## Three-dimensional structured illumination microscopy using Lukosz bound apodization reduces pixel negativity at no resolution cost |

Optics Express, Vol. 22, Issue 9, pp. 11215-11227 (2014)

http://dx.doi.org/10.1364/OE.22.011215

Acrobat PDF (2849 KB)

### Abstract

The quality of the reconstructed image in structured illumination microscopy (SIM) depends on various aspects of the image filtering process. To optimize the trade-off between resolution and ringing artifacts, which lead to negative intensities, we extend Lukosz-bound filtering to 3D SIM and derive the parametrization of the 3D SIM cut-off. We compare the use of the Lukosz-bound as apodization filter to triangular apodization and find a tenfold reduction in the most negative pixel value with a minimal resolution loss. We test this algorithm on experimental SIM images of tubulin filaments and DAPI stained DNA structure in cancer cells and find a substantial reduction in the most negative pixel value and the percentage of pixels with a negative value. This means that there is no longer a need to clip the final image to avoid these negative pixel values.

© 2014 Optical Society of America

## 1. Introduction

1. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. **56**, 1463–1471 (1966). [CrossRef]

2. 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]

7. 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]

8. R. Heintzmann, T. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” J. Opt. Soc. Am. B **19**, 1599–1609 (2002). [CrossRef]

15. O. Mandula, M. Kielhorn, K. Wicker, G. Krampert, I. Kleppe, and R. Heintzmann, “Line scan - structured illumination microscopy super-resolution imaging in thick fluorescent samples,” Opt. Express **20**, 24167–24174 (2012). [CrossRef] [PubMed]

11. 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]

16. C. Berenstein and E. Patrick, “Exact deconvolution for multiple convolution operators–An overview, plus performance characterizations for imaging sensors,” Proc. IEEE **78**, 723–734 (1990). [CrossRef]

18. S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” J. Opt. Soc. Am. A **26**, 413–424 (2009). [CrossRef]

19. C. H. Righolt, J. A. Slotman, I. T. Young, S. Mai, L. J. van Vliet, and S. Stallinga, “Image filtering in structured illumination microscopy using the lukosz bound,” Opt. Express **21**, 24431–24451 (2013). [CrossRef] [PubMed]

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

19. C. H. Righolt, J. A. Slotman, I. T. Young, S. Mai, L. J. van Vliet, and S. Stallinga, “Image filtering in structured illumination microscopy using the lukosz bound,” Opt. Express **21**, 24431–24451 (2013). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Overview of 3D SIM reconstruction

*x*,

*y*,

*z*)

*, with coordinate*

^{T}*z*along the optical axis, are normalized to

**u**= (

*xn*

_{med}sin

*α/λ*,

*yn*

_{med}sin

*α/λ*,

*z*4

*n*

_{med}sin

^{2}(

*α*/2)

*/λ*))

*, with*

^{T}*n*

_{med}the refractive index of the medium in object space,

*λ*the wavelength of the emitted light and

*α*the marginal ray angle in object space. The spatial frequencies are denoted by

**v**, for which we use the Fourier transform convention: in which

*H*(

**u**) is the PSF and

*Ĥ*(

**v**) the optical transfer function (OTF). The illumination function

*W*(

**u**) is periodic and its spectrum consists of delta-peaks: where

*w*denotes the relative weight of the

_{m}*m*

^{th}frequency component. In 3D SIM, the zeroth and first orders of a diffraction grating are projected on the back focal plane of the objective. The interference pattern of the three resulting plane waves in object space leads to a periodic illumination pattern with a spectrum that has seven peaks,

**q**

*∈ (0, 0, 0)*

_{m}*, (±2*

^{T}*q*, 0, 0)

_{r}*, (±*

^{T}*q*, 0, ±

_{r}*q*)

_{z}*, where*

^{T}*q*= 1/

_{r}*p*and with

*p*the normalized grating period. Images are recorded for a set of rotations

**R**

*, (*

_{l}*l*= 1, 2,...

*N*) and translations

_{r}**u**

*=*

_{n}*n*(

*p/N*, 0) (

_{t}*n*= 1, 2,...

*N*). For 3D SIM typically

_{t}*N*∈ {3, 5} rotations and

_{r}*N*= 5 translations are used. In the remainder of the paper we will use

_{t}*N*= 3.

