Abstract

Beamforming and holographic imaging procedures are widely used in many applications such as radar sensing, sonar, and in the area of microwave medical imaging. Nevertheless, an analytical comparison of the methods has not been done. In this paper, the Point Spread Functions pertaining to the two methods are analytically determined. This allows a formal comparison of the two techniques, and to easily highlight how the performance depends on the configuration parameters, including frequency range, number of scatterers, and data discretization. It is demonstrated that the beamforming and holography basically achieve the same resolution but beamforming requires a cheaper (less sensors) configuration..

© 2016 Optical Society of America

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References

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2014 (1)

R. Solimene, I. Catapano, G. Gennarelli, A. Dell’Aversano, A. Cuccaro, and F. Soldovieri, “SAR imaging algorithms and some unconventional applications: a unified mathematical overview,” IEEE Signal Process. Mag. 31(4), 90–98 (2014).
[Crossref]

2013 (1)

Y.-D. Joh, Y. M. Kwon, J. Y. Huh, and W.-K. Park, “Structure analysis of single-and multi-frequency subspace migrations in inverse scattering problems,” Prog. Electromag. Res. 136, 607–622 (2013).
[Crossref]

2011 (1)

D. Flores-Tapia and S. Pistorius, “Real time breast microwave radar image reconstruction using circular holography: a study of experimental feasibility,” Med Phys. 38, 5420–5431 (2011).
[Crossref] [PubMed]

2009 (3)

Y. T. Cho, M. J. Roan, and J. S. Bolton, “A comparison of near-field beamforming and acoustical holography for sound source visualization,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 223, 819–834 (2009).
[Crossref]

M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beam-forming algorithms,” Progress in Electromagnetics Research 97, 141–158 (2009).
[Crossref]

F. Joud, F. Laloe, M. Atlan, J. Hare, and M. Gross, “Imaging a vibrating object by sideband digital holography,” Opt. Express 17, 2774–2779 (2009).
[Crossref] [PubMed]

2008 (2)

H. Lim, N. Nhung, E. Li, and N. Thang, “Confocal microwave imaging for breast cancer detection: delay-multiply-and-sum image reconstruction algorithm,” IEEE Trans. Biomed. Eng. 55(6), 1697–1704 (2008).
[Crossref] [PubMed]

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

2007 (1)

C. P. Oden, H. M. Powers, D. L. Wright, and G. R. Olhoeft, “Improving GPR Image Resolution in Lossy Ground Using Dispersive Migration,” IEEE Trans. Geosc. Rem. Sens. 45, 2492–2500 (2007).
[Crossref]

2006 (1)

B. Drinkwater and P. Wilcox, “Ultrasonic arrays for non-destructive evaluation: a review,” NDT and E Int. 39(6), 525–541 (2006).
[Crossref]

2005 (3)

A. B. Baggeroer, “Sonar arrays and array processing,” Rev. Prog. Quant. Nondestr. Eval. 13(3), 137(2005).
[Crossref]

E. Julliard, S. Pauzin, F. Simon, and D. Biron, “Acoustic sources localization in presence of reverberation,” J. Acoust. Soc. Am. 118, 1886 (2005).
[Crossref]

M. Takeda, W. Wamg, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[Crossref] [PubMed]

2002 (1)

J. Chen, K. Yao, and R. Hudson, “Source localization and beamforming,” IEEE Signal Process. Mag. 19(2), 30–39 (2002).
[Crossref]

2000 (1)

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-D Radar imaging using range migration techniques,” IEEE Trans. Antennas Propag.,  48(5), 728–737 (2000).
[Crossref]

1999 (1)

P. N. T. Wells, “Ultrasonic imaging of the human body,” Rep. Prog. Phys. 62(5), 671–722 (1999).
[Crossref]

1998 (2)

S. C. Hagness, A. Taove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna array sensors,” IEEE Trans. Biomed. Eng. 45(12), 1470–1479 (1998).
[Crossref] [PubMed]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: Results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[Crossref]

1997 (1)

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

1996 (1)

