Abstract

A depth-variant (DV) image restoration algorithm for wide field fluorescence microscopy, using an orthonormal basis decomposition of DV point-spread functions (PSFs), is investigated in this study. The efficient PSF representation is based on a previously developed principal component analysis (PCA), which is computationally intensive. We present an approach developed to reduce the number of DV PSFs required for the PCA computation, thereby making the PCA-based approach computationally tractable for thick samples. Restoration results from both synthetic and experimental images show consistency and that the proposed algorithm addresses efficiently depth-induced aberration using a small number of principal components. Comparison of the PCA-based algorithm with a previously-developed strata-based DV restoration algorithm demonstrates that the proposed method improves performance by 50% in terms of accuracy and simultaneously reduces the processing time by 64% using comparable computational resources.

© 2015 Optical Society of America

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References

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  1. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
    [Crossref] [PubMed]
  2. J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).
  3. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
    [Crossref] [PubMed]
  4. D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [Crossref] [PubMed]
  5. T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6(7), 1006–1014 (1989).
    [Crossref]
  6. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5(5), 666–673 (1988).
    [Crossref]
  7. S. F. Gibson, Modeling the Three-Dimensional Imaging Properties of the Fluorescence Light Microscope (Carnegie Mellon University, 1990).
  8. C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004).
    [Crossref] [PubMed]
  9. V. Myneni and C. Preza, “3-D reconstruction of fluorescence microscopy image intensities using multiple depth-variant point-spread functions,” Digital Image Processing and Analysis (DIPA), Imaging and Applied Optics, OSA Optics and Photonics Congress, DTuA2 (2010).
    [Crossref]
  10. C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” Proc. SPIE 7570, 757003 (2010).
    [Crossref]
  11. M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principal component analysis,” Opt. Express 18(7), 6461–6476 (2010).
    [Crossref] [PubMed]
  12. S. Yuan and C. Preza, “Performance evaluation of an image estimation method based on principal component analysis (PCA) developed for quantitative depth-variant fluorescence microscopy imaging,” Proc. SPIE 8227, 82270H (2012).
    [Crossref]
  13. S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” Proc. SPIE 7904, 79040M (2011).
    [Crossref]
  14. S. Yuan and C. Preza, “3D computational microscopy with depth-varying point-spread functions using a principal component analysis method,” in Imaging and Applied Optics, OSA Technical Digest (online), paper IM3E.4 (2013).
    [Crossref]
  15. N. Patwary and C. Preza, “Computationally tractable approach to PCA-based depth-variant PSF representation for 3D microscopy image restoration,” in Classical Optics 2014, OSA Technical Digest (online), paper CW1C.5 (2014).
    [Crossref]
  16. J.-A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A 15(10), 2609–2619 (1998).
    [Crossref] [PubMed]
  17. J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, Three-Dimensional Microscopy: Image Acquisition and Processing III, 199 (1996).
  18. I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nat. Phys. Sci (Lond.) 229(1), 29–30 (1971).
    [Crossref]
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    [Crossref] [PubMed]
  21. O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
    [Crossref]
  22. D. L. Snyder, C. W. Helstrom, A. D. Lanterman, M. Faisal, and R. L. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12(2), 272–283 (1995).
    [Crossref]
  23. N. Patwary, “Performance Aanalysis of PCA-based Image Reconstruction in 3D Wide Field Fluorescence Microscopy (MS Thesis),” University of Memphis, https://umwa.memphis.edu/etd/index.php (2014).
  24. J.-B. Sibarita, Deconvolution Microscopy (Springer Berlin / Heidelberg, 2005).
  25. S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging,” Opt. Express 19(23), 23298–23314 (2011).
    [Crossref] [PubMed]
  26. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC press, 2010).

2012 (1)

S. Yuan and C. Preza, “Performance evaluation of an image estimation method based on principal component analysis (PCA) developed for quantitative depth-variant fluorescence microscopy imaging,” Proc. SPIE 8227, 82270H (2012).
[Crossref]

2011 (2)

S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” Proc. SPIE 7904, 79040M (2011).
[Crossref]

S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging,” Opt. Express 19(23), 23298–23314 (2011).
[Crossref] [PubMed]

2010 (2)

2004 (1)

2003 (1)

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

1999 (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

1998 (1)

1997 (1)

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).

