Abstract

Many biological structures of interest are beyond the diffraction limit of conventional microscopes and their visualization requires application of super-resolution techniques. Such techniques have found remarkable success in surpassing the diffraction limit to achieve sub-diffraction limited resolution; however, they are predominantly limited to fluorescent samples. Here, we introduce a non-fluorescent analogue to structured illumination microscopy, termed structured oblique illumination microscopy (SOIM), where we use simultaneous oblique illuminations of the sample to multiplex high spatial-frequency content into the frequency support of the system. We introduce a theoretical framework describing how to demodulate this multiplexed information to reconstruct an image with a spatial-frequency support exceeding that of the system’s classical diffraction limit. This approach allows enhanced-resolution imaging of non-fluorescent samples. Experimental confirmation of the approach is obtained in a reflection test target with moderate numerical aperture.

© 2012 OSA

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References

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2012

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

2010

2009

2008

M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opin. Chem. Biol.12(5), 505–514 (2008).
[CrossRef] [PubMed]

2007

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

2006

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

A. Mudassar, A. R. Harvey, A. H. Greenaway, and J. D. C. Jones, “Resolution beyond classical limites with spatial frequency heterodyning,” Chin. Opt. Lett.4(3), 148–151 (2006).

2005

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A.102(37), 13081–13086 (2005).
[CrossRef] [PubMed]

2002

2000

M. Gustafsson, D. Agard, and J. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE3919, 141–150 (2000).
[CrossRef]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc.198(2), 82–87 (2000).
[CrossRef] [PubMed]

1995

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. (Berl.)60(5), 495–497 (1995).
[CrossRef]

1994

1992

1966

Agard, D.

M. Gustafsson, D. Agard, and J. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE3919, 141–150 (2000).
[CrossRef]

Bates, M.

M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opin. Chem. Biol.12(5), 505–514 (2008).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Cremer, C.

Davidson, M. W.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Dubertret, B.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

Fienup, J. R.

Greenaway, A. H.

Gustafsson, M.

M. Gustafsson, D. Agard, and J. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE3919, 141–150 (2000).
[CrossRef]

Gustafsson, M. G.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A.102(37), 13081–13086 (2005).
[CrossRef] [PubMed]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc.198(2), 82–87 (2000).
[CrossRef] [PubMed]

Gustafsson, M. G. L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Hajek, K. M.

Harvey, A. R.

Heckenberg, N.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Heintzmann, R.

Hell, S. W.

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. (Berl.)60(5), 495–497 (1995).
[CrossRef]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett.19(11), 780–782 (1994).
[CrossRef] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Huang, B.

M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opin. Chem. Biol.12(5), 505–514 (2008).
[CrossRef] [PubMed]

Johansson, G. A.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Jones, J. D. C.

Jovin, T. M.

Kamps-Hughes, N.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Kroug, M.

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. (Berl.)60(5), 495–497 (1995).
[CrossRef]

Lai, K.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Leith, E. N.

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Littleton, B.

K. M. Hajek, B. Littleton, D. Turk, T. J. McIntyre, and H. Rubinsztein-Dunlop, “A method for achieving super-resolved widefield CARS microscopy,” Opt. Express18(18), 19263–19272 (2010).
[CrossRef] [PubMed]

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Longstaff, D.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Loriette, V.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

Lukosz, W.

Macklin, J. J.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

McIntyre, T. J.

Mudassar, A.

Munroe, P.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Olivo-Marin, J. C.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

Orieux, F.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Rego, E. H.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

K. M. Hajek, B. Littleton, D. Turk, T. J. McIntyre, and H. Rubinsztein-Dunlop, “A method for achieving super-resolved widefield CARS microscopy,” Opt. Express18(18), 19263–19272 (2010).
[CrossRef] [PubMed]

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Sarafis, V.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Sedat, J.

M. Gustafsson, D. Agard, and J. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE3919, 141–150 (2000).
[CrossRef]

Sepulveda, E.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

Shao, L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Shroff, S. A.

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Sun, P. C.

Turk, D.

Wichmann, J.

Williams, D. R.

Winoto, L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Zhuang, X.

