Abstract

Various aspects of image filtering affect the final image quality in Structured Illumination Microscopy, in particular the regularization parameter and type of regularization function, the relative height of the side bands, and the shape of the apodization function. We propose an apodization filter without adjustable parameters based on the application of the Lukosz bound in order to guarantee a non-negative point spread function. Simulations of digital resolution charts and experimental data of chromatin structures and of actin filaments show artefact free reconstructions for a wide range of filter parameters. In general, a trade-off is observed between sharpness and noise suppression.

© 2013 Optical Society of America

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2013 (1)

2012 (2)

2011 (2)

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods12, 1044–1046 (2011).
[CrossRef]

L. Wang, M. C. Pitter, and M. G. Somekh, “Wide-field high-resolution structured illumination solid immersion fluorescence microscopy,” Opt. Lett.36, 2794–2796 (2011).
[CrossRef] [PubMed]

2009 (3)

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” J. Opt. Soc. Am. A26, 413–424 (2009).
[CrossRef]

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods6, 24–32 (2009).
[CrossRef] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods6, 339–342 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (1)

P. Pankajakshan, B. Zhang, L. Blanc-Feraud, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Parametric blind deconvolution for confocal laser scanning microscopy,” Conf. Proc. IEEE Eng. Med. Biol. Soc.2007, 6532–6535 (2007).

2006 (1)

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

2005 (1)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A.102, 13081–13086 (2005).
[CrossRef] [PubMed]

2002 (1)

2000 (6)

G. M. P. van Kempen and L. J. van Vliet, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A17, 425–434 (2000).
[CrossRef]

G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett.25, 46–48 (2000).
[CrossRef]

M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens, and T. Wilson, “Wide-field optically sectioning fluorescence microscopy with laser illumination,” J. Microsc.197, 1–4 (2000).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc.198, 82–87 (2000).
[CrossRef] [PubMed]

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A.97, 7232–7236 (2000).
[CrossRef] [PubMed]

G. M. P. van Kempen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. Microsc.198, 63–75 (2000).
[CrossRef] [PubMed]

1999 (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE3568, 185–196 (1999).
[CrossRef]

1997 (2)

G. M. P. van Kempen, L. J. van Vliet, P. Verveer, and H. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc.185, 354–365 (1997).
[CrossRef]

M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett.22, 1905–1907 (1997).
[CrossRef]

1994 (1)

1992 (1)

N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process.1, 322–336 (1992).
[CrossRef] [PubMed]

1990 (1)

C. Berenstein and E. Patrick, “Exact deconvolution for multiple convolution operators–an overview, plus performance characterizations for imaging sensors,” Proc. IEEE78, 723–734 (1990).
[CrossRef]

1986 (1)

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

1966 (1)

1962 (2)

W. Lukosz, “Properties of linear low-pass filters for nonnegative signals,” J. Opt. Soc. Am.52, 827–829 (1962).
[CrossRef]

W. Lukosz, “Übertragung nicht-negativer signale durch lineare filter,” J. Mod. Opt.9, 335–364 (1962).

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Arsenin, V. A.

A. N. Tikhonov and V. A. Arsenin, Solution of Ill-posed Problems (Winston-Wiley, 1977).

Bastiaens, P. I. H.

M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens, and T. Wilson, “Wide-field optically sectioning fluorescence microscopy with laser illumination,” J. Microsc.197, 1–4 (2000).
[CrossRef] [PubMed]

Beck, M.

Berenstein, C.

C. Berenstein and E. Patrick, “Exact deconvolution for multiple convolution operators–an overview, plus performance characterizations for imaging sensors,” Proc. IEEE78, 723–734 (1990).
[CrossRef]

Best, G.

Blanc-Feraud, L.

P. Pankajakshan, B. Zhang, L. Blanc-Feraud, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Parametric blind deconvolution for confocal laser scanning microscopy,” Conf. Proc. IEEE Eng. Med. Biol. Soc.2007, 6532–6535 (2007).

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Caulfield, H. J.

Chhun, B. B.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods6, 339–342 (2009).
[CrossRef] [PubMed]

Cragg, G. E.

Cremer, C.

R. Heintzmann, T. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” J. Opt. Soc. Am. B19, 1599–1609 (2002).
[CrossRef]

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE3568, 185–196 (1999).
[CrossRef]

Dey, N.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

Diehl, V.

