Abstract

In this paper, wavefront-encoded (WFE) computational optical sectioning microscopy (COSM) using a fabricated square cubic (SQUBIC) phase mask, designed to render the system less sensitive to depth-induced aberration, is investigated. The WFE-COSM system is characterized by a point spread function (PSF) that does not vary as rapidly with imaging depth compared to the conventional system. Thus, in WFE-COSM, image restoration from large volumes can be achieved using computationally efficient space-invariant (SI) algorithms, thereby avoiding the use of depth-variant algorithms. The fabricated SQUBIC phase mask was first evaluated and found to have a 75% fidelity compared to the theoretical design; it was then integrated in a commercial wide-field microscope to implement a WFE-COSM system. Evaluation of the WFE-COSM system is demonstrated with comparative studies of theoretical and experimental PSFs and simulated and measured images of spherical shells located at different depths in a test sample. These comparisons show that PSF and imaging models capture major trends in experimental data with a 99% correlation between forward image intensity distribution in experimental and simulated images of spherical shells. Our experimental SI restoration results demonstrate that the WFE-COSM system achieves more than a twofold performance improvement over the conventional system of up to a 65 μm depth below the coverslip investigated in this study.

© 2017 Optical Society of America

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Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy

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Appl. Opt. 54(29) 8587-8595 (2015)

References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (2)

S. Ghosh and C. Preza, “Three-dimensional block-based restoration integrated with wide-field fluorescence microscopy for the investigation of thick specimens with spatially variant refractive index,” J. Biomed. Opt. 21, 046010 (2016).
[Crossref]

N. Patwary, S. V. King, G. Saavedra, and C. Preza, “Reducing effects of aberration in 3D fluorescence imaging using wavefront coding with a radially symmetric phase mask,” Opt. Express 24, 12905–12921 (2016).
[Crossref]

2015 (2)

2014 (1)

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

2011 (1)

2010 (1)

2009 (2)

G. Saavedra, I. Escobar, R. Martínez-Cuenca, E. Sánchez-Ortiga, and M. Martínez-Corra, “Reduction of spherical-aberration impact in microscopy by wavefront coding,” Opt. Express 17, 13810–13818 (2009).
[Crossref]

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

2004 (2)

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

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

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, 55–63 (2003).
[Crossref]

1998 (1)

1996 (1)

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

1995 (1)

1993 (1)

1992 (1)

1989 (1)

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

1984 (1)

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

1971 (1)

I. J. Good, “Non-parametric roughness penalty for probability densities,” Nat. Phys. Sci. 229, 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, 6461–6476 (2010).
[Crossref]

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

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

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” in Deconvolution of Images and Spectra (Academic, 1997), Chap. 9.

Arigovindan, M.

Aubert, G.

S. Ben Hadj, L. Blanc-Féraud, G. Aubert, and G. 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), pp. 915–919.

Bäumer, S.

S. Bäumer, Handbook of Plastic Optics (Wiley, 2011).

Ben Hadj, S.

S. Ben Hadj, L. Blanc-Féraud, G. Aubert, and G. 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), pp. 915–919.

Bertero, M.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Blanc-Féraud, L.

S. Ben Hadj, L. Blanc-Féraud, G. Aubert, and G. 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), pp. 915–919.

Boccacci, P.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Cathey, W. T.

Conchello, J.-A.

Desiderà, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Doblas, A.

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54, 8587–8595 (2015).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

Dowski, E. R.

Engler, G.

S. Ben Hadj, L. Blanc-Féraud, G. Aubert, and G. 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), pp. 915–919.

Escobar, I.

Ghosh, S.

S. Ghosh and C. Preza, “Three-dimensional block-based restoration integrated with wide-field fluorescence microscopy for the investigation of thick specimens with spatially variant refractive index,” J. Biomed. Opt. 21, 046010 (2016).
[Crossref]

Gibson, S. F.

Good, I. J.

I. J. Good, “Non-parametric roughness penalty for probability densities,” Nat. Phys. Sci. 229, 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, 55–63 (2003).
[Crossref]

Hammoud, A. M.

Hiraoka, Y.

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

King, S. V.

N. Patwary, S. V. King, G. Saavedra, and C. Preza, “Reducing effects of aberration in 3D fluorescence imaging using wavefront coding with a radially symmetric phase mask,” Opt. Express 24, 12905–12921 (2016).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54, 8587–8595 (2015).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, S. V. King, H. Shabani, and C. Preza, “Experimental implementation of wavefront encoding in 3D widefield fluorescence microscopy using a fabricated phase mask designed to reduce system depth variability,” in Classical Optics 2014, OSA Technical Digest (online) (2016), paper CW2D-3.

Lanni, F.

Martínez-Corra, M.

Martínez-Corral, M.

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54, 8587–8595 (2015).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

Martínez-Cuenca, R.

McGowan, J.

McNally, J. G.

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

Patwary, N.