X. Zhuang, “Nano-imaging with Storm,” Nat. Photonics3(7), 365–367 (2009).
[CrossRef] [PubMed]

M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opin. Chem. Biol.12(5), 505–514 (2008).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. (Berl.)

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. (Berl.)60(5), 495–497 (1995).
[CrossRef]

Chin. Opt. Lett.

Curr. Opin. Chem. Biol.

M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opin. Chem. Biol.12(5), 505–514 (2008).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

F. Orieux, E. Sepulveda, V. Loriette, B. Dubertret, and J. C. Olivo-Marin, “Bayesian estimation for optimized structured illumination microscopy,” IEEE Trans. Image Process.21(2), 601–614 (2012).
[CrossRef] [PubMed]

J. Microsc.

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc.198(2), 82–87 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Micron

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron38(2), 150–157 (2007).
[CrossRef] [PubMed]

Nat. Methods

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Nat. Photonics

X. Zhuang, “Nano-imaging with Storm,” Nat. Photonics3(7), 365–367 (2009).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A.102(37), 13081–13086 (2005).
[CrossRef] [PubMed]

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Proc. SPIE

M. Gustafsson, D. Agard, and J. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE3919, 141–150 (2000).
[CrossRef]

Science

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Other

J. Pawley, Handbook of Biological Confocal Microscopy (Springer Science+Business Media, New York, 1989).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1959).

J. Goodman, Introduction to Fourier Optics (Roberts & Company, Greenwood Village, CO, 2005).

J. Prince and J. Links, Medical Imaging Signals and Systems (Pearson Prentice Hall, Upper Saddle River, NJ, 2006).

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Figures (4)

Fig. 1
Fig. 1

Numerical simulation showing the extended transfer function and its enhanced-resolution reconstruction ability. (a) The extended transfer function given by Eq. (4) and the intensity of the associated structured illumination field given by Eq. (7). The transfer function’s axes are given in multiples of �� c . The green dashed circle outlines the frequency support of the original diffraction limit. (b,c,d) True, orthogonally illuminated, and enhanced-resolution images, respectively, of a sample USAF test chart. (e,f,g) Magnified view of Group −1 El 4 set of bars at 0.71 lpmm from (b,c,d), respectively. Note the enhanced-resolution capabilities shown in (d,g).

Fig. 2
Fig. 2

Schematic structured illumination imaging system with moderate numerical aperture. SM: single mode fiber at 405 nm; CL: collimating lens; DLP: pixel-addressable diffractive element; L1, L2, L3, L4: lens (f = 120, 200, 150, 50 mm); M: mask; BS: beam splitter; LA: limiting aperture; S: coherently scattering sample.

Fig. 3
Fig. 3

Experimental data showing (a) an orthogonally-illuminated (BF) image and (b) an enhanced-resolution (SI) reconstruction. (c) Horizontal cross-sectional profiles taken from (a),(b). (d) Vertical cross-sectional profiles taken from (a),(b). (e) Intensity modulations vs bar freq compared between BF and SI.

Fig. 4
Fig. 4

Top: (a) Experimental orthogonally-illuminated (BF) image and (b) enhanced-resolution (SI) reconstruction of 20 µm polystyrene beads. (c) and (d) represent enlarged regions of (a) and (b). (e) shows a comparison of the cross-sectional intensity profiles between the BF and SI images at the locations marked in yellow in images (c) and (d) Bottom: Same as above, but for a histological sample of a mouse joint.

Equations (33)

