Abstract

Structured illumination imaging uses multiple images of an object having different phase shifts in the sinusoidally patterned illumination to obtain lateral superresolution in stationary specimens in microscopy. In our recent work we have discussed a method to estimate these phase shifts a posteriori, allowing us to apply this technique to non-stationary objects such as in vivo tissue. Here we show experimental verification of our earlier simulations for phase shift estimation a posteriori. We estimated phase shifts in fluorescence microscopy images for an object having unknown, random translational motion and used them to obtain an artifact-free reconstruction having the expected superresolution.

© 2010 Optical Society of America

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2009 (2)

2008 (2)

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[CrossRef] [PubMed]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images,” Proc. SPIE 7094, 709402-1–11 (2008).

2006 (2)

2005 (1)

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

2004 (1)

L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

2001 (1)

2000 (2)

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

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

1999 (3)

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

M. Gustafsson, “Extended resolution fluorescence microscopy,” Curr. Opin. Struct. Biol. 9, 627–634 (1999).
[CrossRef] [PubMed]

X. Chen and S. R. J. Brueck, “Imaging interferometric lithography: approaching the resolution limits of optics,” Opt. Lett. 24, 124–126 (1999).
[CrossRef]

1997 (1)

1996 (1)

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

1994 (2)

1992 (1)

D. J. Tolhurst, Y. Tadmore, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

1967 (2)

Agard, D. A.

M. Gustafsson, L. Shao, D. A. Agard, and J. W. Sedat, “Fluorescence microscopy without resolution limit,” in Biophotonics/Optical Interconnects and VLSI Photonics/WBM Microcavities, 2004 Digest of the LEOS Summer Topical Meetings (IEEE, 2004), Vol. 2, pp. 28–30.

Brueck, S. R. J.

Calef, B.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

Caulfield, H. J.

Chao, T.

D. J. Tolhurst, Y. Tadmore, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Chen, X.

Chung, E.

Cragg, G. E.

Cremer, C.

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

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

Fienup, J. R.

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]

S. T. Thurman and J. R. Fienup, “Wiener reconstruction of undersampled imagery,” J. Opt. Soc. Am. A 26, 283–288 (2009).
[CrossRef]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[CrossRef] [PubMed]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images,” Proc. SPIE 7094, 709402-1–11 (2008).

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Estimation of phase shifts in structured illumination for high resolution imaging,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper FMH4.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase shift estimation in structured illumination imaging for lateral resolution enhancement,” in Signal Recovery and Synthesis, Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper SMA2.

García, J.

Gerwe, D. R.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

Guizar-Sicairos, M.

Gustaffson, M.

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

Gustafsson, M.

M. Gustafsson, “Extended resolution fluorescence microscopy,” Curr. Opin. Struct. Biol. 9, 627–634 (1999).
[CrossRef] [PubMed]

M. Gustafsson, “Extended-resolution reconstruction of structured illumination microscopy data,” in Computational Optical Sensing and Imaging Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JMA2.

M. Gustafsson, L. Shao, D. A. Agard, and J. W. Sedat, “Fluorescence microscopy without resolution limit,” in Biophotonics/Optical Interconnects and VLSI Photonics/WBM Microcavities, 2004 Digest of the LEOS Summer Topical Meetings (IEEE, 2004), Vol. 2, pp. 28–30.

Heintzmann, R.

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

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

Hell, S. W.

Helstrom, C. W.

Jain, M.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

Jovin, T. M.

Kim, D.

Kiryuschev, I.

Konforti, N.

Kuznetsova, Y.

Lohmann, A. W.

Lukosz, W.

Luna, C.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

Marom, E.

Mendlovic, D.

Mico, V.

Sabo, E.

Schaefer, L. H.

L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
[CrossRef] [PubMed]

Schaffer, J.

L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
[CrossRef] [PubMed]

Schuster, D.

L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
[CrossRef] [PubMed]

Schwarz, C. J.

Sedat, J. W.

M. Gustafsson, L. Shao, D. A. Agard, and J. W. Sedat, “Fluorescence microscopy without resolution limit,” in Biophotonics/Optical Interconnects and VLSI Photonics/WBM Microcavities, 2004 Digest of the LEOS Summer Topical Meetings (IEEE, 2004), Vol. 2, pp. 28–30.