_{r}24. K. Wicker, “Non-iterative determination of pattern phase in structured illumination microscopy using autocorrelations in Fourier space,” Opt. Express **21**, 24692–24701 (2013). [CrossRef] [PubMed]

*T̂*(

**v**). A final reconstructed image can now be obtained by moving each band in frequency space and filtering it with kernel

*F̂*(

_{lm}**v**): where

*s*is an additional weight factor. The generalized Tikhonov-Miller functional to be minimized in SIM reconstruction [19

_{m}19. C. H. Righolt, J. A. Slotman, I. T. Young, S. Mai, L. J. van Vliet, and S. Stallinga, “Image filtering in structured illumination microscopy using the lukosz bound,” Opt. Express **21**, 24431–24451 (2013). [CrossRef] [PubMed]

**v**

*=*

_{lm}**v**+

**R**

*·*

_{l}**q**

*is the band-shifted spatial frequency vector,*

_{m}*B̂*(

_{lm}**v**) =

*s*(

_{m}w_{m}Ĥ**v**

*) denotes the ideal or desired response of the imaging system,*

_{lm}*κ*the regularization parameter,

*Â*(

**v**) the regularization weight function,

*Ĉ*(

**v**) the apodization function, and

*ĝ*(

**v**) a data misfit weight function, that can be used to suppress the hexagonal imprint artifact that is typical for SIM [23

23. K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express **21**, 2032–2049 (2013). [CrossRef] [PubMed]

*α*is the amplitude,

*σ*the lateral frequency width and

_{l}*σ*the axial frequency width. The functional is minimized with respect to the reconstructed image

_{a}*Î*

_{gen}(

**v**), leading to the filter kernels: and the effective OTF for the reconstructed image:

### 2.2. The 3D SIM cut-off

*ρ*,

*θ*,

*ϕ*), such that

*v*=

_{x}*ρ*cos

*ϕ*sin

*θ*,

*v*=

_{y}*ρ*sin

*ϕ*sin

*θ*and

*v*=

_{z}*ρ*cos

*θ*. The SIM OTF of Eq. (9) is non-zero when ∑

_{lm}*Ĥ*(

**v**+

**R**

*·*

_{l}**q**

*) is non-zero. The cut-off of the SIM OTF is, therefore, given by the outer boundary segments of the incoherent OTFs moved to positions {*

_{m}**R**

*·*

_{l}**q**

*}. The cut-off of the incoherent OTF has been derived by Sheppard [25*

_{m}25. C. J. R. Sheppard, M. Gu, Y. Kawata, and S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A **11**, 593–598 (1994). [CrossRef]

*ρ*(

*θ*,

*ϕ*), is thus given by the intersection of the radial line in direction (

*θ*,

*ϕ*) and one of the moved elliptical arcs of the incoherent OTF. Using the translation

**q**

*= (*

_{m}*q*,

_{m,x}*q*,

_{m,y}*q*), see Eq. (3), and using our coordinate normalization, we then find: Substituting the lateral coordinates in Eq. (10) with

_{m,z}*v*=

_{x}*v*cos

_{z}*ϕ*tan

*θ*and

*v*=

_{y}*v*sin

_{z}*ϕ*tan

*θ*and squaring this equation leads to a quartic equation in

*v*after several rearrangements: The polynomial coefficients are: with the constants: Equation (11) has four solutions

_{z}*v*(

_{z,m,j}*θ*,

*ϕ*) for each peak in the illumination pattern. The

*z*-solutions can be transformed to spherical coordinates leading to: The

*j*th solution represents an actual intersection when it is positive. Negative solutions match positive solutions for a different

*m*because of symmetry in the system, and imaginary solutions occur when there is no intersection. The maximum of all solutions, excluding imaginary and negative numbers yields the SIM cut-off: The above derivation is valid, except when

*θ*∈ {0,

*π*/2,

*π*}. For

*θ*=

*π*/2,

*v*= 0; for standard 3D SIM this leads to: For

_{z}*θ*∈ {0,

*π*},

*v*=

_{x}*v*= 0 and Eq. (10) can be solved directly as The resulting parametrization of the SIM cut-off is shown in Fig. 1.

