Abstract

In this article, we characterize the lateral field distortions in a low numerical aperture and large field-of-view (FOV) fluorescence imaging system. To this end, we study a commercial fluorescence MACROscope setup, which is a zooming microscope. The versatility of this system lies in its ability to image at different zoom ranges, so that sample preparations can be examined in three-dimensions, at cellular, organ and whole body levels. Yet, we found that the imaging system’s optics are optimized only for high magnifications where the observed FOV is small. When we studied the point-spread function (PSF) by using fluorescent polystyrene beads as “guide-stars”, we noticed that the PSF is spatially varying due to field distortions. This variation was found to be laterally symmetrical and the distortions were found to increase with the distance from the center of the FOV. In this communication, we investigate the idea of using the field at the back focal plane of an optical system for characterizing distortions. As this field is unknown, we develop a theoretical framework to retrieve the amplitude and phase of the field at the back focal pupil plane, from the empirical bead images. By using the retrieved amplitude, we can understand and characterize the underlying cause of these distortions. We also propose a few approaches, before acquisition, to either avoid it or correct it at the optical design level.

© 2012 OSA

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References

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  1. P. Sendrowski and C. Kress, “Arrangement for analyzing microscopic and macroscopic preparations,” WO 2009/04711 (2009). PCT/EP2008/062749.
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  3. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  4. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  5. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. A 59, 1314–1321 (1969).
    [CrossRef]
  6. P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.
  7. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
    [CrossRef] [PubMed]
  8. M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
    [CrossRef] [PubMed]
  9. Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
    [CrossRef] [PubMed]
  10. M. J. Booth, “Adaptive optics in microscopy,” Philos. Transact. A Math. Phys. Eng. Sci. 365, 2829–2843 (2007).
    [CrossRef] [PubMed]
  11. R. Juškaitis and T. Wilson, “The measurement of the amplitude point spread function of microscope objective lenses,” J. Microsc. 189, 8–11 (1998).
    [CrossRef]
  12. P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).
  13. P. Pankajakshan, “Blind Deconvolution for Confocal Laser Scanning Microscopy,” Ph.D. thesis, Université de Nice Sophia-Antipolis (2009).
  14. T. J. Holmes, D. Biggs, and A. Abu-Tarif, “Blind Deconvolution,” in Handbook of Biological Confocal Microscopy, 3rd ed, J. B. Pawley, ed. (Springer, New York, 2006), Chap. 24, pp. 468–487.
    [CrossRef]
  15. B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
    [CrossRef] [PubMed]
  16. J. E. Webb, “Distortion tuning of quasi-telecentric lens,” US Patent 7646543 (2010).
  17. J. Winterot and T. Kaufhold, “Optical arrangement and method for the imaging of depth-structured objects,” US Patent 7564620 (2009).

2009 (1)

J. Winterot and T. Kaufhold, “Optical arrangement and method for the imaging of depth-structured objects,” US Patent 7564620 (2009).

2007 (2)

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

M. J. Booth, “Adaptive optics in microscopy,” Philos. Transact. A Math. Phys. Eng. Sci. 365, 2829–2843 (2007).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

1998 (1)

R. Juškaitis and T. Wilson, “The measurement of the amplitude point spread function of microscope objective lenses,” J. Microsc. 189, 8–11 (1998).
[CrossRef]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. A 59, 1314–1321 (1969).
[CrossRef]

Abu-Tarif, A.

T. J. Holmes, D. Biggs, and A. Abu-Tarif, “Blind Deconvolution,” in Handbook of Biological Confocal Microscopy, 3rd ed, J. B. Pawley, ed. (Springer, New York, 2006), Chap. 24, pp. 468–487.
[CrossRef]

Agard, D.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

Agard, D. A.

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Biggs, D.

T. J. Holmes, D. Biggs, and A. Abu-Tarif, “Blind Deconvolution,” in Handbook of Biological Confocal Microscopy, 3rd ed, J. B. Pawley, ed. (Springer, New York, 2006), Chap. 24, pp. 468–487.
[CrossRef]

Blanc-Feraud, L.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Blanc-Féraud, L.

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

Booth, M. J.

M. J. Booth, “Adaptive optics in microscopy,” Philos. Transact. A Math. Phys. Eng. Sci. 365, 2829–2843 (2007).
[CrossRef] [PubMed]

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Dieterlen, A.

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Engler, G.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gustafsson, M. G.

Hanser, B. M.

Holmes, T. J.

T. J. Holmes, D. Biggs, and A. Abu-Tarif, “Blind Deconvolution,” in Handbook of Biological Confocal Microscopy, 3rd ed, J. B. Pawley, ed. (Springer, New York, 2006), Chap. 24, pp. 468–487.
[CrossRef]

Juškaitis, R.