Shao, L.

M. Gustafsson, L. Shao, D. A. Agard, and J. W. Sedat, “Fluorescence microscopy without resolution limit,” in Biophotonics/Optical Interconnects and VLSI Photonics/WBM Microcavities, 2004 Digest of the LEOS Summer Topical Meetings (IEEE, 2004), Vol. 2, pp. 28–30.

Shemer, A.

Sheppard, C. J. R.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1983).

Shroff, S. A.

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]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images,” Proc. SPIE 7094, 709402-1–11 (2008).

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Estimation of phase shifts in structured illumination for high resolution imaging,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper FMH4.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase shift estimation in structured illumination imaging for lateral resolution enhancement,” in Signal Recovery and Synthesis, Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper SMA2.

So, P. T.

So, P. T. C.

Strang, G.

G. Strang, Linear Algebra and Its Applications (Thomson Learning, Inc., 1998).

Tadmore, Y.

D. J. Tolhurst, Y. Tadmore, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Thurman, S. T.

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmore, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

van der Schaaf, A.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

Wichmann, J.

Williams, D. R.

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]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images,” Proc. SPIE 7094, 709402-1–11 (2008).

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase shift estimation in structured illumination imaging for lateral resolution enhancement,” in Signal Recovery and Synthesis, Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper SMA2.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Estimation of phase shifts in structured illumination for high resolution imaging,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper FMH4.

Wilson, T.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1983).

Yaroslavsky, L. P.

Zalevsky, Z.

Appl. Opt. (3)

Curr. Opin. Struct. Biol. (1)

M. Gustafsson, “Extended resolution fluorescence microscopy,” Curr. Opin. Struct. Biol. 9, 627–634 (1999).
[CrossRef] [PubMed]

J. Microsc. (2)

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

L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Ophthalmic Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmore, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (6)

Proc. SPIE (3)

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images,” Proc. SPIE 7094, 709402-1–11 (2008).

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for nonlinear image restoration using a prior on the object power spectrum,” in Proc. SPIE 5896, 1–15 (2005).

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

Vision Res. (1)

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

Other (6)

M. Gustafsson, “Extended-resolution reconstruction of structured illumination microscopy data,” in Computational Optical Sensing and Imaging Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JMA2.

M. Gustafsson, L. Shao, D. A. Agard, and J. W. Sedat, “Fluorescence microscopy without resolution limit,” in Biophotonics/Optical Interconnects and VLSI Photonics/WBM Microcavities, 2004 Digest of the LEOS Summer Topical Meetings (IEEE, 2004), Vol. 2, pp. 28–30.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase shift estimation in structured illumination imaging for lateral resolution enhancement,” in Signal Recovery and Synthesis, Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper SMA2.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Estimation of phase shifts in structured illumination for high resolution imaging,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper FMH4.

G. Strang, Linear Algebra and Its Applications (Thomson Learning, Inc., 1998).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1983).

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Figures (8)

Fig. 1
Fig. 1

Visualization of sinusoidal grid patterned image in the Fourier domain [17].

Fig. 2
Fig. 2

Visualization of extension of OTF passband with structured illumination.

Fig. 3
Fig. 3

(a) Conventional image, (b) deconvolved conventional image, (c) sinusoidal grid patterned image, (d) deconvolved noise-reduced conventional image.

Fig. 4
Fig. 4

Stretched Fourier transform of (a) conventional image, (b) deconvolved conventional image, (c) sinusoidal grid patterned image, (d) deconvolved noise-reduced conventional image.

Fig. 5
Fig. 5

(a)–(e) show the 47 % superresolved image obtained using two sets of (a) 20, (b) 12, (c) 6, (d) 4, (e) 3 sinusoidal grid patterned images in each orientation with random phase shifts in the sinusoidal illumination. (f)–(h) used two sets of (f) 6, (g) 4, and (h) 3 images with phase shifts well spaced over 360°.

Fig. 6
Fig. 6

(a)–(e) show the stretched Fourier transform of a 47 % superresolved image obtained using two sets of (a) 20, (b) 12, (c) 6, (d) 4, (e) 3 sinusoidal grid patterned images in each orientation with random phase shifts in the sinusoidal illumination. (f)–(h) used two sets of (f) 6, (g) 4, and (h) 3 images with phase shifts well spaced over 360°.