N. Patwary, S. V. King, G. Saavedra, and C. Preza, “Reducing effects of aberration in 3D fluorescence imaging using wavefront coding with a radially symmetric phase mask,” Opt. Express 24, 12905–12921 (2016).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54, 8587–8595 (2015).
[Crossref]

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6, 3826–3841 (2015).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

N. Patwary, S. V. King, H. Shabani, and C. Preza, “Experimental implementation of wavefront encoding in 3D widefield fluorescence microscopy using a fabricated phase mask designed to reduce system depth variability,” in Classical Optics 2014, OSA Technical Digest (online) (2016), paper CW2D-3.

Preza, C.

N. Patwary, S. V. King, G. Saavedra, and C. Preza, “Reducing effects of aberration in 3D fluorescence imaging using wavefront coding with a radially symmetric phase mask,” Opt. Express 24, 12905–12921 (2016).
[Crossref]

S. Ghosh and C. Preza, “Three-dimensional block-based restoration integrated with wide-field fluorescence microscopy for the investigation of thick specimens with spatially variant refractive index,” J. Biomed. Opt. 21, 046010 (2016).
[Crossref]

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6, 3826–3841 (2015).
[Crossref]

S. V. King, A. Doblas, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Spatial light modulator phase mask implementation of wavefront encoded 3D computational-optical microscopy,” Appl. Opt. 54, 8587–8595 (2015).
[Crossref]

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[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, 23298–23314 (2011).
[Crossref]

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

N. Patwary, S. V. King, H. Shabani, and C. Preza, “Experimental implementation of wavefront encoding in 3D widefield fluorescence microscopy using a fabricated phase mask designed to reduce system depth variability,” in Classical Optics 2014, OSA Technical Digest (online) (2016), paper CW2D-3.

Saavedra, G.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

Sánchez-Ortiga, E.

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, 6461–6476 (2010).
[Crossref]

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

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” in Deconvolution of Images and Spectra (Academic, 1997), Chap. 9.

Sha, P.

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

Shabani, H.

N. Patwary, S. V. King, H. Shabani, and C. Preza, “Experimental implementation of wavefront encoding in 3D widefield fluorescence microscopy using a fabricated phase mask designed to reduce system depth variability,” in Classical Optics 2014, OSA Technical Digest (online) (2016), paper CW2D-3.

Shaevitz, J.

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Snyder, D. L.

Swedlow, J. R.

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” in Deconvolution of Images and Spectra (Academic, 1997), Chap. 9.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

Vicidomini, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

White, R. L.

Yuan, S.

Annu. Rev. Biophys. Bioeng. (1)

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

Appl. Opt. (2)

Biomed. Opt. Express (1)

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Inverse Probl. (1)

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

J. Biomed. Opt. (1)

S. Ghosh and C. Preza, “Three-dimensional block-based restoration integrated with wide-field fluorescence microscopy for the investigation of thick specimens with spatially variant refractive index,” J. Biomed. Opt. 21, 046010 (2016).
[Crossref]

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

Methods Cell Biol. (1)

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

Nat. Phys. Sci. (1)

I. J. Good, “Non-parametric roughness penalty for probability densities,” Nat. Phys. Sci. 229, 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, 55–63 (2003).
[Crossref]

Opt. Express (4)

Proc. SPIE (2)

A. Doblas, S. V. King, N. Patwary, G. Saavedra, M. Martínez-Corral, and C. Preza, “Investigation of the SQUBIC phase mask design for depth-invariant widefield microscopy point-spread function engineering,” Proc. SPIE 8949, 894914 (2014).
[Crossref]

J.-A. Conchello and J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE 2655, 199–208 (1996).
[Crossref]

Other (6)

N. Patwary, S. V. King, H. Shabani, and C. Preza, “Experimental implementation of wavefront encoding in 3D widefield fluorescence microscopy using a fabricated phase mask designed to reduce system depth variability,” in Classical Optics 2014, OSA Technical Digest (online) (2016), paper CW2D-3.

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

J. R. Swedlow, J. W. Sedat, and D. A. Agard, “Deconvolution in optical microscopy,” in Deconvolution of Images and Spectra (Academic, 1997), Chap. 9.

S. Bäumer, Handbook of Plastic Optics (Wiley, 2011).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

S. Ben Hadj, L. Blanc-Féraud, G. Aubert, and G. 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), pp. 915–919.

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Figures (7)

Fig. 1.
Fig. 1.

SQUBIC ( A = 50 ) PM fabrication for a 63 × / 1.4    NA oil immersion objective lens at the design wavelength of 515 nm. (a) The fabricated PM mounted in a 25.4 mm diameter lens holder for use in the WFE system implementation. (b) Surface rendering of the PM showing the unwrapped phase computed with Eq. (1). (c) Mask height as a function of physical distance used in the fabrication process. (d) and (e) show schematics of the side and top view of the PM, respectively.

Fig. 2.
Fig. 2.