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

R. Juškaitis and T. Wilson, “The measurement of the amplitude point spread function of microscope objective lenses,” J. Microsc. 189, 8–11 (1998).
[CrossRef]

Kam, Z.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Kaufhold, T.

J. Winterot and T. Kaufhold, “Optical arrangement and method for the imaging of depth-structured objects,” US Patent 7564620 (2009).

Kner, P.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

Kress, C.

P. Sendrowski and C. Kress, “Arrangement for analyzing microscopic and macroscopic preparations,” WO 2009/04711 (2009). PCT/EP2008/062749.

Neil, M. A.

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Olivo-Marin, J.-C.

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Pankajakshan, P.

P. Pankajakshan, “Blind Deconvolution for Confocal Laser Scanning Microscopy,” Ph.D. thesis, Université de Nice Sophia-Antipolis (2009).

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sedat, J. W.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[CrossRef] [PubMed]

Sendrowski, P.

P. Sendrowski and C. Kress, “Arrangement for analyzing microscopic and macroscopic preparations,” WO 2009/04711 (2009). PCT/EP2008/062749.

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Stokseth, P. A.

P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. A 59, 1314–1321 (1969).
[CrossRef]

Webb, J. E.

J. E. Webb, “Distortion tuning of quasi-telecentric lens,” US Patent 7646543 (2010).

Wilson, T.

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

R. Juškaitis and T. Wilson, “The measurement of the amplitude point spread function of microscope objective lenses,” J. Microsc. 189, 8–11 (1998).
[CrossRef]

Winterot, J.

J. Winterot and T. Kaufhold, “Optical arrangement and method for the imaging of depth-structured objects,” US Patent 7564620 (2009).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Zerubia, J.

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

Appl. Opt. (1)

J. Microsc. (3)

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef] [PubMed]

R. Juškaitis and T. Wilson, “The measurement of the amplitude point spread function of microscope objective lenses,” J. Microsc. 189, 8–11 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. A 59, 1314–1321 (1969).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Philos. Transact. A Math. Phys. Eng. Sci. (1)

M. J. Booth, “Adaptive optics in microscopy,” Philos. Transact. A Math. Phys. Eng. Sci. 365, 2829–2843 (2007).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

M. J. Booth, M. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Other (8)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

P. Pankajakshan, Z. Kam, A. Dieterlen, G. Engler, L. Blanc-Féraud, J. Zerubia, and J.-C. Olivo-Marin, “Point-spread function model for fluorescence macroscopy imaging,” in Proc. of Asilomar Conference on Signals, Systems and Computers, (2010), 1364–1368.

J. E. Webb, “Distortion tuning of quasi-telecentric lens,” US Patent 7646543 (2010).

J. Winterot and T. Kaufhold, “Optical arrangement and method for the imaging of depth-structured objects,” US Patent 7564620 (2009).

P. Sendrowski and C. Kress, “Arrangement for analyzing microscopic and macroscopic preparations,” WO 2009/04711 (2009). PCT/EP2008/062749.

P. Pankajakshan, A. Dieterlen, G. Engler, Z. Kam, L. Blanc-Feraud, J. Zerubia, and J.-C. Olivo-Marin, “Wavefront sensing for aberration modeling in fluorescence macroscopy,” in Proc. IEEE International Symposium on Biomedical Imaging (ISBI), IEEE (IEEE, Chicago, USA, 2011).

P. Pankajakshan, “Blind Deconvolution for Confocal Laser Scanning Microscopy,” Ph.D. thesis, Université de Nice Sophia-Antipolis (2009).

T. J. Holmes, D. Biggs, and A. Abu-Tarif, “Blind Deconvolution,” in Handbook of Biological Confocal Microscopy, 3rd ed, J. B. Pawley, ed. (Springer, New York, 2006), Chap. 24, pp. 468–487.
[CrossRef]

Supplementary Material (1)

» Media 1: AVI (23072 KB)     

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic of a simple wide-field fluorescence MACROscope (Reproduced from [1]); Best of two worlds: maximum intensity projection along the optical axis of a Convallaria majalis sample taken using a Leica™ ZAPO16, fit with a confocal scanning head, at (b) a minimum zoom setting with lateral pixel size of 1.09 μm and (c) a sub-region of the sample at the maximum zoom setting with lateral pixel size of 0.89 μm (Courtesy of INRA). The scale bars are 100 μm in length.