Fig. 7
Fig. 7

Phase shift estimates for sinusoidally patterned images with random phase shifts having 20, 12, 6, 4, and 3 images in each set and 2 orientations for each set.

Fig. 8
Fig. 8

Phase shift estimates for sinusoidally patterned images with phase shifts spread over 360° having 6, 4, and 3 images in each set and 2 orientations for each set.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

I s , n ( x , y ) = 1 2 [ 1 + m cos ( 4 π f o x + 2 φ n ) ] ,
G ̂ n ( f x , f y ) = G n ( f x , f y ) + N n ( f x , f y ) = 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) + m 4 H 1 ( 2 f o , 0 ) e i 2 φ n H 2 ( f x , f y ) G o ( f x 2 f o , f y ) + m 4 H 1 ( 2 f o , 0 ) e i 2 φ n H 2 ( f x , f y ) G o ( f x + 2 f o , f y ) + N n ( f x , f y ) .
A = [ 1 e i 2 φ 1 e i 2 φ 1 1 e i 2 φ 2 e i 2 φ 2 . . . . . . 1 e i 2 φ N e i 2 φ N ] N × 3 ,
X = [ 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x , f y ) G o ( f x 2 f o , f y ) 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x , f y ) G o ( f x + 2 f o , f y ) ] 3 × 1 ,
B = [ G ̂ 1 ( f x , f y ) G ̂ 2 ( f x , f y ) . . G ̂ N ( f x , f y ) ] N × 1 .
I c 1 ( f x , f y ) = 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) ,
otf 1 ( f x , f y ) = 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) .
I c 2 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) G o ( f x , f y )
I c 3 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x 2 f o , f y ) G o ( f x , f y ) ,
otf 2 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) ,
SNR c 1 = 1 σ 2 | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) ,
SNR c 2 = 1 4 σ 2 | m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) | 2 Φ o ( f x , f y ) .
SNR c 1 ( f x , f y ) = [ N ( 4 σ 2 ) ] | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) .
SNR c i = η i | otf i ( f x , f y ) | 2 Φ o Φ N i ,
I rec ( x , y ) = IFT { i = 1 M [ I ̂ c i ( f x , f y ) otf i ( f x , f y ) × SNR c i c + j = 1 M SNR c j ] } ,
I rec ( x , y ) = IFT { i = 1 M [ I ̂ c i ( f x , f y ) otf i * ( f x , f y ) η i Φ o Φ N i c + j = 1 M η j | otf j ( f x , f y ) | 2 Φ o Φ N j ] } .
G n ( 2 f o , 0 ) = 1 2 H 1 ( 0 , 0 ) H 2 ( 2 f o , 0 ) G o ( 2 f o , 0 ) + e i 2 φ n m 4 H 1 ( 2 f o , 0 ) H 2 ( 2 f o , 0 ) G o ( 0 , 0 ) + e i 2 φ n m 4 H 1 ( 2 f o , 0 ) H 2 ( 2 f o , 0 ) G o ( 4 f o , 0 ) + N n ( 2 f o , 0 ) .
2 φ ̂ n = tan 1 { imag [ G n ( 2 f o , 0 ) ] real [ G n ( 2 f o , 0 ) ] } ,
I ̂ c 1 ( f x , f y ) = 1 4 [ G ̂ 1 ( f x , f y ) + G ̂ 2 ( f x , f y ) + G ̂ 3 ( f x , f y ) + G ̂ 4 ( f x , f y ) ] = 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) + 1 4 [ N 1 ( f x , f y ) + N 2 ( f x , f y ) + N 3 ( f x , f y ) + N 4 ( f x , f y ) ] .
Φ S 1 ( f x , f y ) = | 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) | 2 = 1 4 | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) ,
