Abstract

We investigated the deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method and the depth-variant point spread function (DV-PSF). We build the DV-PSF by fitting a parameterized theoretical PSF model to an experimental microbead image. On the basis of the constructed DV-PSF, we restore the 3D widefield microscopy by minimizing an objective function consisting of a negative Poisson likelihood function and a total variation regularization function. In simulations and experiments, the proposed method showed better performance than existing methods.

© 2013 OSA

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    [CrossRef]
  25. J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process.4, 1417–1429 (1995).
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    [CrossRef]
  30. S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl.27, 095001 (2011).
    [CrossRef]
  31. P. Huber, Robust Statistics (Wiley, 1974).
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    [CrossRef]
  33. N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process.10, 1299–1308 (2001).
    [CrossRef]
  34. F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express13, 10503–10522 (2005).
    [CrossRef] [PubMed]

2013 (1)

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

2011 (1)

S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl.27, 095001 (2011).
[CrossRef]

2009 (2)

J. F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vis.34, 307–327 (2009).
[CrossRef]

S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl.25, 015002 (2009).
[CrossRef]

2007 (1)

2006 (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag.23, 32–45 (2006).
[CrossRef]

2005 (1)

2004 (2)

J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag.23, 1165–1175 (2004).
[CrossRef]

C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A21, 1593–1601 (2004).
[CrossRef]

2001 (2)

J. Markham and J. A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A18, 1062–1071 (2001).
[CrossRef]

N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process.10, 1299–1308 (2001).
[CrossRef]

1999 (2)

J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999).
[CrossRef] [PubMed]

J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A16, 2377–2391 (1999).
[CrossRef]

1997 (1)

1996 (1)

J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE2655, 199–208 (1996)
[CrossRef]

1995 (2)

A. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imag.14, 132–137 (1995).
[CrossRef]

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process.4, 1417–1429 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

P. J. Green, “On use of the EM for penalized likelihood estimation,” J. R. Stat. Soc. B52, 443–452 (1990).

1989 (1)

J. Llacer and E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imag.8, 186–193 (1989).
[CrossRef]

1982 (1)

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

1972 (1)

Aguet, F.

F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express13, 10503–10522 (2005).
[CrossRef] [PubMed]

F. Aguet, D. Van De Ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 157–160.

Anderson, J.

J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag.23, 1165–1175 (2004).
[CrossRef]

Aubert, G.

S. Ben Hadj, G. Blanc-Feraud, G. Aubert, and Engler, “Blind restoration of confocal microscopy images in presence of a depth-variant blur and poisson noise,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2013), 915–919.

Aujol, J. F.

J. F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vis.34, 307–327 (2009).
[CrossRef]

Ben Hadj, S.

S. Ben Hadj, G. Blanc-Feraud, G. Aubert, and Engler, “Blind restoration of confocal microscopy images in presence of a depth-variant blur and poisson noise,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2013), 915–919.

S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2012), 1671–1674.

S. Ben Hadj and L. Blanc-Feraud, “Modeling and removing depth variant blur in 3D fluorescence microscopy,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2012), 689–692.

Bertero, M.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

Blanc-Feraud, G.

S. Ben Hadj, G. Blanc-Feraud, G. Aubert, and Engler, “Blind restoration of confocal microscopy images in presence of a depth-variant blur and poisson noise,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2013), 915–919.

Blanc-Feraud, L.

S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2012), 1671–1674.

S. Ben Hadj and L. Blanc-Feraud, “Modeling and removing depth variant blur in 3D fluorescence microscopy,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2012), 689–692.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2004), 1223–1226

Blanc-Féraud, L.

P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 740–743.

Boccacci, P.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

Bonettini, S.

S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl.27, 095001 (2011).
[CrossRef]

S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl.25, 015002 (2009).
[CrossRef]

Cavicchioli, R.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

Chang, J. H.

J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag.23, 1165–1175 (2004).
[CrossRef]

Colicchio, B.

S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2012), 1671–1674.

Conchello, J.

J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999).
[CrossRef] [PubMed]

Conchello, J. A.

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999).
[CrossRef] [PubMed]

De Pierro, A.

A. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imag.14, 132–137 (1995).
[CrossRef]

Dey, N.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2004), 1223–1226

Dieterlen, A.

S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2012), 1671–1674.

Engler,

S. Ben Hadj, G. Blanc-Feraud, G. Aubert, and Engler, “Blind restoration of confocal microscopy images in presence of a depth-variant blur and poisson noise,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2013), 915–919.

Fessler, J.

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process.4, 1417–1429 (1995).
[CrossRef] [PubMed]

J. Fessler, “Image reconstruction: Algorithms and analysis,” Online preprint of book in preparation.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Fletcher, D.