H. Krim and M. Viberg, “Two decades of array signal processing research: the parametric approach,” IEEE Signal Process. Mag. 13(4), 67–94 (1996).
[Crossref]

1988 (1)

B. D. Van Veen and K. M. Buckley, “Beamforming: a versatile approach to spatial filtering,” IEEE ASSP Mag. 5(2), 4–24 (1988).
[Crossref]

1984 (1)

J. Gazdag and P. Sguazzero, “Migration of Seismic Data,” Proc. IEEE 72(10), 1302–1315 (1984).
[Crossref]

1978 (1)

R. H. Stolt, “Migration by Fourier transform,” Geophys. 43(1), 23–48 (1978).
[Crossref]

1970 (1)

1957 (1)

L. Gatteschi, “Sul comportamento asintotico delle funzioni di Bessel di prima specie di ordine ed argomento quasi uguali,” Ann. Mat. Pura Appl. 43(4), 97–117 (1957).
[Crossref]

Andre, M. P.

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

Atlan, M.

Baggeroer, A. B.

A. B. Baggeroer, “Sonar arrays and array processing,” Rev. Prog. Quant. Nondestr. Eval. 13(3), 137(2005).
[Crossref]

Benjamin, R.

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

Biron, D.

E. Julliard, S. Pauzin, F. Simon, and D. Biron, “Acoustic sources localization in presence of reverberation,” J. Acoust. Soc. Am. 118, 1886 (2005).
[Crossref]

Blinstein, N.

N. Blinstein and R. A. Handelsman, Asymtotic Expansions of Integrals, (Dover Publications, 1986).

Bolton, J. S.

Y. T. Cho, M. J. Roan, and J. S. Bolton, “A comparison of near-field beamforming and acoustical holography for sound source visualization,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 223, 819–834 (2009).
[Crossref]

Brancaccio, A.

Bridges, J. E.

S. C. Hagness, A. Taove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna array sensors,” IEEE Trans. Biomed. Eng. 45(12), 1470–1479 (1998).
[Crossref] [PubMed]

Buckley, K. M.

B. D. Van Veen and K. M. Buckley, “Beamforming: a versatile approach to spatial filtering,” IEEE ASSP Mag. 5(2), 4–24 (1988).
[Crossref]

Catapano, I.

R. Solimene, I. Catapano, G. Gennarelli, A. Dell’Aversano, A. Cuccaro, and F. Soldovieri, “SAR imaging algorithms and some unconventional applications: a unified mathematical overview,” IEEE Signal Process. Mag. 31(4), 90–98 (2014).
[Crossref]

Chen, J.

J. Chen, K. Yao, and R. Hudson, “Source localization and beamforming,” IEEE Signal Process. Mag. 19(2), 30–39 (2002).
[Crossref]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media, (Van Nostrand Reinhold, 1990).

Cho, Y. T.

Y. T. Cho, M. J. Roan, and J. S. Bolton, “A comparison of near-field beamforming and acoustical holography for sound source visualization,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 223, 819–834 (2009).
[Crossref]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, (Springer-VerlagDover, 1992).
[Crossref]

Craddock, I. J.

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

Cuccaro, A.

R. Solimene, I. Catapano, G. Gennarelli, A. Dell’Aversano, A. Cuccaro, and F. Soldovieri, “SAR imaging algorithms and some unconventional applications: a unified mathematical overview,” IEEE Signal Process. Mag. 31(4), 90–98 (2014).
[Crossref]

Dell’Aversano, A.

R. Solimene, I. Catapano, G. Gennarelli, A. Dell’Aversano, A. Cuccaro, and F. Soldovieri, “SAR imaging algorithms and some unconventional applications: a unified mathematical overview,” IEEE Signal Process. Mag. 31(4), 90–98 (2014).
[Crossref]

Drinkwater, B.

B. Drinkwater and P. Wilcox, “Ultrasonic arrays for non-destructive evaluation: a review,” NDT and E Int. 39(6), 525–541 (2006).
[Crossref]

Duan, Z.

Flores-Tapia, D.