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

Dong, C. Y.

Drexler, H. G.

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

Fienup, J. R.

Fiolka, R.

Frohn, J. T.

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A.97, 7232–7236 (2000).
[CrossRef] [PubMed]

Gaedicke, G.

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

Galatsanos, N. P.

N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process.1, 322–336 (1992).
[CrossRef] [PubMed]

Golubovskaya, I. N.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Griffis, E. R.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods6, 339–342 (2009).
[CrossRef] [PubMed]

Gustafsson, M. G. L.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods12, 1044–1046 (2011).
[CrossRef]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods6, 339–342 (2009).
[CrossRef] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A.102, 13081–13086 (2005).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc.198, 82–87 (2000).
[CrossRef] [PubMed]

Heintzmann, R.

Hell, S. W.

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods6, 24–32 (2009).
[CrossRef] [PubMed]

Hsu, K.

Jovin, T.

Juskaitis, R.

M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens, and T. Wilson, “Wide-field optically sectioning fluorescence microscopy with laser illumination,” J. Microsc.197, 1–4 (2000).
[CrossRef] [PubMed]

M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett.22, 1905–1907 (1997).
[CrossRef]

Kam, Z.

P. Pankajakshan, B. Zhang, L. Blanc-Feraud, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Parametric blind deconvolution for confocal laser scanning microscopy,” Conf. Proc. IEEE Eng. Med. Biol. Soc.2007, 6532–6535 (2007).

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

Katsaggelos, A. K.

N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process.1, 322–336 (1992).
[CrossRef] [PubMed]

Kielhorn, M.

Kleppe, I.

Knapp, H. F.

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A.97, 7232–7236 (2000).
[CrossRef] [PubMed]

Kner, P.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods12, 1044–1046 (2011).
[CrossRef]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods6, 339–342 (2009).
[CrossRef] [PubMed]

Krampert, G.

Kwon, H.-S.

Leonhardt, H.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol.190, 165–175 (2012).
[CrossRef]

Lok, M. S.

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

Lukosz, W.

Mandula, O.

Minowada, J.

H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, and J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leuk. Res.10, 487–500 (1986).
[CrossRef]

Neil, M. A. A.

M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens, and T. Wilson, “Wide-field optically sectioning fluorescence microscopy with laser illumination,” J. Microsc.197, 1–4 (2000).
[CrossRef] [PubMed]

M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett.22, 1905–1907 (1997).
[CrossRef]

Olivo-Marin, J. C.

P. Pankajakshan, B. Zhang, L. Blanc-Feraud, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Parametric blind deconvolution for confocal laser scanning microscopy,” Conf. Proc. IEEE Eng. Med. Biol. Soc.2007, 6532–6535 (2007).

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

Pankajakshan, P.

P. Pankajakshan, B. Zhang, L. Blanc-Feraud, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Parametric blind deconvolution for confocal laser scanning microscopy,” Conf. Proc. IEEE Eng. Med. Biol. Soc.2007, 6532–6535 (2007).

Patrick, E.

C. Berenstein and E. Patrick, “Exact deconvolution for multiple convolution operators–an overview, plus performance characterizations for imaging sensors,” Proc. IEEE78, 723–734 (1990).
[CrossRef]

Pitter, M. C.

Rego, E. H.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods12, 1044–1046 (2011).
[CrossRef]

Roux, P.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J. C. Olivo-Marin, and J. Zerubia, “Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech.69, 260–266 (2006).
[CrossRef] [PubMed]

Schermelleh, L.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol.190, 165–175 (2012).
[CrossRef]

Sedat, J. W.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Shao, L.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods12, 1044–1046 (2011).
[CrossRef]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J.94, 4957–4970 (2008).
[CrossRef] [PubMed]

Shroff, S. A.

So, P. T. C.

Somekh, M. G.

Squire, A.

M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens, and T. Wilson, “Wide-field optically sectioning fluorescence microscopy with laser illumination,” J. Microsc.197, 1–4 (2000).
[CrossRef] [PubMed]

Stemmer, A.