1995 (1)

1992 (1)

1989 (2)

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6(7), 1006–1014 (1989).
[Crossref]

1988 (1)

1984 (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

1971 (1)

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nat. Phys. Sci (Lond.) 229(1), 29–30 (1971).
[Crossref]

Agard, D. A.

M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principal component analysis,” Opt. Express 18(7), 6461–6476 (2010).
[Crossref] [PubMed]

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

Arigovindan, M.

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Conchello, J.-A.

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Faisal, M.

Gibson, S. F.

Good, I. J.

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nat. Phys. Sci (Lond.) 229(1), 29–30 (1971).
[Crossref]

Haeberlé, O.

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

Helstrom, C. W.

Hiraoka, Y.

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Holmes, T. J.

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Lanni, F.

Lanterman, A. D.

McGowan, J.

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Myneni, V.

C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” Proc. SPIE 7570, 757003 (2010).
[Crossref]

Preza, C.

S. Yuan and C. Preza, “Performance evaluation of an image estimation method based on principal component analysis (PCA) developed for quantitative depth-variant fluorescence microscopy imaging,” Proc. SPIE 8227, 82270H (2012).
[Crossref]

S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” Proc. SPIE 7904, 79040M (2011).
[Crossref]

S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging,” Opt. Express 19(23), 23298–23314 (2011).
[Crossref] [PubMed]

C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” Proc. SPIE 7570, 757003 (2010).
[Crossref]

C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004).
[Crossref] [PubMed]

Sedat, J. W.

M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principal component analysis,” Opt. Express 18(7), 6461–6476 (2010).
[Crossref] [PubMed]

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Shaevitz, J.

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Snyder, D. L.

Swedlow, J. R.

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).

White, R. L.

Yuan, S.

S. Yuan and C. Preza, “Performance evaluation of an image estimation method based on principal component analysis (PCA) developed for quantitative depth-variant fluorescence microscopy imaging,” Proc. SPIE 8227, 82270H (2012).
[Crossref]

S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” Proc. SPIE 7904, 79040M (2011).
[Crossref]

S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging,” Opt. Express 19(23), 23298–23314 (2011).
[Crossref] [PubMed]

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

Deconvolution of Images and Spectra (1)

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” Deconvolution of Images and Spectra 285, 284–309 (1997).

J. Opt. Soc. Am. A (6)

Methods (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Methods Cell Biol. (1)

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Nat. Phys. Sci (Lond.) (1)

I. J. Good, “Non-Parametric Roughness Penalty for Probability Densities,” Nat. Phys. Sci (Lond.) 229(1), 29–30 (1971).
[Crossref]

Opt. Commun. (1)

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216(1), 55–63 (2003).
[Crossref]

Opt. Express (2)

Proc. SPIE (3)

S. Yuan and C. Preza, “Performance evaluation of an image estimation method based on principal component analysis (PCA) developed for quantitative depth-variant fluorescence microscopy imaging,” Proc. SPIE 8227, 82270H (2012).
[Crossref]

S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” Proc. SPIE 7904, 79040M (2011).
[Crossref]

C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” Proc. SPIE 7570, 757003 (2010).
[Crossref]

Other (9)

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC press, 2010).

N. Patwary, “Performance Aanalysis of PCA-based Image Reconstruction in 3D Wide Field Fluorescence Microscopy (MS Thesis),” University of Memphis, https://umwa.memphis.edu/etd/index.php (2014).

J.-B. Sibarita, Deconvolution Microscopy (Springer Berlin / Heidelberg, 2005).

S. Yuan and C. Preza, “3D computational microscopy with depth-varying point-spread functions using a principal component analysis method,” in Imaging and Applied Optics, OSA Technical Digest (online), paper IM3E.4 (2013).
[Crossref]

N. Patwary and C. Preza, “Computationally tractable approach to PCA-based depth-variant PSF representation for 3D microscopy image restoration,” in Classical Optics 2014, OSA Technical Digest (online), paper CW1C.5 (2014).
[Crossref]

Computational Imaging Research Laboratory, Computational Optical Sectioning Microscopy Open Source (COSMOS) software package; http://cirl.memphis.edu/COSMOS.

