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

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References

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    [CrossRef]

2013 (3)

2012 (1)

2011 (2)

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]

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]

2009 (2)

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]

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]

2008 (2)

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]

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 (2)

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

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]

2000 (4)

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]

1999 (1)

R. Heintzmann, C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[CrossRef]

1997 (1)

1994 (2)

1990 (1)

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]

1986 (1)

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]

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, J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[CrossRef] [PubMed]

Bakx, J. L.

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

Bastiaens, P. I. H.

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]

Beck, M.

Berenstein, C.

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]

Best, G.

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, 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, 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, M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6, 339–342 (2009).
[CrossRef] [PubMed]

Cragg, G. E.

Cremer, C.

Cremer, C. G.

R. Heintzmann, C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[CrossRef]

Diehl, V.

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]

Drexler, H. G.

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]

Fienup, J. R.

Fiolka, R.

Frohn, J. T.

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]

Gaedicke, G.

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]

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, 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, M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6, 339–342 (2009).
[CrossRef] [PubMed]

Gu, M.

Gustafsson, M. G. L.

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]

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]

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]

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.

Jovin, T.

Juskaitis, R.

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. 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]

Kawata, S.

Kawata, Y.

Kielhorn, M.

Kleppe, I.

Knapp, H. F.

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]

Kner, P.

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]

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]

Krampert, G.

Lok, M. S.

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]

Lukosz, W.

Mai, S.

Mandula, O.

Minowada, J.

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]

Neil, M. A. A.

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. 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]

Patrick, E.

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]

Pitter, M. C.

Rego, E. H.

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]

Righolt, C. H.

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, 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, M. G. L. Gustafsson, “Super-resolution 3d microscopy of live whole cells using structured illumination,” Nat. Methods 12, 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, J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Shroff, S. A.

Slotman, J. A.

So, P. T.

Somekh, M. G.

Squire, A.

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]

Stallinga, S.

Stemmer, A.

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]

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]

van Vliet, L. J.

Wang, C. J. R.

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]

Wang, L.

Wicker, K.

Williams, D. R.

Wilson, T.

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. 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]

Winoto, L.

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]

Yaroslavsky, L. P.

Young, I. T.

Appl. Opt. (2)

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

L. P. Yaroslavsky, H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162 (1994).
[CrossRef] [PubMed]

Biophys. J. (1)

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]

J. Microsc. (2)

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. Mod. Opt. (1)

W. Lukosz, “Übertragung Nicht-negativer Signale Durch Lineare Filter,” J. Mod. Opt. 9, 335–364 (1962).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

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

Leukemia Res. (1)

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]

Nat. Methods (2)

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]

Opt. Express (4)

Opt. Lett. (4)

Proc. IEEE (1)

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]

Proc. Natl. Acad. Sci. U. S. A. (2)

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]

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]

Proc. SPIE (1)

R. Heintzmann, C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[CrossRef]

Other (1)

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

Supplementary Material (4)

» Media 1: AVI (3046 KB)     
» Media 2: AVI (3170 KB)     
» Media 3: AVI (2528 KB)     
» Media 4: AVI (3687 KB)     

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Figures (5)

Fig. 1
Fig. 1

Plot of the frequency cut-off in normalized frequency coordinates for standard 3D SIM in which the pattern is created by the zeroth and first order diffraction orders with three equally spaced rotation angles for qr = 0.7 and qz = 0.2.

Fig. 2
Fig. 2

Cross-sectional plots of the PSF’s corresponding to CDT (u) in blue and ΛFR (u) in red for 3D-SIM with three rotation angles. The PSF’s are the inverse Fourier transforms of ĈDT (v) and Λ̂FR (v). The left plot depicts the line through the origin in the lateral vz = 0 plane in one of the pattern directions of Rl. The middle plot shows a similar line in the vz = 0 plane in the direction midway between the directions of Rl and Rl+1. The right plot shows the values of the PSF’s along the optical axis.

Fig. 3
Fig. 3

3D SIM image of tubulin filaments from a BPEA cell sample reconstructed using the full 3D SIM Lukosz-bound as apodization function. The intensity of the reconstructed image is linearly stretched, without clipping, between gray values 0 and 255 for display purposes. Shown is the result for a lateral z-slice (A) and an axial y-slice (B) for regularization parameter κ = 10−5. The lateral slice is shown for a range of values for the regularization parameters in Media 1 and for all z-positions in Media 2. The scale bar is 5 μm.

Fig. 4
Fig. 4

3D SIM image of the DNA structure (DAPI) in a multinucleated Reed-Sternberg cell reconstructed using the 3D SIM Lukosz-bound as apodization function. The intensity of the reconstructed image is linearly stretched, without clipping, between gray values 0 and 255 for display purposes. Shown is the result for a lateral z-slice (A) and an axial y-slice (B) for regularization parameter κ = ×10−3. The lateral slice is shown for a range of values for the regularization parameters in Media 3 and for all z-positions in Media 4. The scale bar is 5 μm.

Fig. 5
Fig. 5

Probability density function (pdf) of the normalized pixel values in reconstructed 3D SIM images. For each image, the object pixels are normalized by linearly scaling the background level to 0 and maximum pixel value to 1. The blue lines denote the pdf’s, the bin-widths are 0.01, the red lines indicate the zero-intensity levels. This is done for: the ZEN reconstruction with triangular apodization of the tubulin sample (a); our reconstruction with Lukosz-bound apodization of the tubulin sample (b); the triangular apodized ZEN reconstruction of the DAPI cell sample (c); and our Lukosz-bound apodization in the reconstruction of the DAPI cell sample (d).

