Abstract

RESCH (refocusing after scanning using helical phase engineering) microscopy is a scanning technique using engineered point spread functions which provides volumetric information. We present a strategy for processing the collected raw data with a multi-view maximum likelihood deconvolution algorithm, which inherently comprises the resolution gain of pixel-reassignment microscopy. The method, which we term MD-RESCH (for multi-view deconvolved RESCH), achieves in our current implementation a 20% resolution advantage along all three axes compared to RESCH and confocal microscopy. Along the axial direction, the resolution is comparable to that of image scanning microscopy. However, because the method inherently reconstructs a volume from a single 2D scan, a significantly higher optical sectioning becomes directly visible to the user, which would otherwise require collecting multiple 2D scans taken at a series of axial positions. Further, we introduce the use of a single-helical detection PSF to obtain an increased post-acquisition refocusing range. We present data from numerical simulations as well as experiments to confirm the validity of our approach.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (1)

2014 (1)

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

2013 (2)

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

G. M. R. De Luca, R. M. P. Breedijk, R. A. J. Brandt, C. H. C. Zeelenberg, B. E. de Jong, W. Timmermans, L. N. Azar, R. A. Hoebe, S. Stallinga, and E. M. M. Manders, “Re-scan confocal microscopy: scanning twice for better resolution,” Biomed. Opt. Express 4(11), 2644–2656 (2013).
[Crossref] [PubMed]

2011 (1)

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett 36(2), 202–204 (2011).
[Crossref] [PubMed]

2010 (1)

C. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

2007 (1)

R. Heintzmann, “Estimating missing information by maximum likelihood deconvolution,” Micron. 38(2), 136–144 (2007).
[Crossref]

2003 (1)

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

2001 (1)

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

2000 (2)

R. Piestun, Y. Y. Schechner, and J. J. Shamir, “Propagation-invariant wave fields with finite energy,” Opt. Soc. Am. A. 17(2), 294–303 (2000).
[Crossref]

R. Heintzmann, G. Kreth, and C. Cremer, “Reconstruction of Axial Tomographic High Resolution Data from Confocal Fluorescence Microscopy: A Method for Improving 3D FISH Images,” Anal. Cell. Path. 20(1), 7–15 (2000).
[Crossref]

1997 (1)

1996 (1)

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. of modern optics 43(7), 1495–1507 (1996).
[Crossref]

1994 (1)

1988 (1)

C. J. R. Sheppard, “Super-resolution in Confocal Imaging,” Optik 80(2), 53–54 (1988).

1987 (1)

M. Bertero, P. Brianzi, and E. R. Pike, “Super-resolution in confocal scanning microscopy,” Inverse Problems 3(2), 195–212 (1987).
[Crossref]

1984 (1)

M. Bertero, C. De Mol, E. R. Pike, and J. G. Walker, “Resolution in Diffraction-limited Imaging, a Singular Value Analysis,” Optica Acta: International Journal of Optics 31(8), 923–946 (1984).
[Crossref]

1972 (1)

Andrews, M.

Arndt-Jovin, D.

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

Azar, L. N.

Badieirostami, M.

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett 36(2), 202–204 (2011).
[Crossref] [PubMed]

Bertero, M.

M. Bertero, P. Brianzi, and E. R. Pike, “Super-resolution in confocal scanning microscopy,” Inverse Problems 3(2), 195–212 (1987).
[Crossref]

M. Bertero, C. De Mol, E. R. Pike, and J. G. Walker, “Resolution in Diffraction-limited Imaging, a Singular Value Analysis,” Optica Acta: International Journal of Optics 31(8), 923–946 (1984).
[Crossref]

Biggs, D. S. C.

Brandt, R. A. J.

Breedijk, R. M. P.

Brianzi, P.

M. Bertero, P. Brianzi, and E. R. Pike, “Super-resolution in confocal scanning microscopy,” Inverse Problems 3(2), 195–212 (1987).
[Crossref]

Chandris, P.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Chitnis, A.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Cremer, C.

R. Heintzmann, G. Kreth, and C. Cremer, “Reconstruction of Axial Tomographic High Resolution Data from Confocal Fluorescence Microscopy: A Method for Improving 3D FISH Images,” Anal. Cell. Path. 20(1), 7–15 (2000).
[Crossref]

Dalle Nogare, D.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

de Jong, B. E.

De Luca, G. M. R.

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, and J. G. Walker, “Resolution in Diffraction-limited Imaging, a Singular Value Analysis,” Optica Acta: International Journal of Optics 31(8), 923–946 (1984).
[Crossref]

Enderlein, J.

C. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Fischer, R. S.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Gu, M.

M. Gu, Advanced optical imaging theory (Springer Science & Business Media, 1999).

Hanley, Q. S.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

Head, J.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Heintzmann, R.

R. Heintzmann, “Estimating missing information by maximum likelihood deconvolution,” Micron. 38(2), 136–144 (2007).
[Crossref]

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

R. Heintzmann, G. Kreth, and C. Cremer, “Reconstruction of Axial Tomographic High Resolution Data from Confocal Fluorescence Microscopy: A Method for Improving 3D FISH Images,” Anal. Cell. Path. 20(1), 7–15 (2000).
[Crossref]

Hoebe, R. A.