Gibson, S. F.

Golub, G.

N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process.10, 1299–1308 (2001).
[CrossRef]

Green, P. J.

P. J. Green, “On use of the EM for penalized likelihood estimation,” J. R. Stat. Soc. B52, 443–452 (1990).

Hero, A.

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process.4, 1417–1429 (1995).
[CrossRef] [PubMed]

Huber, P.

P. Huber, Robust Statistics (Wiley, 1974).

Joshi, S.

Jovin, T. M.

Kam, Z.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2004), 1223–1226

P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 740–743.

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999).
[CrossRef] [PubMed]

Lanni, F.

Lewis, J.

Llacer, J.

J. Llacer and E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imag.8, 186–193 (1989).
[CrossRef]

Maalouf, E.

S. Ben Hadj, L. Blanc-Feraud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3D fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2012), 1671–1674.

Markham, J.

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19, 373–385 (1999).
[CrossRef] [PubMed]

J. A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” Proc. SPIE2655, 199–208 (1996)
[CrossRef]

J. G. McNally, C. Preza, J. A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A11, 1056–1067 (1994).
[CrossRef]

C. Preza, M. I. Miller, J. Lewis, J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A9, 219–228 (1992).
[CrossRef] [PubMed]

Milanfar, P.

N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process.10, 1299–1308 (2001).
[CrossRef]

Miller, M. I.

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag.23, 32–45 (2006).
[CrossRef]

Nguyen, N.

N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Trans. Image Process.10, 1299–1308 (2001).
[CrossRef]

Olivo-Marin, J.

P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 740–743.

Olivo-Marin, J. C.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2004), 1223–1226

Pankajakshan, P.

P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 740–743.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Preza, C.

Richardson, W. H.

Ruggiero, V.

S. Bonettini and V. Ruggiero, “An alternating extragradient method for total variation-based image restoration from poisson data,” Inverse Probl.27, 095001 (2011).
[CrossRef]

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Proc. Mag.23, 32–45 (2006).
[CrossRef]

Shaevitz, J.

Shepp, L. A.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Thomas, J.

Thomas, L. J.

Unser, M.

F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express13, 10503–10522 (2005).
[CrossRef] [PubMed]

F. Aguet, D. Van De Ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 157–160.

Van De Ville, D.

F. Aguet, D. Van De Ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 157–160.

Vardi, Y.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

Veklerov, E.

J. Llacer and E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imag.8, 186–193 (1989).
[CrossRef]

Verveer, P. J.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Vicidomini, G.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

Ville, D. V. D.

Votaw, J.

J. H. Chang, J. Anderson, and J. Votaw, “Regularized image reconstruction algorithms for positron emission tomography,” IEEE Trans. Med. Imag.23, 1165–1175 (2004).
[CrossRef]

Zanella, R.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl.25, 015002 (2009).
[CrossRef]

Zanghirati, G.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

Zanni, L.

R. Zanella, G. Zanghirati, R. Cavicchioli, L. Zanni, P. Boccacci, M. Bertero, and G. Vicidomini, “Towards real-time image deconvolution: application to confocal and STED microscopy,” Sci. Rep.3, 2523(2013).
[CrossRef] [PubMed]

S. Bonettini, R. Zanella, and L. Zanni, “A scaled gradient projection method for constrained image deblurring,” Inverse Probl.25, 015002 (2009).
[CrossRef]

Zerubia, J.

N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2004), 1223–1226

P. Pankajakshan, B. Zhang, L. Blanc-Féraud, Z. Kam, J. Olivo-Marin, and J. Zerubia, “Blind deconvolution for diffraction-limited fluorescence microscopy,” in Proceedings of IEEE International Symposium on Biomedical Imaging (IEEE, 2008), 740–743.

Zhang, B.

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Figures (5)

Fig. 1
Fig. 1

Microbead image and fitted PSF: (a) xz profile of acquired microbead image; (b) xz profile of estimated PSF image; (c) xy profile of acquired image at z = 36.7 μm; (d) xy profile of estimated PSF image at z = 36.7 μm.

Fig. 2
Fig. 2

Images of xz profiles in simulation: (a) true image; (b) blurred and noisy image (SNR=15dB); (c) INV-RL; (d) INV-RL (stopping) (e) INV-GEM; (f) DV-RL; (g) DV-RL (stopping); (f) DV-GEM.

Fig. 3
Fig. 3

Images of xy profiles in simulation: (a) true image; (b) blurred and noisy image (SNR=15dB); (c) INV-RL; (d) INV-RL (stopping) (e) INV-GEM; (f) DV-RL; (g) DV-RL (stopping); (f) DV-GEM.