_{y}### 2.3. The 3D SIM Lukosz-bound

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

**21**, 24431–24451 (2013). [CrossRef] [PubMed]

*q*is the cut-off frequency. For the 2D case, the product of two 1D bounds along two orthogonal directions has to be taken. The actual bound is the minimum over all possible orientations of the 2D coordinate system. This formalism can be extended into 3D: where the minimum is taken over a set

_{c}*𝒢*of rotations

*ℛ*of the 3D coordinate frame, where

**v′**are the spatial frequencies in the rotated frame and

*q*(

_{c,j}*ℛ*) are the cut-offs along the rotated cardinal directions. The axial and lateral directions are fundamentally different, which requires care in implementing this recipe. We have devised two implementations that work well in practice. In the first, only rotations around the optical axis are considered; in the second, the full group of 3D rotations, SO(3), is considered in combination with an anisotropic pre-strechting of the spatial frequency space.

*ϕ*around the

*v*-axis. The coordinate system is now described by orthogonal axes (cos

_{z}*ϕ*, sin

*ϕ*, 0), (−sin

*ϕ*, cos

*ϕ*, 0) and (0, 0, 1). A spatial frequency vector

**v**= (

*v*sin

*ξ*cos

*ψ*,

*v*sin

*ξ*sin

*ψ*,

*v*cos

*ξ*) will have coordinates

**v′**= (

*v*sin

*ξ*cos(

*ψ*−

*ϕ*),

*v*sin

*ξ*sin(

*ψ*−

*ϕ*), cos

*ξ*) in the rotated frame. The minimum for all rotations for this bound is then given by: where the respective cut-offs are described by Eqs. (22) and (23). The 3D bound should still adhere to the 1D bound in the direction (

*ξ*,

*ψ*). The 3D Lukosz-bound for rotations around the optical axis, Λ̂

*(*

_{OAR}**v**), therefore becomes: Note that Λ̂

*(*

_{OAR}*v*,

_{x}*v*, 0) is the 2D bound we described previously in [19

_{y}**21**, 24431–24451 (2013). [CrossRef] [PubMed]

*v*-axis by a factor: Note that this scale factor can be changed accordingly when a different spatial normalization is chosen; 2 and 1/2 are the lateral and axial frequency cut-offs of the lens OTF for our normalization. This scaling leads to several changed coordinates, marked with a tilde: The cut-off can also be obtained in the rescaled coordinate system: All of the cardinal direction cut-offs

_{z}*q*(

_{c,j}*ℛ*) and spatial frequency components

*v′*in the rotated frame can be calculated using a parametrization of the rotation matrices

_{j}*ℛ*with the three Euler angles, and can then be plugged into Eq. (28). This then yields Λ̂

*(*

_{FR}**v**), the 3D Lukosz-bound based on the full set of SO(3) rotations.

### 2.4. Comparison of apodization functions

11. 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]

23. K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express **21**, 2032–2049 (2013). [CrossRef] [PubMed]

*𝒟*denotes the distance transform and ∑

_{lm}*Ĥ*(

**v**+

**R**

*·*

_{l}**q**

*) > 0 is the footprint of the SIM OTF. The triangle can alternatively be formed over every line originating in the origin to the cut-off in that direction. This leads to an apodization function:*

_{m}*q*= 0.65 and

_{r}*q*= 0.16 for three equidistant rotation angles. The corresponding PSF’s, the inverse Fourier transforms of the respective functions were computed using the chirp z-transform algorithm [26

_{z}26. J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. **41**, 4879–4903 (2002). [CrossRef]

*C*(

_{DT}**u**), yields negative values of −9% of the value at the origin. Negative PSF values lead to ringing artifacts in the reconstructed SIM image. The minimum occurs around

*u*≈

_{x}*u*≈ 0.25, which causes these artifacts to be at

_{y}*the*length scale of interest for this type of microscopy. Using a different method to create the triangle, namely by parametrizing the SIM cut-off, and drawing a triangle in each direction leads to a twofold reduction of the negativity. An order of magnitude improvement can, however, be gained by switching to the 3D SIM Lukosz-bound with the full SO(3) rotation group. This comes at the price of a small (around ≈ 10%) increase in the lateral FWHM. The axial FWHM is hardly affected. Note that, as for the 2D case, these 3D Lukosz-bounds have a small residual amount of negativity, because the conditions on which they are based are necessary conditions, not sufficient conditions.

*(*

_{FR}**v**) is computationally intensive because it needs to be performed in a 6D-space (three spatial frequency components, three Euler angles parametrizing all rotations). Fortunately, this function does not change from image to image when the imaging system parameters, in particular the pitches and rotation angles of the illumination pattern, remain constant.