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, Three-Dimensional Microscopy: Image Acquisition and Processing III, 199 (1996).

V. Myneni and C. Preza, “3-D reconstruction of fluorescence microscopy image intensities using multiple depth-variant point-spread functions,” Digital Image Processing and Analysis (DIPA), Imaging and Applied Optics, OSA Optics and Photonics Congress, DTuA2 (2010).
[Crossref]

S. F. Gibson, Modeling the Three-Dimensional Imaging Properties of the Fluorescence Light Microscope (Carnegie Mellon University, 1990).

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Figures (10)

Fig. 1
Fig. 1

The PCA coefficients of the first three components as a function of depth (axial location z ) for a 20X/0.8 NA dry lens in the presence of spherical aberration. (a), (b), and (c) are the coefficients of the first, second, and third components, respectively. (d) Normalized mean square error (NMSE3D) between the true PSFs and PSFs computed from the PCA-IC technique, and the full PCA computation as a function of significant PCs.

Fig. 2
Fig. 2

Determination of the OPR for: (a) SNR = 11.2 dB; (b) SNR = 10 dB; (c) SNR = 6.8 dB. The ORP is the value of γ for which the NMSE3D between the true object and the restored image becomes the minimum. The ORPs for these three cases were found to be equal to 0.00004, 0.0001, and 0.003, respectively.

Fig. 3
Fig. 3

Evaluation of the PCA forward imaging model. XZ medial section of: (a) the numerical spherical shell (object 1 as described in Section 3.2.1); (b) the simulated image of (a) obtained by the PCA model (described in Section 3.2.1); (c) experimentally acquired image of a 6-μm bead test sample (described in Section 3.3.1). (d) Axial normalized intensity profile through the center of the images shown in (b) and (c). To plot the intensity profile, both simulated and experimental images were normalized, and were registered properly to match the peak intensities. Lens: 63X/oil immersion; Emission Wavelength: 515 nm.

Fig. 4
Fig. 4

Algorithm performance comparison of the PCA-EM and strata-EM algorithms using a simulated image from a 6 µm in diameter spherical shell (object 2) shown in (a). (b)-(d) the PCA-EM restored images using 2, 4, and 6 PCs, respectively; (e)-(g) the strata-EM restored images using 2, 4, and 6 strata, respectively. (h) Algorithm performance quantified by the NMSE3D (Eq. (14)) as a function of the number of PCs/strata used in the restoration. (f) Time needed to compute the restoration result after 1000 iterations of each algorithm as a function of the number of PCs/strata used in each case. Lens: 63X/1.4 oil immersion; Emission Wavelength: 515 nm.

Fig. 5
Fig. 5

Performance evaluation of the regularized PCA-EM algorithm demonstrated by restoring noisy simulated images using PCA-EM (with 10 PCs) and a roughness regularization penalty (Eq. (11)). XZ medial section of: (a) the synthetic object 3; and (b) the noisy simulated image with SNR 6.8 db. PCA-EM restored images computed from noisy images using an ORP: (c) γ = 0.0004 for SNR = 11.2 db; (d) γ = 0.0001 for SNR = 10 db; and (e) γ = 0.003 for SNR = 6.8 db. Specimen embedding medium is water. Lens: 63X/1.4 NA oil immersion. Emission wavelength: 515 nm.

Fig. 6
Fig. 6

Performance comparison between the PCA-EM restorations (after 600 iterations) from simulated data using 3 PCs computed with and without interpolated coefficients (IC). XZ sectional view from: (a) the numerical object; (b) the simulated image of the numerical object; (c) the PCA-EM restored image using PCA data from a full set of PSFs; and (d) the PCA-EM restored image using PCA data from a reduced set of PSFs computed with PCA-IC. A 20X/0.8 NA air objective lens was simulated assuming that the specimen is embedded in water.

Fig. 7
Fig. 7

Restoration performance evaluation of the PCA-EM algorithm using 3 PCs. (a) XZ medial section of the measured image of a 6 µm in diameter spherical shell with a shell thickness equal to 1 µm. Restoration using PCs computed from the set of: (b) 200PSFs; and (c) 40 PSFs. The restorations were carried out with a regularization parameter equal to 0.0001. The restored images after 1000 iterations are displayed. Images were resampled in the axial direction to compensate for the axial shift due to SA (Eq. (12)).