D. Flores-Tapia and S. Pistorius, “Real time breast microwave radar image reconstruction using circular holography: a study of experimental feasibility,” Med Phys. 38, 5420–5431 (2011).
[Crossref] [PubMed]

D. Flores-Tapia and S. Pistorius, “Spatial sampling constraints on breast microwave radar scan acquired along circular geometries,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (2011) pp. 496–499.

Fortuny-Guasch, J.

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-D Radar imaging using range migration techniques,” IEEE Trans. Antennas Propag.,  48(5), 728–737 (2000).
[Crossref]

Gatteschi, L.

L. Gatteschi, “Sul comportamento asintotico delle funzioni di Bessel di prima specie di ordine ed argomento quasi uguali,” Ann. Mat. Pura Appl. 43(4), 97–117 (1957).
[Crossref]

Gazdag, J.

J. Gazdag and P. Sguazzero, “Migration of Seismic Data,” Proc. IEEE 72(10), 1302–1315 (1984).
[Crossref]

Gennarelli, G.

R. Solimene, I. Catapano, G. Gennarelli, A. Dell’Aversano, A. Cuccaro, and F. Soldovieri, “SAR imaging algorithms and some unconventional applications: a unified mathematical overview,” IEEE Signal Process. Mag. 31(4), 90–98 (2014).
[Crossref]

Glavin, M.

M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beam-forming algorithms,” Progress in Electromagnetics Research 97, 141–158 (2009).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw Hill, 1968).

Gross, M.

Hagness, S. C.

S. C. Hagness, A. Taove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna array sensors,” IEEE Trans. Biomed. Eng. 45(12), 1470–1479 (1998).
[Crossref] [PubMed]

Handelsman, R. A.

N. Blinstein and R. A. Handelsman, Asymtotic Expansions of Integrals, (Dover Publications, 1986).

Hare, J.

Hudson, R.

J. Chen, K. Yao, and R. Hudson, “Source localization and beamforming,” IEEE Signal Process. Mag. 19(2), 30–39 (2002).
[Crossref]

Huh, J. Y.

Y.-D. Joh, Y. M. Kwon, J. Y. Huh, and W.-K. Park, “Structure analysis of single-and multi-frequency subspace migrations in inverse scattering problems,” Prog. Electromag. Res. 136, 607–622 (2013).
[Crossref]

Janee, H. S.

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

Joh, Y.-D.

Y.-D. Joh, Y. M. Kwon, J. Y. Huh, and W.-K. Park, “Structure analysis of single-and multi-frequency subspace migrations in inverse scattering problems,” Prog. Electromag. Res. 136, 607–622 (2013).
[Crossref]

Jones, E.

M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beam-forming algorithms,” Progress in Electromagnetics Research 97, 141–158 (2009).
[Crossref]

Joud, F.

Julliard, E.

E. Julliard, S. Pauzin, F. Simon, and D. Biron, “Acoustic sources localization in presence of reverberation,” J. Acoust. Soc. Am. 118, 1886 (2005).
[Crossref]

Klemm, M.

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, (Springer-VerlagDover, 1992).
[Crossref]

Krim, H.

H. Krim and M. Viberg, “Two decades of array signal processing research: the parametric approach,” IEEE Signal Process. Mag. 13(4), 67–94 (1996).
[Crossref]

Kwon, Y. M.

Y.-D. Joh, Y. M. Kwon, J. Y. Huh, and W.-K. Park, “Structure analysis of single-and multi-frequency subspace migrations in inverse scattering problems,” Prog. Electromag. Res. 136, 607–622 (2013).
[Crossref]

Laloe, F.

Langenberg, K. J.

K. J. Langenberg, Applied Inverse Problems for Acoustic, Electromagnetic and Elastic Wave Scattering. Basic Methods for Tomography and Inverse Problems (Hilger, 1987).

Leendertz, J. A.

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

Leone, G.

Li, E.

H. Lim, N. Nhung, E. Li, and N. Thang, “Confocal microwave imaging for breast cancer detection: delay-multiply-and-sum image reconstruction algorithm,” IEEE Trans. Biomed. Eng. 55(6), 1697–1704 (2008).
[Crossref] [PubMed]

Lim, H.