Fig. 2
Fig. 2

Haemocytometer grid used for illustrating and measuring the distortion in the field. The square area indicated in red is of size 1 mm2, in green is 0.0625 mm2, in yellow is 0.04 mm2 and finally the smallest in blue is 0.0025 mm2. Reproduced from Wikimedia Commons

Fig. 3
Fig. 3

The focal plane of the observed transmitted volume of the Haemocytometer (top) and the maximum intensity projection along the y-direction (bottom). The object is imaged using a 2x/air PlanApo objective fit to a Leica™ MacroFluo™ APOZ16. The zoom for this acquisition was set at 12.7x, the lateral sampling at 390nm, the slice thickness at 50 μm and the scale bar length is 50 μm. The total size of the displayed volume is 343×343×6200 μm.

Fig. 4
Fig. 4

Empirical PSFs are shown at the different positions (denoted by a cross) in the lateral field of the lens. Here, the PSFs are shown as the MIP along the y-direction.

Fig. 5
Fig. 5

The schematic to measure the maximum object spread for constraining the iterative algorithm.

Fig. 6
Fig. 6

(a) The first section of the observed intensity, with z = −57 μm, and the scale bar is 10 μm, and (b) retrieved unwrapped pupil phase, φ̂a. The bead image was cropped from the right peripherary intensity image of Fig. 4 at a zoom 1.6× (radial sampling of 998.3nm and axial sampling of 1000nm). τ = 0.6, the maximum number of iteration is 32 and the phase scale is between [−π, +π] radians.

Fig. 7
Fig. 7

A schematic showing the effect of two limiting apertures (here zoom and objective lenses) at the back focal plane of the optical system. Here the on-axis and off-axis positions are shown.

Fig. 8
Fig. 8

Diversity sections, i(x), taken at four symmetrical positions with defocus at (a) z = −36 μm, (b) z = −15 μm, (c) z = +15 μm and (d) z = +36 μm. The objective is a 2x/air PlanApo and the zoom is set at 4.6x. The slice width is fixed at 3 μm and the effective NA was calculated to be 0.17.

Fig. 9
Fig. 9

Retrieved back focal pupil amplitude, |P̂(kx, ky, z = 0)|, from the defocus sections in Fig. 8. The algorithm was run with variations in the effective NA (a) 0.05, (b) 0.17, (c) 0.20.

Tables (1)

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Algorithm 1: Proposed Algorithm.

Equations (12)

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h A ( x , y , z ) = 2 D 1 { P ( k x , k y , z ; NA ) } ,
P ( k x , k y , z ; NA , φ a ) = { exp ( j z ( ( k 0 n i ) 2 ( k x 2 + k y 2 ) ) 1 2 + j φ a ) , if ( k x 2 + k y 2 ) 1 2 k 0 < NA , 0 , otherwise ,
h ( x ) = | h A ( x ; λ ex ) | 2 .
γ i ( x ) = 𝒫 { γ | h A ( x ) | 2 + b ( x ) } , x Ω s ,
Pr ( h A | i ) = Pr ( i | h A ) Pr ( h A ) Pr ( i ) ,
Pr ( i | h A ) = x Ω s ( | h A | 2 + b ) ( x ) i ( x ) exp ( ( | h A | 2 + b ) ( x ) ) i ( x ) ! .
h ^ A ( x ) = argmax h A ( x ) Pr ( h A | i ) , s. t.  k MAX < 2 π λ ex NA , = argmin h A ( x ) log [ Pr ( h A | i ) ] , s.t.  k MAX < 2 π λ ex NA ,
𝒥 ( h A ) = log [ Pr ( h A | i ) ] = 𝒥 obs ( h A ) Image energy + 𝒥 reg ( h A ) Prior energy .
h ^ A ( x ) = argmin h A ( x ) 𝒥 obs ( h A ) , s. t. k MAX < 2 π λ ex NA = argmin h A ( x ) log [ Pr ( i | h A ) ] , s . t . k MAX < 2 π λ ex NA , = argmax h A ( x ) | h A ( x ) | 2 i ( x ) log ( | h A ( x ) | 2 + b ( x ) ) , s . t. k MAX < 2 π λ ex NA.
h ^ A ( n + 1 ) ( x ) = h ^ A ( n ) ( x ) τ 2 𝒥 obs ( h A ) .
𝒥 obs ( h A ) = 𝒥 obs ( h A ) ( h A ( x ) ) + j 𝒥 obs ( h A ) ( h A ( x ) ) = 2 × ( h A ( x ) i ( x ) ( | h A ( x ) | 2 + b ( x ) ) h A ( x ) ) , x Ω s .
h ^ A ( n + 1 ) ( x ) = ( 1 τ ) h ^ A ( n ) ( x ) + τ ( i ( x ) | h ^ A ( n ) ( x ) | 2 + b ( x ) h ^ A ( n ) ( x ) ) , x Ω s ,

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