Fig. 4
Fig. 4

Images of xz profiles in real experiments: (a) observed; (b) INV-RL; (c) INV-GEM; (d) DV-RL; (e) DV-GEM.

Fig. 5
Fig. 5

Images of xy profiles at z = 17.97 μm in real experiments: (a) observed; (b) INV-RL; (c) INV-GEM; (d) DV-RL; (e) DV-GEM.

Tables (2)

Tables Icon

Table 1 COR values of true object image and restored images obtained by using the four methods.

Tables Icon

Table 2 Computation time of the four methods per iteration (unit is second).

Equations (22)

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g ( x i ) = Poisson { o s ( x o ) h ( x i x o ; z o , θ ) } ,
h ( x i x o ; z o , θ ) = | A 0 1 e i k o Λ ( ρ , x i x o , z o ; θ ) J 0 ( k 0 ( ( x i x o ) 2 + ( y i y o ) 2 ) NA ρ ) ρ d ρ | 2 ,
Λ ( ρ , x i x o ; z o , θ ) = z o n s 2 NA 2 ρ 2 t i * n i * 2 NA 2 ρ 2 + ( z o z i + n i ( z o n s + t i * n i * ) ) n i 2 NA 2 ρ 2 ,
Φ ( s ; s k ) L ( s ) ,
Φ ( s k ; s k ) = L ( s k ) .
L ( s ; g ) = i o s ( x o ) h ˜ o ( x i x o ) i g ( x i ) log ( o s ( x o ) h ˜ o ( x i x o ) ) ,
o s ( x o ) h ˜ o ( x i x o ) = o s k ( x o ) h ˜ o ( x i x o ) o s k ( x o ) h ˜ o ( x i x o ) s ( x o ) s k ( x o ) o s k ( x o ) h ˜ o ( x i x o ) .
log ( o s ( x o ) h ˜ o ( x i x o ) ) o s k ( x o ) h ˜ o ( x i x o ) o s k ( x o ) h ˜ o ( x i x o ) log ( s ( x o ) s k ( x o ) o s k ( x o ) h ˜ o ( x i x o ) ) .
Φ ( s ; s k ) = i o s ( x o ) h ˜ o ( x i x o ) + i g ( x i ) o s k ( x o ) h ˜ o ( x i x o ) g k ( x i ) log ( s ( x o ) s k ( x o ) g k ( x i ) ) ,
g k ( x i ) = o s k ( x o ) h ˜ o ( x i x o ) .
Φ ( s ; s k ) s ( x o ) = i h ˜ o ( x i x o ) + s k ( x o ) s ( x o ) i g ( x i ) h ˜ o ( x i x o ) g k ( x i ) .
s k + 1 ( x o ) = s k ( x o ) i h ˜ o ( x i x o ) [ i g ( x i ) h ˜ o ( x i x o ) g k ( x i ) ] ,
( θ ^ , x ^ o ) = argmax θ , x o L ( θ , x o ; g ) ,
L ( θ , x o ; g ) = i h ( x i x o , z o ; θ ) + g ( x i ) log ( h ( x i x o ) ) .
s k + 1 ( x o ) = argmin s ( x o ) Φ ( s ; s k ) + γ R ( s ) ,
R ( s ) = n ψ ( [ D s ] n ) ,
ψ ( s ) = s 2 + ε 2 ,
ψ ( [ D s ] n ) q ( [ D s ] n ; [ D s ( k , m ) ] n ) = q ( o | d n o | d n ( d n | d n o | d n o ( s ( x o ) s ( k , m ) ( x o ) ) + α ) ; α ) o | d n o | d n q ( d n | d n o | d n o ( s ( x o ) s ( k , m ) ( x o ) ) + α ; α ) ,
R ( s ) n o | d n o | d n q ( d n | d n o | d n o ( s ( x o ) s ( k , m ) ( x o ) ) + α ; α ) = R s ( s ; s ( k , m ) ) .
R s ( s ( x o ) ; s ( k , m ) ) s ( x o ) = R ( s ) s ( x o ) | s = s ( k , m ) + ( s ( x o ) s ( k , m ) ( x o ) ) č .
s ( k , m + 1 ) = argmin s ( x o ) Φ ( s ( x o ) ; s ( k , m ) ( x o ) ) + γ R s ( s ( x o ) ; s ( k , m ) ) .
0 = i h ˜ o ( x i x o ) + s k ( x o ) s ( x o ) [ i g ( x i ) h o ( x i x o ) g k ( x i ) ] + γ ( R ( s ) s ( x o ) | s = s ( k , m ) + ( s ( x o ) s ( k , m ) ( x o ) ) č ) .

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