Φ N 1 ( f x , f y ) = 1 16 [ σ N 1 2 + σ N 2 2 + σ N 3 2 + σ N 4 2 ] = 1 4 σ 2 ,
SNR c 1 = Φ S 1 ( f x , f y ) Φ N 1 = 1 σ 2 | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) .
[ G 1 ( f x , f y ) G 3 ( f x , f y ) ] i [ G 2 ( f x , f y ) G 4 ( f x , f y ) ] = m H 1 ( 2 f o , 0 ) H 2 ( f x , f y ) G o ( f x 2 f o , f y ) + [ N 1 ( f x , f y ) N 3 ( f x , f y ) ] i [ N 2 ( f x , f y ) N 4 ( f x , f y ) ] .
I ̂ c 2 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) G o ( f x , f y ) + 1 4 { [ N 1 ( f x + 2 f o , f y ) N 3 ( f x + 2 f o , f y ) ] i [ N 2 ( f x + 2 f o , f y ) N 4 ( f x + 2 f o , f y ) ] } .
Φ S 2 ( f x , f y ) = | 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) G o ( f x , f y ) | 2 = 1 16 | m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) | 2 Φ o ( f x , f y ) .
Φ N 2 ( f x , f y ) = 1 16 ( σ N 1 2 + σ N 2 2 + σ N 3 2 + σ N 4 2 ) = 1 4 σ 2 ,
SNR c 2 = Φ S 2 ( f x , f y ) Φ N 2 = 1 4 σ 2 | m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) | 2 Φ o ( f x , f y ) .
I ̂ c 1 ( f x , f y ) = ( 1 + 2 2 2 ) [ G ̂ 1 ( f x , f y ) + G ̂ 4 ( f x , f y ) ] 1 2 2 [ G ̂ 2 ( f x , f y ) + G ̂ 3 ( f x , f y ) ] = 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) + ( 1 + 2 2 2 ) [ N 1 ( f x , f y ) + N 4 ( f x , f y ) ] 1 2 2 [ N 2 ( f x , f y ) + N 3 ( f x , f y ) ] .
Φ S 1 ( f x , f y ) = | 1 2 H 1 ( 0 , 0 ) H 2 ( f x , f y ) G o ( f x , f y ) | 2 = 1 4 | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) .
Φ N 1 ( f x , f y ) = ( 1 + 2 2 2 ) 2 ( σ N 1 2 + σ N 4 2 ) + ( 1 2 2 ) 2 ( σ N 2 2 + σ N 3 2 ) = 2 + 1 2 σ 2 = 1.7 σ 2 ,
SNR c 1 = Φ S 1 ( f x , f y ) Φ N 1 = 1 6.8 σ 2 | H 1 ( 0 , 0 ) H 2 ( f x , f y ) | 2 Φ o ( f x , f y ) ,
( 0.04 + i 0.5 ) G ̂ 1 ( f x , f y ) + ( 0.3 i 0.4 ) G ̂ 2 ( f x , f y ) + ( 0.1 i 0.5 ) G ̂ 3 ( f x , f y ) + ( 0.4 + i 0.3 ) G ̂ 4 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x , f y ) G o ( f x 2 f o , f y ) + ( 0.04 + i 0.5 ) N 1 ( f x , f y ) + ( 0.3 i 0.4 ) N 2 ( f x , f y ) + ( 0.1 i 0.5 ) N 3 ( f x , f y ) + ( 0.4 + i 0.3 ) N 4 ( f x , f y ) .
I ̂ c 2 ( f x , f y ) = 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) G o ( f x , f y ) + ( 0.04 + i 0.5 ) N 1 ( f x + 2 f o , f y ) + ( 0.3 i 0.4 ) N 2 ( f x + 2 f o , f y ) + ( 0.1 i 0.5 ) N 3 ( f x + 2 f o , f y ) + ( 0.4 + i 0.3 ) N 4 ( f x + 2 f o , f y ) .
Φ S 2 ( f x , f y ) = | 1 4 m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) G o ( f x , f y ) | 2 = 1 16 | m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) | 2 Φ o ( f x , f y ) .
Φ N 2 ( f x , f y ) = ( 0.04 + i 0.5 ) 2 σ N 1 2 + ( 0.3 i 0.4 ) 2 σ N 2 2 + ( 0.1 i 0.5 ) 2 σ N 3 2 + ( 0.4 + i 0.3 ) 2 σ N 4 2 = 0.98 σ 2 ,
SNR c 2 = Φ S 2 ( f x , f y ) Φ N 2 = 1 15.7 σ 2 | m H 1 ( 2 f o , 0 ) H 2 ( f x + 2 f o , f y ) | 2 Φ o ( f x , f y ) .

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