## 3. Experiments

27. 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,” Leukemia Res. **10**, 487–500 (1986). [CrossRef]

*n*

_{med}= 1.518 immersion oil equipped with an Andor EM-CCD iXon 885 camera with 8

*μ*m pixel size. The back-projected pixel size was 79 nm, because a 1.6x tube lens was used as well. The tubulin image was acquired using 488 nm laser excitation, a projected grating period of 444 nm and an emission passband of 495–575 nm. The DAPI image was acquired using 405 nm laser excitation, a projected grating period of 365 nm and a 420–480 nm passband for the emitted light.

24. K. Wicker, “Non-iterative determination of pattern phase in structured illumination microscopy using autocorrelations in Fourier space,” Opt. Express **21**, 24692–24701 (2013). [CrossRef] [PubMed]

*Ĉ*(

**v**). We used conventional total energy regularization,

*Â*(

**v**) = 1 and the same amplitude

*α*= 0.95 for the Gaussian data misfit function as for the 2D reconstruction. We fit a Gaussian to the lens PSF, Fourier transformed this Gaussian and took half the widths in Fourier space as the widths for the data misfit function:

*σ*= 0.9 and

_{l}*σ*= 0.3. The images were reconstructed for a range of regularization parameters

_{a}*κ*without boosting the side-bands, i.e.

*s*= 1 for all

_{m}*m*. A slice of the 3D volume depicting tubulin filaments is shown for

*κ*= 1 × 10

^{−5}in Fig. 3. A slice depicting the DNA structure in the multinucleated cell for

*κ*= 1×10

^{−3}is shown in Fig. 4. We choose these values because they visually led to the best trade-off between noise suppression and image sharpness and contrast. The influence of the regularization parameter on these slices is shown in Media 1 and Media 3 for both images. A

*z*-stack movie, going through the slices along the optical axis is visible in Media 2 and Media 4 for both samples. Note that no clipping of negative pixel values has been applied to any of the reconstructions.

^{−5}and 10

^{−3}respectively, as for Lukosz-bound apodization. The ratio of the minimum to the maximum pixel value and the percentage of negative pixels were determined for both reconstructions as follows. First, we estimated the zero-level in the ZEN reconstructions from the background, because the ZEN software maps the images to 16-bit files. The foreground object was subsequently found by a coarse segmentation – an isodata threshold followed by a binary dilation – for the same slices as shown in Figs. 3 and 4. We took the minimum as the 1st and the maximum as the 99th percentile over the region of interest (ROI) thus found in order to be independent of outliers. We also estimated the number of pixels in the ROI with value below the background level. The distribution of normalized pixel values over the ROI’s are plotted in Fig. 5.

## 4. Discussion

23. K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express **21**, 2032–2049 (2013). [CrossRef] [PubMed]

## References and links

1. | W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. |

2. | M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. |

3. | R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE |

4. | G. E. Cragg and P. T. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. |

5. | 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. |

6. | M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. |

7. | 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. |

8. | R. Heintzmann, T. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” J. Opt. Soc. Am. B |

9. | M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U. S. A. |

10. | R. Fiolka, M. Beck, and A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. |

11. | 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. |

12. | 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. Methods |

13. | L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods |

14. | L. Wang, M. C. Pitter, and M. G. Somekh, “Wide-field high-resolution structured illumination solid immersion fluorescence microscopy,” Opt. Lett. |

15. | O. Mandula, M. Kielhorn, K. Wicker, G. Krampert, I. Kleppe, and R. Heintzmann, “Line scan - structured illumination microscopy super-resolution imaging in thick fluorescent samples,” Opt. Express |

16. | C. Berenstein and E. Patrick, “Exact deconvolution for multiple convolution operators–An overview, plus performance characterizations for imaging sensors,” Proc. IEEE |

17. | L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. |

18. | S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” J. Opt. Soc. Am. A |

19. | C. H. Righolt, J. A. Slotman, I. T. Young, S. Mai, L. J. van Vliet, and S. Stallinga, “Image filtering in structured illumination microscopy using the lukosz bound,” Opt. Express |

20. | W. Lukosz, “Übertragung Nicht-negativer Signale Durch Lineare Filter,” J. Mod. Opt. |

21. | W. Lukosz, “Properties of linear low-pass filters for nonnegative signals,” J. Opt. Soc. Am. |

22. | K. Wicker, “Increasing resolution and light efficiency in fluorescence microscopy,” Ph.D. thesis, King’s College, London (2010). |