Fig. 8
Fig. 8

Evaluation of the optical sectioning capability of the PCA-EM algorithm. (a)-(d) are four XY sections (planes at different axial locations) from the WF image of mouse lung epithelial cells captured using a 20X/0.8 NA dry lens; (e)-(h) are corresponding planes captured using the Zeiss ApoTome attachment (SIM); (i)-(l) are corresponding planes from the PCA-EM restored image computed with 3 PCs and a regularization parameter equal to 0.0001 after 1000 iterations. Image intensities are displayed on a separate scale for the best visualization of features in each image.

Fig. 9
Fig. 9

Comparison of PCA-EM restoration with results obtained from other optical sectioning methods. XY sectional image from a Glioblastoma cell acquired with two different modalities and restored using different methods: (a) WF unprocessed image with three marker beads enclosed by a rectangle, a circle, and a triangle; (b) Deconvolved SIM image captured using ApoTome; (c) Restoration with ZEN iterative deconvolution algorithm; (d) Restoration with PCA-EM algorithm with 3 PCs. In the case of iterative restoration, results were obtained after 1000 iterations using a regularization parameter equal to 0.0001. Image intensities are displayed on a separate scale to facilitate visualization of features in each image. Lens: 63x/1.4 NA oil; Wavelength: 509 nm.

Fig. 10
Fig. 10

Quantitative comparison between the processed images of 175-nm in diameter microspheres (PSF beads) cultured in the Glioblastoma cell. (a-c) Axial profile through the three beads marked in Fig. 9(a), which demonstrate that the PCA-EM restored images have better contrast and axial resolution compared to SIM images and ZEN (ZEISS deconvolution software) restored images; (d) Lateral intensity profile through bead#2 for all the cases. Legend shown in (a) corresponds to all the figures. Lens: 63x/1.4 NA oil; Wavelength: 509 nm.

Equations (14)

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g ( x , y , z ) = z ' y ' x ' h ( x x ' , y y ' , z ; z ' ) f ( x ' , y ' , z ' ) ,
g ( x , y , z ) = m = 1 M { h m ( x , y , z ) [ a m ( z ) f m ( x , y , z ) ] + h m + 1 ( x , y , z ) [ ( 1 a m ( z ) ) f m ( x , y , z ) ] } ,
a m ( z o ) = { Z m + 1 z o Z m + 1 Z m for z o O m 0 otherwise ,
h ^ ( x , y , z ; z o ) = h ¯ ( x , y , z ) + n = 1 B C ( n , z o ) P n ( x , y , z ) ,
g ( x , y , z ) = h ¯ ( x , y , z ) f ( x , y , z ) + n = 1 B P n ( x , y , z ) [ C ( n , z ) f ( x , y , z ) ] ,
f ( k + 1 ) ( x , y , z ) = f ( k ) ( x , y , z ) H z o [ h ^ ( x , y , z ; z o ) d ( x , y , z ) g ( k ) ( x , y , z ) ] ,
f ( k + 1 ) ( x ) = f ( k ) ( x ) H B ( z ) { h ¯ ( x ) d ( x ) g ( k ) ( x ) + n = 1 B C ( n , z ) [ P n ( x ) d ( x ) g ( k ) ( x ) ] }
g ( k ) ( x ) = h ¯ ( x ) f ( k ) ( x ) + n = 1 B P n ( x ) [ C ( n , z ) f ( k ) ( x ) ]
H B ( z ) = z ' y ' x ' h ^ ( x ' , y ' , z ' ; z )
L [ f ( x ) | g ( x ) ] γ R [ f ( x ) ] ,
R [ f ( x ) ] = x | f ( x ) | 2 .
d z = tan [ sin 1 ( N A / n ) ] tan [ sin 1 ( N A / n s ) ] d z '
S N R = 10 × log 10 ( σ s σ N ) ,
N M S E 3 D = a ( x , y , z ) b ( x , y , z ) 2 b ( x , y , z ) 2 .

Metrics