Tables (1)

Tables Icon

Table 1 Comparison of the PSF’s corresponding to the four different apodization options for 3D-SIM with three rotation angles. In addition to the absolute values of the measures, a normalized measure with respect to the value for CDT (u) is included as well. The relative minimum value of the PSF (compared to the maximum) is listed as min(PSF)/max(PSF). The lateral and axial full width at half maximum (FWHM lat. and FWHM ax.) are tabulated in the last four columns. The lateral anisotropy leads to a lateral FWHM difference between the Rl and (Rl + Rl+1)/2 directions of a half percent or less in all four cases. Note that the lateral and axial coordinates are normalized differently.

Equations (33)

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H ^ ( v ) = H ( u ) exp ( 2 π i u v ) d u ,
W ^ ( v ) = m w m δ ( v q m ) ,
q z = 1 4 sin 2 ( α / 2 ) ( 1 1 sin 2 α p 2 )
I ^ bands , l m ( v ) = w m H ^ ( v ) T ^ ( v R l q m ) ,
I ^ gen ( v ) = l m s m F ^ l m ( v ) I ^ band , l m ( v + R l q m ) ,
gen = [ l m g ^ ( v l m ) | B ^ l m ( v ) I ^ gen ( v ) s m C ^ ( v ) I ^ band , l m ( v l m ) | 2 + κ | A ^ ( v ) I ^ gen ( v ) | 2 ] d v .
g ^ ( v ) = 1 α exp [ v x 2 + v y 2 2 σ l 2 v z 2 2 σ a 2 ] ,
F ^ l m ( v ) = C ^ ( v ) g ^ ( v + R l q m ) B ^ l m ( v ) * κ A ^ ( v ) 2 + l m g ^ ( v + R l q m ) | B ^ l m ( v ) | 2
H ^ gen ( v ) = C ^ ( v ) l m s m w m g ^ ( v + R l q m ) B ^ l m ( v ) * H ^ ( v + R l q m ) κ A ^ ( v ) 2 + l m g ^ ( v + R l q m ) | B ^ l m ( v ) | 2 .
sin 2 α ( ( v x q m , x ) 2 + ( v y q m , y ) 2 1 ) 2 + 16 sin 4 ( α 2 ) ( v z q m , z cos α 4 sin 2 ( α / 2 ) ) 2 = 1 .
a 4 v z 4 + a 3 v z 3 + a 2 v z 2 + a 1 v z + a 0 = 0 .
a 0 = Q C 2 4 q m , x 2 4 q m , y 2 ,
a 1 = 4 Q B Q C + 8 q m , x cos ϕ tan θ + 8 q m , y sin ϕ tan θ ,
a 2 = 2 Q A Q C + 4 Q B 2 4 tan 2 θ ,
a 3 = 4 Q A Q B ,
a 4 = Q A 2 ,
Q A = 4 tan 2 ( α 2 ) + tan 2 θ ,
Q B = q m , x cos ϕ tan θ + q m , y sin ϕ tan θ + ( 1 + 4 q m , z ) tan 2 ( α 2 ) 1 ,
Q C = 1 sin 2 α ( 1 tan α 2 q m , z tan ( α 2 ) ) 2 q m , x 2 q m , y 2 1 .
ρ m , j ( θ , ϕ ) = v z , m , j ( θ , ϕ ) 1 + tan 2 θ .
ρ ( θ , ϕ ) = max m , j { ρ m , j ( θ , ϕ ) > 0 } .
ρ ( π 2 , ϕ ) = 2 q r cos ϕ r + 2 1 q r 2 sin 2 ϕ r .
ρ ( 0 , ϕ ) = ρ ( π , ϕ ) = q z + 1 ( q r 1 ) 2 sin 2 α cos α 4 sin 2 ( α / 2 ) .
Λ ^ 1 ( v , q c ) = cos ( π | v | | v | + q c ) for | v | q c and Λ ^ 1 ( v ) = 0 elsewhere ,
Λ ^ 3 ( v ) = min { j = 1 3 Λ ^ 1 ( v j , q c , j ( ) ) | 𝒢 } ,
Λ ^ 3 , ϕ ( v ) = min { Λ ^ 1 ( v x , ρ ( π 2 , ϕ ) ) Λ ^ 1 ( v y , ρ ( π 2 , ϕ + π 2 ) ) Λ ^ 1 ( v z , ρ ( 0 , ϕ ) ) | ϕ [ 0 , 2 π ) } ,
Λ ^ OAR ( v ) = min { Λ ^ 3 , ϕ ( v ) , Λ ^ 1 ( v , ρ ( ξ , ψ ) ) } .
Λ ^ F R ( v ) = min { j = 1 3 Λ ^ 1 ( v j , q c , j ( ) ) | SO ( 3 ) } .
ζ = q max , r q max , z = q r + 2 q z + 1 / 2 .
v ˜ z = ζ v z , v ˜ = v x 2 + v y 2 + ζ 2 v z 2 and θ ˜ = arccos ( ζ v z v x 2 + v y 2 + ζ 2 v z 2 ) .
ρ ˜ ( θ ˜ , ϕ ) = ρ ( θ , ϕ ) sin 2 θ + ζ 2 cos 2 θ ,
C ^ D T ( v ) = 𝒟 { l m H ^ ( v + R l q m ) > 0 } max 𝒟 { l m H ^ ( v + R l q m ) > 0 } ,
C ^ line ( v ) = 1 v ρ ( θ , ϕ ) for v ρ ( θ , ϕ ) and C ^ line ( v ) = 0 elsewhere .

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