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y( r )= | h c ( r )[ x( r )[ h c ( r )i( r ) ] ] | 2
Y( 𝝎 )=autocorr( H c ( 𝝎 )[ X( 𝝎 )( H c ( 𝝎 )I( 𝝎 ) ) ] )
Y BF ( 𝝎 )=autocorr( H c ( 𝝎 )X( 𝝎 ) )
H SI ( 𝝎 )= H c ( 𝝎 𝝎 0 )+ H c ( 𝝎+ 𝝎 0 )+ H c ( 𝝎𝝎 ' 0 )+ H c ( 𝝎+𝝎 ' 0 )
Y SI ( 𝝎 )=[ autocorr[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) ]+autocorr[ H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] +autocorr[ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ]+autocorr[ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) ] ] +[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) +[ H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) ] +[ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c (𝝎𝝎 ' 0 )X( 𝝎 ) +[ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ] +[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) + H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] +[ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) + H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) ] +[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) + H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] +[ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) + H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ]
Y SI ( 𝝎 )= n=0 8 G n ( 𝝎 )
i( r )=cos( 𝝎 0 r+ ϕ k )+cos( 𝝎 ' 𝝎 r+ϕ ' k )
Y k ( 𝝎 )=[ autocorr[ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) ]+autocorr[ H c ( 𝝎 )X( 𝝎 𝝎 0 ) ] +autocorr[ H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) ]+autocorr[ H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) ] ] + e j2 ϕ k [ H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) + e j2 ϕ k [ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 ) ] + e j2 ϕ k ' [ H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) ] + e j2 ϕ k ' [ H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) ] + e j( ϕ k ϕ k ' ) [ H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) + H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) ]+ e j( ϕ k ϕ k ' ) [ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) + H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 ) ] + e j( ϕ k + ϕ k ' ) [ H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) + H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) ] + e j( ϕ k + ϕ k ' ) [ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) + H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 ) ]
Y k ( 𝝎 )= n=0 8 e j Φ n F n ( 𝝎 )
Y k ( 𝝎 )= G 0 ( 𝝎 )+ e j2 ϕ k G 1 ( 𝝎2 𝝎 0 )+ e j2 ϕ k G 2 ( 𝝎+2 𝝎 0 ) + e j2 ϕ k ' G 3 ( 𝝎2𝝎 ' 0 )+ e j2 ϕ k ' G 4 (𝝎+2𝝎 ' 0 + e j( ϕ k ϕ k ' ) G 5 ( 𝝎( 𝝎 0 𝝎 ' 0 ) )+ e j( ϕ k ϕ k ' ) G 6 ( 𝝎+( 𝝎 0 𝝎 ' 0 ) ) + e j( ϕ k + ϕ k ' ) G 7 ( 𝝎( 𝝎 0 +𝝎 ' 0 ) )+ e j( ϕ k + ϕ k ' ) G 8 ( 𝝎+( 𝝎 0 +𝝎 ' 0 ) )
m= I max I min I max + I min
m * = I max I min I max + I min
F v ( 𝝎 )= H c ( 𝝎 )X( 𝝎+ 𝝎 1 ) H c ( 𝝎 )X( 𝝎+ 𝝎 2 )
G v ( 𝝎 )= H c ( 𝝎 𝝎 1 )X( 𝝎 ) H c ( 𝝎 𝝎 2 )X( 𝝎 )