Hoogendoorn, E.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

Ingaramo, M.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

Jesacher, A.

Jovin, T. M.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

Kreth, G.

R. Heintzmann, G. Kreth, and C. Cremer, “Reconstruction of Axial Tomographic High Resolution Data from Confocal Fluorescence Microscopy: A Method for Improving 3D FISH Images,” Anal. Cell. Path. 20(1), 7–15 (2000).
[Crossref]

Lee, S. F.

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett 36(2), 202–204 (2011).
[Crossref] [PubMed]

Lew, M. D.

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett 36(2), 202–204 (2011).
[Crossref] [PubMed]

Manders, E. M. M.

Moerner, W. E.

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett 36(2), 202–204 (2011).
[Crossref] [PubMed]

Müller, C.

C. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Munroe, P.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

Nailon, J.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

Patterson, G. G.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

Piestun, R.

A. Jesacher, M. Ritsch-Marte, and R. Piestun, “Three-dimensional information from two-dimensional scans: a scanning microscope with postacquisition refocusing capability,” Optica 2(3), 210–213 (2015).
[Crossref]

R. Piestun, Y. Y. Schechner, and J. J. Shamir, “Propagation-invariant wave fields with finite energy,” Opt. Soc. Am. A. 17(2), 294–303 (2000).
[Crossref]

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. of modern optics 43(7), 1495–1507 (1996).
[Crossref]

R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19(11), 771–773 (1994).
[Crossref] [PubMed]

S. Quirin and R. Piestun, “3-D Imaging Using Helical Point Spread Functions,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper IWC1.

R. Piestun and S. Quirin, “Methods and systems for three dimensional optical imaging, sensing, particle localization and manipulation,” U.S. Patent No. 8,620, 065. 31Dec.2013.

Pike, E. R.

M. Bertero, P. Brianzi, and E. R. Pike, “Super-resolution in confocal scanning microscopy,” Inverse Problems 3(2), 195–212 (1987).
[Crossref]

M. Bertero, C. De Mol, E. R. Pike, and J. G. Walker, “Resolution in Diffraction-limited Imaging, a Singular Value Analysis,” Optica Acta: International Journal of Optics 31(8), 923–946 (1984).
[Crossref]

Postma, M.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

Quirin, S.

R. Piestun and S. Quirin, “Methods and systems for three dimensional optical imaging, sensing, particle localization and manipulation,” U.S. Patent No. 8,620, 065. 31Dec.2013.

S. Quirin and R. Piestun, “3-D Imaging Using Helical Point Spread Functions,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper IWC1.

Richardson, W. H.

Ritsch-Marte, M.

Sarafis, V.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

Schechner, Y. Y.

R. Piestun, Y. Y. Schechner, and J. J. Shamir, “Propagation-invariant wave fields with finite energy,” Opt. Soc. Am. A. 17(2), 294–303 (2000).
[Crossref]

Shamir, J.

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. of modern optics 43(7), 1495–1507 (1996).
[Crossref]

R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19(11), 771–773 (1994).
[Crossref] [PubMed]

Shamir, J. J.

R. Piestun, Y. Y. Schechner, and J. J. Shamir, “Propagation-invariant wave fields with finite energy,” Opt. Soc. Am. A. 17(2), 294–303 (2000).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, “Super-resolution in Confocal Imaging,” Optik 80(2), 53–54 (1988).

Shroff, H.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Sibarita, J.-B.

J.-B. Sibarita, “Deconvolution Microscopy,” in Microscopy Techniques, (SpringerBerlin Heidelberg2005).
[Crossref]

Spektor, B.

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. of modern optics 43(7), 1495–1507 (1996).
[Crossref]

Stallinga, S.

Timmermans, W.

Walker, J. G.

M. Bertero, C. De Mol, E. R. Pike, and J. G. Walker, “Resolution in Diffraction-limited Imaging, a Singular Value Analysis,” Optica Acta: International Journal of Optics 31(8), 923–946 (1984).
[Crossref]

Wawrzusin, P.

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Wilson, T.

T. Wilson, Confocal Microscopy (Academic: London, 1990).

York, A. G.

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Zeelenberg, C. H. C.