23. | K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express |

24. | K. Wicker, “Non-iterative determination of pattern phase in structured illumination microscopy using autocorrelations in Fourier space,” Opt. Express |

25. | C. J. R. Sheppard, M. Gu, Y. Kawata, and S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A |

26. | J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. |

27. | 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,” Leukemia Res. |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(100.6640) Image processing : Superresolution

(110.4280) Imaging systems : Noise in imaging systems

(180.2520) Microscopy : Fluorescence microscopy

(070.2615) Fourier optics and signal processing : Frequency filtering

**ToC Category:**

Microscopy

**History**

Original Manuscript: January 24, 2014

Revised Manuscript: March 20, 2014

Manuscript Accepted: March 23, 2014

Published: May 1, 2014

**Virtual Issues**

Vol. 9, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Christiaan H. Righolt, Sabine Mai, Lucas J. van Vliet, and Sjoerd Stallinga, "Three-dimensional structured illumination microscopy using Lukosz bound apodization reduces pixel negativity at no resolution cost," Opt. Express **22**, 11215-11227 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-11215

Sort: Year | Journal | Reset

### References

- W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1471 (1966). [CrossRef]
- M. A. A. Neil, R. Juskaitis, T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997). [CrossRef]
- R. Heintzmann, C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999). [CrossRef]
- G. E. Cragg, P. T. 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, 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, 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]
- R. Heintzmann, T. Jovin, C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” J. Opt. Soc. Am. B 19, 1599–1609 (2002). [CrossRef]
- 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]
- R. Fiolka, M. Beck, A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. 33, 1629–1631 (2008). [CrossRef] [PubMed]
- M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008). [CrossRef] [PubMed]
- P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6, 339–342 (2009). [CrossRef] [PubMed]
- L. Shao, P. Kner, E. H. Rego, M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods 12, 1044–1046 (2011). [CrossRef]
- L. Wang, M. C. Pitter, M. G. Somekh, “Wide-field high-resolution structured illumination solid immersion fluorescence microscopy,” Opt. Lett. 36, 2794–2796 (2011). [CrossRef] [PubMed]
- O. Mandula, M. Kielhorn, K. Wicker, G. Krampert, I. Kleppe, R. Heintzmann, “Line scan - structured illumination microscopy super-resolution imaging in thick fluorescent samples,” Opt. Express 20, 24167–24174 (2012). [CrossRef] [PubMed]
- C. Berenstein, E. Patrick, “Exact deconvolution for multiple convolution operators–An overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990). [CrossRef]
- L. P. Yaroslavsky, H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162 (1994). [CrossRef] [PubMed]
- S. A. Shroff, J. R. Fienup, D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” J. Opt. Soc. Am. A 26, 413–424 (2009). [CrossRef]
- C. H. Righolt, J. A. Slotman, I. T. Young, S. Mai, L. J. van Vliet, S. Stallinga, “Image filtering in structured illumination microscopy using the lukosz bound,” Opt. Express 21, 24431–24451 (2013). [CrossRef] [PubMed]
- W. Lukosz, “Übertragung Nicht-negativer Signale Durch Lineare Filter,” J. Mod. Opt. 9, 335–364 (1962).
- W. Lukosz, “Properties of linear low-pass filters for nonnegative signals,” J. Opt. Soc. Am. 52, 827–829 (1962). [CrossRef]
- K. Wicker, “Increasing resolution and light efficiency in fluorescence microscopy,” Ph.D. thesis, King’s College, London (2010).
- K. Wicker, O. Mandula, G. Best, R. Fiolka, R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21, 2032–2049 (2013). [CrossRef] [PubMed]
- K. Wicker, “Non-iterative determination of pattern phase in structured illumination microscopy using autocorrelations in Fourier space,” Opt. Express 21, 24692–24701 (2013). [CrossRef] [PubMed]
- C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994). [CrossRef]
- J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. 41, 4879–4903 (2002). [CrossRef]
- H. G. Drexler, G. Gaedicke, M. S. Lok, V. Diehl, J. Minowada, “Hodgkin’s disease derived cell lines HDLM-2 and L-428: comparison of morphology, immunological and isoenzyme profiles,” Leukemia Res. 10, 487–500 (1986). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: AVI (3046 KB)

» Media 2: AVI (3170 KB)

» Media 3: AVI (2528 KB)

» Media 4: AVI (3687 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.