F v ( 𝝎 )= H c ( 𝝎 )X( 𝝎+ 𝝎 1 ) H c ( 𝝎 )X( 𝝎+ 𝝎 2 ) ={ 1 { H c ( 𝝎 )X( 𝝎+ 𝝎 1 ) H c ( 𝝎 )X( 𝝎+ 𝝎 2 ) } } ={ 1 { H c ( 𝝎 )X( 𝝎+ 𝝎 1 ) } 1 { H c ( 𝝎 )X( 𝝎+ 𝝎 2 ) } * } ={ [ h c ( r )x( r )exp( j 𝝎 1 r ) ] [ h c ( r )x( r )expexp( j 𝝎 2 r ) ] * } ={ [ h c ( τ )x( rτ )exp( j 𝝎 1 ( rτ ) )dτ ] [ h c ( τ )x( rτ )exp( j 𝝎 2 (rτ) )dτ ] * } ={ [ h c ( τ )x( rτ )exp( j 𝝎 1 r )exp( j 𝝎 1 τ )dτ ] [ h c ( τ )x( rτ )exp( j 𝝎 2 r )exp( j 𝝎 2 τ )dτ ] * } ={ exp( j( 𝝎 1 𝝎 2 )r )[ h c ( τ )expexp( j 𝝎 1 τ )x( rτ )dτ ] [ h c ( τ )exp( j 𝝎 2 τ )x( rτ )dτ ] * } ={ exp( j( 𝝎 1 𝝎 2 )r )[ h c ( r )exp( j 𝝎 1 τ )x( r ) ] [ h c ( r )exp( j 𝝎 2 τ )x( r ) ] * } ={ exp( j( 𝝎 1 𝝎 2 )r ) }{ [ h c ( r )exp( j 𝝎 1 τ )x( r ) ] [ h c ( r )exp( j 𝝎 2 τ )x( r ) ] * } =δ( 𝝎+( 𝝎 1 𝝎 2 ) )[ H c ( 𝝎 𝝎 1 )X( 𝝎 ) H c ( 𝝎 𝝎 2 )X( 𝝎 ) ] = G v ( 𝝎+( 𝝎 1 𝝎 2 ) )
F 0 ( 𝝎 )=autocorr[ H c ( 𝝎 )X( 𝝎 𝝎 0 ) ]+autocorr[ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) ] +autocorr[ H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) ] +autocorr[ H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) ] =autocorr[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) ]+autocorr[ H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] +autocorr[ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ]+autocorr[ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) ] = G 0 ( 𝝎 )
F 1 ( 𝝎 )= H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) =δ( 𝝎2 𝝎 0 )[ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] = G 1 ( 𝝎2 𝝎 0 )
F 2 ( 𝝎 )= H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 ) =δ( 𝝎+2 𝝎 0 )[ H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) ] = G 2 ( 𝝎+2 𝝎 0 )
F 3 ( 𝝎 )= H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) =δ( 𝝎2𝝎 ' 0 )[ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) ] = G 3 ( 𝝎2𝝎 ' 0 )
F 4 ( 𝝎 )= H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) =δ( 𝝎+2𝝎 ' 0 )[ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ] = G 4 ( 𝝎+2𝝎 ' 0 )
F 5 ( 𝝎 )= H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 )+ H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) =δ( 𝝎( 𝝎 0 𝝎 ' 0 ) ) [ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) + H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] = G 5 ( 𝝎( 𝝎 0 𝝎 ' 0 ) )
F 6 ( 𝝎 )= H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 )+ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) =δ( 𝝎+( 𝝎 0 𝝎 ' 0 ) ) [ H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) + H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) ] = G 6 ( 𝝎+( 𝝎 0 𝝎 ' 0 ) )
F 7 ( 𝝎 )= H c ( 𝝎 )X( 𝝎 𝝎 0 ) H c ( 𝝎 )X( 𝝎+𝝎 ' 0 )+ H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) =δ( 𝝎( 𝝎 0 +𝝎 ' 0 ) ) [ H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) + H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎 𝝎 0 )X( 𝝎 ) ] = G 7 ( 𝝎( 𝝎 0 +𝝎 ' 0 ) )
F 8 ( 𝝎 )= H c ( 𝝎 )X( 𝝎+𝝎 ' 0 ) H c ( 𝝎 )X( 𝝎 𝝎 0 )+ H c ( 𝝎 )X( 𝝎+ 𝝎 0 ) H c ( 𝝎 )X( 𝝎𝝎 ' 0 ) =δ( 𝝎+( 𝝎 0 +𝝎 ' 0 ) ) [ H c ( 𝝎𝝎 ' 0 )X( 𝝎 ) H c ( 𝝎+ 𝝎 0 )X( 𝝎 ) + H c ( 𝝎 𝝎 0 )X( 𝝎 ) H c ( 𝝎+𝝎 ' 0 )X( 𝝎 ) ] = G 8 ( 𝝎+( 𝝎 0 +𝝎 ' 0 ) )
F 0 ( 𝝎 )= G 0 ( 𝝎 )
F 1 ( 𝝎 )= G 1 ( 𝝎2 𝝎 0 )
F 2 ( 𝝎 )= G 2 ( 𝝎+2 𝝎 0 )
F 3 ( 𝝎 )= G 3 ( 𝝎2𝝎 ' 0 )
F 4 ( 𝝎 )= G 4 ( 𝝎+2𝝎 ' 0 )
F 5 ( 𝝎 )= G 5 ( 𝝎( 𝝎 0 𝝎 ' 0 ) )
F 6 ( 𝝎 )= G 6 ( 𝝎+( 𝝎 0 𝝎 ' 0 ) )
F 7 ( 𝝎 )= G 7 ( 𝝎( 𝝎 0 +𝝎 ' 0 ) )
F 8 ( 𝝎 )= G 8 ( 𝝎+( 𝝎 0 +𝝎 ' 0 ) )

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