Experimental setups for the investigation studies. (a) Schematic of the modified Mach–Zehnder interferometer used to evaluate the fabricated PM. The L2 ( f 2 = 50    mm ) and L3 ( f 3 = 100    mm ) lenses were used in a telecentric-afocal configuration to provide a magnification equal to 2. The laser wavelength used was equal to 532 nm. (b) WFE system implementation using the SQUBIC ( A = 50 ) PM by modifying the side imaging port of a commercial microscope (Zeiss Axio imager.Z1) through a 4F optical setup composed by two achromatic doublet lenses (L4 and L5) with equal focal length ( f = 75    mm ) yielding unity magnification.

Fig. 3.
Fig. 3.

Performance evaluation of the fabricated SQUBIC PM through interferometric measurement. (a) Theoretical PM computed from Eq. (1) using A = 50 , NA = 1.4 , and n = 1.518 . Phase extracted from interferometric measurement of the SQUBIC ( A = 50 ) using Eqs. (5) and (6): (b) fabricated PM; and (c) SLM-based implementation. Phase values along a horizontal line passing through the center of the phase images (a)–(c) are plotted in panel (d) along the radius of the PM. Panels (e) and (f) show details from these profiles over two selected regions.

Fig. 4.
Fig. 4.

(a) Evaluation of depth invariance (or robustness to depth-induced SA) of the SQUBIC WFE imaging system (for different A values) as a function of the imaging depth through SI restoration with a PSF at a depth of 50 μm. Robustness is measured in terms of the correlation coefficient ( r ) between the restored image of a spherical shell centered at a depth of 50 μm and the images of shells centered at the other depths. (b) Restoration performance of the SQUBIC WFE system for the fabricated PM at different emission wavelengths. EB, emission bandwidth of emission filter used.

Fig. 5.
Fig. 5.

Evaluation of the experimental WFE PSF using images of 170 nm beads located on the coverslip. XZ section image of the (a) theoretical PSF at 27 μm depth below the coverslip that addresses a residual aberration observed at the coverslip [15]; experimental PSF of a SQUBIC ( A = 50 ) WFE system implemented using: (b) the fabricated PM, and (c) the SLM-based PM. Image intensity is displayed using a nonlinear scale to show details in low values. (d) Axial normalized intensity profiles through the center of the XZ section images of the PSFs. Lens: 63 × / 1.4    NA oil immersion; emission wavelength: 515 nm; specimen mounting medium is optical cement ( RI = 1.56 ).

Fig. 6.
Fig. 6.

Evaluation of experimental WFE images using a test sample. XZ section from the intermediate WFE image of a 6 μm spherical shell with shell thickness equal to 1 μm centered at a depth of 3 μm: (a) simulated image; (b) experimental image when the fabricated PM is used; (c) experimental image when the SLM is used. The red arrow indicates an artifact in experimental image intensity. (d) Comparison of the normalized axial intensity profiles between panels (a)–(c). (e) Comparison of the normalized axial intensity profile between the experimental intermediated images from the WFE system using the fabricated PM captured from spherical shells at three different depths. Lens: 63 × / 1.4    NA oil immersion; wavelength: 515 nm; sample mounting medium is ProLong Diamond ( RI = 1.47 ). At a 65 μm depth below the coverslip the SA is quantified by W 40 = 2.47 .

Fig. 7.
Fig. 7.

Comparison of experimental restored images of the test object computed from intermediate images acquired with the conventional (a) and SQUBIC WFE imaging systems: (b) SLM-based ( A = 20 ), (c) SLM-based ( A = 50 ), and (d) using a fabricated PM ( A = 50 ). Restorations obtained with the SIEM algorithm using an unaberrated PSF from images of spherical shells located at depths of 3, 15, 30, and 65 μm below the coverslip are shown in each row, respectively, for each system. The correlation coefficient value computed between each restored image and a reference image for each system is overlaid at the bottom right of each image. For conventional imaging, the reference for this computation is the restored image obtained using an aberrated PSF that addresses a residual aberration at the coverslip [15]: (e) XZ section image, and (f) XY section image. The numerical spherical shell (g) is provided as the ground truth and its intensity profile labeled as Simulated Object is shown in panel (h) where profiles of normalized axial intensity values through the center of the restored experimental images of shells located at 65 μm below the coverslip are shown for each case.

Equations (7)

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ϕ ( u ˜ , v ˜ ; α ) = 2 π A [ 1 ( u ˜ 2 + v ˜ 2 ) sin 2 α 1 1 cos α + 1 2 ] 3 ,
h ( x , y , z ) = | F 1 { H ( u , v ; z ) e j ϕ ( u , v ) } | 2 ,
g ( x ) = f ( x ) h ( x ) ,
g n ( x ) = Poisson [ g ( x ) + b ] ,
cos [ ϕ ( x , y ) ] = I ( x , y ) I ref ( x , y ) I PM ( x , y ) 2 I ref ( x , y ) I PM ( x , y ) ,
φ ( ρ ) = { cos 1 φ ( ρ ) , when    d cos φ ( ρ ) d ρ < 0 , cos 1 φ ( ρ ) , when    d cos φ ( ρ ) d ρ 0 .
SNR = 10 × log 10 ( g ( x ) 2 [ g n ( x ) g ( x ) ] 2 ) ,

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