R. Fiolka, M. Beck, and A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett.33, 1629–1631 (2008).
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Supplementary Material (14)

» Media 1: AVI (1029 KB)     
» Media 2: AVI (1020 KB)     
» Media 3: AVI (709 KB)     
» Media 4: AVI (729 KB)     
» Media 5: AVI (750 KB)     
» Media 6: AVI (745 KB)     
» Media 7: AVI (513 KB)     
» Media 8: AVI (492 KB)     
» Media 9: AVI (582 KB)     
» Media 10: AVI (502 KB)     
» Media 11: AVI (679 KB)     
» Media 12: AVI (641 KB)     
» Media 13: AVI (727 KB)     
» Media 14: AVI (650 KB)     

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Figures (9)

Fig. 1
Fig. 1

(a) Plot of the Lukosz bound OTF and incoherent OTF in 1D. (b) Plot of the Lukosz bound PSF and incoherent PSF in 1D. (c) Plot of the Lukosz bound OTF and incoherent OTF in 2D. (d) Plot of the Lukosz bound PSF and incoherent PSF in 2D.

Fig. 2
Fig. 2

(a) Plot of the 2D SIM Lukosz bound OTF. (b) Plot of the 2D SIM Lukosz bound PSF. (c) Cross-sections of the 2D SIM Lukosz bound OTF along the x and y-direction, and the incoherent OTF for cutoff 2 + 2q as reference. (d) Cross-sections of the 2D SIM Lukosz bound PSF along the x and y-direction, and the incoherent PSF for cut-off 2 + 2q as reference.

Fig. 3
Fig. 3

(a) Digital resolution chart used in the simulations. The numbers with the bar patterns indicate the pitch in pixel units, i.e. in units of λ/16NA. (b) Simulated widefield image, (c) Screenshot of a series of SIM image reconstructions with different side band height parameters using a weighted sum of bands method ( Media 1). The screenshot is the reconstruction with high side band height parameter s = 3 showing noise amplification. (d) Screenshot of a series of SIM image reconstructions with generalized Wiener filtering without apodization for different regularization parameter values ( Media 2). Shown is the reconstruction at low regularization parameter κ = 10−4, which demonstrates halo artefacts and low-frequency noise structures.

Fig. 4
Fig. 4

(a) Screenshot of a series of SIM image reconstructions with generalized Wiener filtering with signal energy (p = 0) regularization and Lukosz bound apodization ( Media 3). Shown is the reconstruction for κ = 5.62 × 10−4. (b) Screenshot of a series of SIM image reconstructions with generalized Wiener filtering with signal gradient energy (p = 1) regularization and Lukosz bound apodization ( Media 4). Shown is the reconstruction for κ = 10−3. (c) Screenshot of a series of SIM image reconstructions with integrated Lukosz filtering with signal energy (p = 0) regularization ( Media 5). Shown is the reconstruction for κ = 10−1. (d) Screenshot of a series of SIM image reconstructions with integrated Lukosz filtering with signal gradient energy (p = 1) regularization and Lukosz bound apodization ( Media 6). Shown is the reconstruction for κ = 1.78 ×10−1. The regularization parameters for the different screenshots have been manually optimized.

Fig. 5
Fig. 5

Simulated image quality measures as a function of regularization parameter for different side band heights. The columns show the FWHM, the SNR of uniform image patches, and the peak SNR of single spots for generalized Wiener filtering with Lukosz bound apodization with p = 0 (signal energy) regularization ((a)–(c)), for Lukosz apodization with p = 1 (signal gradient energy) regularization ((d)–(f)), integrated Lukosz filtering with p = 0 regularization ((g)–(i)), and for integrated Lukosz filtering with p = 1 regularization ((j)–(l)). The SNR and PSNR are normalized by the values for the corresponding widefield images.

Fig. 6
Fig. 6

Reconstructions from experimental data of the chromatin distribution in a HDLM-2 cell. (a) Screenshot of a series of reconstructions with generalized Wiener filtering with p = 0 (signal energy) regularization and Lukosz bound apodization ( Media 7). Shown is the reconstruction for κ = 3.16. (b) Screenshot of a series of SIM image reconstructions with generalized Wiener filtering with p = 1 (signal gradient energy) regularization and Lukosz bound apodization ( Media 8). Shown is the reconstruction for κ = 1.78 ×10−1. (c) Screenshot of a series of SIM image reconstructions with integrated Lukosz filtering with p = 0 (signal energy) regularization ( Media 9). Shown is the reconstruction for κ = 3.16 × 10−2. (d) Screenshot of a SIM image reconstruction with integrated Lukosz filtering with p = 1 (signal gradient energy) regularization and Lukosz bound apodization ( Media 10). Shown is the reconstruction for κ = 5.62 × 10−3. The regularization parameters for the different screenshots have been manually optimized. The intensities are in arbitrary units, the scale bars are 5 μm.