Anal. Cell. Path. (1)

R. Heintzmann, G. Kreth, and C. Cremer, “Reconstruction of Axial Tomographic High Resolution Data from Confocal Fluorescence Microscopy: A Method for Improving 3D FISH Images,” Anal. Cell. Path. 20(1), 7–15 (2000).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (1)

Chem. Phys. Chem. (1)

M. Ingaramo, A. G. York, E. Hoogendoorn, M. Postma, H. Shroff, and G. G. Patterson, “Richardson-Lucy deconvolution as a general tool for combining images with complementary strengths,” Chem. Phys. Chem. 15(4), 794–800 (2014).
[PubMed]

Inverse Problems (1)

M. Bertero, P. Brianzi, and E. R. Pike, “Super-resolution in confocal scanning microscopy,” Inverse Problems 3(2), 195–212 (1987).
[Crossref]

J. of Microscopy (1)

R. Heintzmann, Q. S. Hanley, D. Arndt-Jovin, and T. M. Jovin, “A dual path programmable array microscope (PAM): simultaneous acquisition of conjugate and non-conjugate images,” J. of Microscopy,  204(2), 119–135 (2001).
[Crossref]

J. of modern optics (1)

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. of modern optics 43(7), 1495–1507 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

Micron. (2)

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron. 34(6), 293–300 (2003).
[Crossref] [PubMed]

R. Heintzmann, “Estimating missing information by maximum likelihood deconvolution,” Micron. 38(2), 136–144 (2007).
[Crossref]

Nat. Meth. (1)

A. G. York, P. Chandris, D. Dalle Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Meth. 10(11), 1122–1126 (2013).
[Crossref]

Opt. Lett (1)

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Supplementary Material (1)

NameDescription
» Visualization 1: AVI (182 KB)      3D volume acquired by RESCH

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Figures (8)

Fig. 1
Fig. 1 Point spread function of a confocal microscope with a point-like off-axis detector: the combined PSF, which is the product of the excitation and detection PSFs, is shifted by half of the distance the detector has from the optical axis. This is however only valid for PSFs whose shapes fulfill certain symmetry conditions, such as e.g. Gaussian PSFs.
Fig. 2
Fig. 2 Phase mask (left) to shape the detection PSF. Gray values correspond to phase values. An iso-intensity surface plot of the corresponding single-helix PSF is shown on the right (NA=1.25, wavelength=660 nm); length scales apply to a medium with refractive index of 1.52 .
Fig. 3
Fig. 3 Flow diagram of the multi-view Richardson-Lucy algorithm.
Fig. 4
Fig. 4 Properties of simulated PSFs for every detector pixel (NA=1.25, wavelength: ex./em.= 640/660 nm). The properties are color-coded in the respective pixels. Compared are three methods: MD-RESCH for different helix orientations and ISM. The red arrows on the left indicate the excitation laser polarization. 1st column: Integrated PSF-strengths; 2nd column: axial PSF shifts, determined as z-centroid coordinates of 3D Gaussian fits to the simulated PSFs; 3rd-5th columns: FWHM-values of 3D Gaussian fits to the PSFs;.
Fig. 5
Fig. 5 Properties of simulated PSFs for three detectors for the MD-RESCH (iso) case. The properties are colour-coded in the respective detector areas. The helical RESCH PSFs are off-centered with respect to the optical axis. 1st column: axial PSF shifts; 2nd-4th columns: FWHM-values of 3D-Gaussian fits to the PSFs; refocusing range and PSF widths apply to a medium with refractive index of 1.52.
Fig. 6
Fig. 6 Measured PSF properties for MD-RESCH (iso) and ISM.
Fig. 7
Fig. 7 Simulated MD-RESCH imaging: the assumed sample is a sketch of an apple tree with branches, leaves and apples in three different axial planes; top left image: confocal image of the strongest detector pixel; 1st image row: ground truth; 2nd row: RESCH based on the three SPs shown in Fig. 5; 3rd and 4th rows: MD-RESCH results (iso) for 3 and 61 detector pixels, after 400 deblurring iterations.
Fig. 8
Fig. 8 a) MD-RESCH images from Alexa 647 stained microtubules in COS-7 cells (NA=1.25). The multi-view deconvolution comprised 200 iterations and is based on 84 views. Projecting the image volume along the optical axis generates a significantly increased depth of focus. The entire image volume is shown in ( Visualization 1). b) Comparison of specimen reconstructions based on 3 and 84 views, respectively. Taking a larger number of views provides improved resolution, which is also shown by cross sections along the same region (marked by white dashed line) of both reconstructions.

Tables (2)

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Table 1 Calculated spatial resolution estimates (in nm) for MD-RESCH (iso and aniso, 61 views), MD-RESCH (iso, 3 views), ISM and confocal microscopy. The numbers are FWHM values of mean PSFs for the respective methods.

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Table 2 Measured FWHM values (in nm) of PSFs for MD-RESCH (iso, 64 views), MDRESCH (iso, 3 views), ISM (38 views), and confocal microscopy (Airy-disc-sized pinhole).

Equations (4)

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h ( x ) G ( x σ ex 2 σ ex 2 + σ det 2 x ^ m , σ ex σ det σ ex 2 + σ det 2 ) ,
h ( x ) G ( x x ^ m 2 , σ ex 2 )
h m ( x , y , z ) = [ δ ( x x ^ m , y y ^ m ) * 2 D h det ( x , y , z ) ] h ex ( x , y , z ) .
E n + 1 ( r ) = E n ( r ) { 1 M m = 1 M [ ( V m ( r ) E n ( r ) * h m ( r ) 1 ) * h m * ( r ) ] + 1 } .

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