP reconstructions (bottom) for different image types.
Fig. 5
Fig. 5 Demonstration of the convergence of the TIP algorithm by comparison of the PSNR of the estimated object at each iteration step. The algorithm shows empirical convergence in the majority of cases. Occasionally, the algorithm switches to another behaviour c.f. the top black line, the cause for this numerical instability is still unknown.
Fig. 6
Fig. 6 Comparison between the object reconstructions for different algorithms with the same a priori information. The sub-region of the first input image is shown on the left-hand side and the object on the right-hand side.
Fig. 7
Fig. 7 Comparison of algorithm performance vs. increasing noise level for four input frames with the same a priori information. The first column is the first image from the set i1. The second column is PL, the third MLE, fourth MCD, and the final column in the TIP algorithm. The object is shown in the top right hand corner.
Fig. 8
Fig. 8 The performance of the algorithm on test images of Jupiter and its moon from a small amateur telescope. The top-row shows four frames from the acquisition; the bottom row shows the PL reconstruction, the TIP reconstruction, the best frame from the sequence, and the MLE reconstruction. The TIP algorithm performs better under these conditions than the other algorithms tested.
Fig. 9
Fig. 9 The performance of the algorithm on experimental images of a crane through horizontal turbulence. The top row shows the first image from the sequence of 16 frames with high turbulence, the middle the MCD solution and the right-hand side is the TIP solution. The second row shows a low turbulence frame, not included in the processing, and a zoomed region-of-interest outlined by the yellow box.
Fig. 10
Fig. 10 A demonstration that single-frame use of the TIP algorithm yields the trivial δ-function PSF due to lack of information in the dataset. Images have been cropped.
Fig. 11
Fig. 11 An example deconvolution problem where the TIP algorithm does not perform better than another algorithm IDAC. This stellar deconvolution problem of 9 frames from [27] cannot be reduced to δ-functions with the TIP algorithm.

Tables (2)

Tables Icon

Table 1 Steps of the TIP algorithm with their descriptions

Tables Icon

Table 2 Comparison between the PSNR for the images shown in Fig. 6. *BSNR is shown for the image.

Equations (21)

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i n = o * h n + w n , i n , o , h n + M × M ,
{ h ^ n , o ^ } = arg min h n , o n = 1 N i n o * h n 2 , s . t . o , h n + M × M ,
h ^ n = i n , o ^ = δ M × M , i n = i n * δ M × M ,
I n ( v m ) = H n ( v m ) O ( v m ) + W n ( v m ) ,
{ H ^ n , O ^ } = arg min H n , O m = 1 M 2 n = 1 N | I n ( v m ) O ( v m ) H n ( v m ) | 2 , s . t . O , H n { ( f ) with f + M × M } .
O ^ = n = 1 N ( H n ) * I n n = 1 N | H n | 2 .
O ^ = n = 1 N ( H n ) * I n n = 1 N | H n | 2 + 1 SNR .
{ H ^ n , O ^ } = arg min H n , O n = 1 N I n O H n 2 , s . t . O O , H n .
{ H ^ n , O ^ } = arg min H n , O n = 1 N I n O H n 2 + λ o Q ( O ) + λ h R ( { H n } ) .
O ^ ( k ) = P O arg min O M × M n = 1 N I n O H ^ n ( k 1 ) 2 .
H ^ n ( k ) = P I n O ^ ( k ) , n = 1 , , N .
o * h n 1 = o 1 h n 1 .
h = h + h , h ( x ) = 0 , if x X , h ( x ) = 0 , if x X .
i 1 = o * h 1 = o 1 h 1 + o 1 h 1 .
h = P 1 I O ,
i / o = h + w / o = ( h + ( w / o ) ) + ( h + ( w / o ) ) ,
PSNR k = 10 log 10 ( o o ^ k 2 2 / M 2 ) ,
BSNR = 10 log 10 ( ( S μ ( S ) ) 2 ( W μ ( W ) ) 2 ) ,
i n = ( 2 16 1 ) h n * o h n * o + randp ( λ = ( 2 16 1 ) * h n * o h n * o ) + randp ( μ = 0 , σ = 2 b ) , i n = ( 2 16 1 ) i n i n .
n = 1 N | H n ( v m ) | > 0 | v m | v max ,
{ H m ( v m ) } ϵ > 0 { O ( v m ) } ϵ > 0 ,

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