Fig. 7
Fig. 7

Reconstructions from experimental data of actin in BPAE cells. (a) Screenshot of a series of reconstructions with generalized Wiener filtering with p = 0 (signal energy) regularization and Lukosz bound apodization ( Media 11). Shown is the reconstruction for κ = 1 ×10−1. (b) Screenshot of a series of SIM image reconstructions with generalized Wiener filtering with p = 1 (signal gradient energy) regularization and Lukosz bound apodization ( Media 12). Shown is the reconstruction for κ = 1.78 ×10−1. (c) Screenshot of a series of SIM image reconstructions with integrated Lukosz filtering with p = 0 (signal energy) regularization ( Media 13). Shown is the reconstruction for κ = 3.16 ×10−2. (d) Screenshot of a SIM image reconstruction with integrated Lukosz filtering with p = 1 (signal gradient energy) regularization and Lukosz bound apodization ( Media 14). Shown is the reconstruction for κ = 5.62 ×10−1. The regularization parameters for the different screenshots have been manually optimized. The intensities are in arbitrary units, the scale bars are 5 μm. The cutouts have been linearly stretched for easier visual inspection.

Fig. 8
Fig. 8

(a) Reconstructed widefield (WF) image of the 2D-slice of the chromatin distribution in a HDLM-2 cell. (b) Power spectrum (on a logarithmic scale) of the reconstructed widefield image. (c) SIM reconstruction with the unfiltered sum of bands method, Eq. (13). (d) Power spectrum (on a logarithmic scale) of the unfiltered sum of bands method image. (e) One of the reconstructed SIM images of Fig. 6 (Lukosz bound apodization with p = 0 signal energy regularization for side band height parameter s = 3 and regularization parameter κ = 3.16) (f) Power spectrum (on a logarithmic scale) of the reconstructed SIM image. The scale of the Fourier space images ((b) and (d, f)) is different due to the difference in number of pixels, consequently the pass-band of the objective lens (the red circles) have a different radius in these figures. The real space images ((a) and (c,e)) are linearly stretched for maximum contrast, the SIM power spectrum (d,f) has been clipped outside the support of the OTF for visibility purposes, scale bars are 5 μm.

Fig. 9
Fig. 9

(a) Reconstructed widefield (WF) image of the 2D-slice of actin in BPAE cells. (b) Power spectrum (on a logarithmic scale) of the reconstructed widefield image. (c) SIM reconstruction with the unfiltered sum of bands method, Eq. (13). (d) Power spectrum (on a logarithmic scale) of the unfiltered sum of bands method image. (e) One of the reconstructed SIM images of Fig. 7 (Lukosz bound apodization with p = 0 signal energy regularization for side band height parameter s = 3 and regularization parameter κ = 1 ×10−1) (f) Power spectrum (on a logarithmic scale) of the reconstructed SIM image. The scale of the Fourier space images ((b) and (d, f)) is different due to the difference in number of pixels, consequently the pass-band of the objective lens (the red circles) have a different radius in these figures. The real space images ((a) and (c,e)) are linearly stretched for maximum contrast, the cutouts are linearly contrast stretched in the insets as well. The SIM power spectrum (d,f) has been clipped outside the support of the OTF for visibility purposes, scale bars are 5 μm.

Equations (34)

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H ^ ( v ) = H ( u ) exp ( 2 π i u v ) d u .
| H ^ ( v ) | Λ 1 stairs ( v ) = cos ( π M + 1 ) for | v | q c M with M = 1 , 2 , 3 ,
Λ ^ 1 ( v / q c ) = cos ( π | v | / q c | v | / q c + 1 ) for | v | q c and Λ ^ 1 ( v ) = 0 elsewhere ,
Λ ^ 2 azi ( v , ϕ ) = Λ ^ 1 ( v x / q c ( ϕ ) ) Λ ^ 1 ( v y / q c ( ϕ + π / 2 ) ) ,
Λ ^ 2 ( v ) = min { Λ ^ 2 azi ( v , ϕ ) | ϕ [ 0 , 2 π ) } .
Λ ^ 2 ( v ) = min { cos ( π | v | | v | + q c ) , cos 2 ( π | v | | v | + 2 q c ) } for | v | q c
W ( u ) = W av [ 1 + 2 w cos ( 4 π q u x ) ] ,
W ^ ( v ) = m w m δ ( v q m ) ,
I rec , ln ( u ) = H ( u ) [ W ( R l u + u n ) T ( u ) ] .
I ^ rec , ln ( v ) = H ^ ( v ) m Φ n m w m T ^ ( v R l q m ) ,
Φ n m = exp ( 2 π i q m u n ) .
I ^ bands , lm ( v ) = n Φ n m * I ^ rec , ln ( v ) = w m H ^ ( v ) T ^ ( v R l q m ) .
I ^ lin ( v ) = lm s m I ^ bands , lm ( v + R l q m ) = lm w m s m H ^ ( v + R l q m ) T ^ ( v ) .
H ^ lin ( v ) = lm w m s m H ^ ( v + R l q m ) ,
I lin ( u ) = ln S ( R l 1 u + u n ) I rec , ln ( u ) ,
S ( u ) = S av [ 1 + 2 s cos ( 4 π q u x ) ] .
I ^ gen ( v ) = lm s m F ^ lm ( v ) I ^ band , lm ( v + R l q m ) .
gen = [ lm | B ^ lm ( v ) I ^ gen ( v ) s m I ^ band , lm ( v + R l q m ) | 2 + κ | A ^ ( v ) I ^ gen ( v ) | 2 ] d v .
A ^ ( v ) = | v | p ,
F ^ lm ( v ) = B ^ lm ( v ) * κ A ^ ( v ) 2 + lm | B ^ lm ( v ) | 2 ,
H ^ gen ( v ) = lm s m w m B ^ lm ( v ) * H ^ ( v + R l q m ) κ A ^ ( v ) 2 + lm | B ^ lm ( v ) | 2 ,
gen = [ lm g ^ ( v + R l q m ) | B ^ lm ( v ) I ^ gen ( v ) s m I ^ band , lm ( v + R l q m ) | 2 + κ | A ^ ( v ) I ^ gen ( v ) | 2 ] d v ,
g ^ ( v ) = 1 α exp ( | v | 2 / 2 σ 2 ) ,
F ^ lm ( v ) = g ^ ( v + R l q m ) B ^ lm ( v ) * κ A ^ ( v ) 2 + lm g ^ ( v + R l q m ) | B ^ lm ( v ) | 2 ,
H ^ gen ( v ) = lm s m w m g ^ ( v + R l q m ) B ^ lm ( v ) * H ^ ( v + R l q m ) κ A ^ ( v ) 2 + lm g ^ ( v + R l q m ) | B ^ lm ( v ) | 2 ,
q c ( ϕ ) = 2 q cos ϕ r + 2 1 q 2 sin 2 ϕ r ,
ϕ r = mod { ϕ + π 6 , π 3 } π 6
Λ ^ ( v ) = Λ ^ ( v ) ( 1 + μ cos ( 6 ϕ r ) ) / ( 1 μ ) ,
H ^ gen ( v ) = Λ ^ ( v ) lm s m 2 w m 2 | H ^ ( v + R l q m ) | 2 κ A ^ ( v ) 2 + lm s m 2 w m 2 | H ^ ( v + R l q m ) | 2 .
H ^ gen ( v ) = Λ ^ ( v ) lm s m 2 w m 2 | H ^ ( v + R l q m ) | 2 κ A ^ ( v ) 2 Λ ^ ( v ) 2 + lm s m 2 w m 2 | H ^ ( v + R l q m ) | 2 .
C V gen ( κ ) = lm | B ^ lm ( v ) I ^ gen ( v ) s m I ^ band , lm ( v + R l q m ) | 2 d v [ ( lm B ^ lm ( v ) F ^ lm ( v ) 1 ) d v ] 2 .
η norm = 1 + κ lm s m 2 w m 2 | H ^ ( R l q m ) | 2 .
z ( v ) = lm s m 2 w m 2 | H ^ ( v + R l q m ) | 2 Λ ^ ( v ) 2 .
s s max = 1 w 3 Λ s 2 H s 2 1 2 ( 3 Λ s 2 1 ) H s 2 ,

Metrics