H. Lim, N. Nhung, E. Li, and N. Thang, “Confocal microwave imaging for breast cancer detection: delay-multiply-and-sum image reconstruction algorithm,” IEEE Trans. Biomed. Eng. 55(6), 1697–1704 (2008).
[Crossref] [PubMed]

Lopez-Sanchez, J. M.

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-D Radar imaging using range migration techniques,” IEEE Trans. Antennas Propag.,  48(5), 728–737 (2000).
[Crossref]

Martin, P. J.

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

Miyamoto, Y.

Nhung, N.

H. Lim, N. Nhung, E. Li, and N. Thang, “Confocal microwave imaging for breast cancer detection: delay-multiply-and-sum image reconstruction algorithm,” IEEE Trans. Biomed. Eng. 55(6), 1697–1704 (2008).
[Crossref] [PubMed]

O’Halloran, M.

M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beam-forming algorithms,” Progress in Electromagnetics Research 97, 141–158 (2009).
[Crossref]

Oden, C. P.

C. P. Oden, H. M. Powers, D. L. Wright, and G. R. Olhoeft, “Improving GPR Image Resolution in Lossy Ground Using Dispersive Migration,” IEEE Trans. Geosc. Rem. Sens. 45, 2492–2500 (2007).
[Crossref]

Olhoeft, G. R.

C. P. Oden, H. M. Powers, D. L. Wright, and G. R. Olhoeft, “Improving GPR Image Resolution in Lossy Ground Using Dispersive Migration,” IEEE Trans. Geosc. Rem. Sens. 45, 2492–2500 (2007).
[Crossref]

Otto, G. P.

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

Palmer, D. A.

M. P. Andre, H. S. Janee, P. J. Martin, G. P. Otto, B. A. Spivey, and D. A. Palmer, “High-speed data acquisition in a diffraction tomography system employing large-scale toroidal arrays,” Int. J. Imaging Syst. Technol. 8(1), 137–147 (1997).
[Crossref]

Park, W.-K.

Y.-D. Joh, Y. M. Kwon, J. Y. Huh, and W.-K. Park, “Structure analysis of single-and multi-frequency subspace migrations in inverse scattering problems,” Prog. Electromag. Res. 136, 607–622 (2013).
[Crossref]

Pauzin, S.

E. Julliard, S. Pauzin, F. Simon, and D. Biron, “Acoustic sources localization in presence of reverberation,” J. Acoust. Soc. Am. 118, 1886 (2005).
[Crossref]

Pierri, R.

Pistorius, S.

D. Flores-Tapia and S. Pistorius, “Real time breast microwave radar image reconstruction using circular holography: a study of experimental feasibility,” Med Phys. 38, 5420–5431 (2011).
[Crossref] [PubMed]

D. Flores-Tapia and S. Pistorius, “Spatial sampling constraints on breast microwave radar scan acquired along circular geometries,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (2011) pp. 496–499.

Powers, H. M.

C. P. Oden, H. M. Powers, D. L. Wright, and G. R. Olhoeft, “Improving GPR Image Resolution in Lossy Ground Using Dispersive Migration,” IEEE Trans. Geosc. Rem. Sens. 45, 2492–2500 (2007).
[Crossref]

Preece, A.

M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Improved delay-and-sum beamforming algorithm for breast cancer detection,” Int. J. Ant. Propag. 2008, 761402 (2008).

Roan, M. J.

Y. T. Cho, M. J. Roan, and J. S. Bolton, “A comparison of near-field beamforming and acoustical holography for sound source visualization,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 223, 819–834 (2009).
[Crossref]

Sguazzero, P.

J. Gazdag and P. Sguazzero, “Migration of Seismic Data,” Proc. IEEE 72(10), 1302–1315 (1984).
[Crossref]

Simon, F.

E. Julliard, S. Pauzin, F. Simon, and D. Biron, “Acoustic sources localization in presence of reverberation,” J. Acoust. Soc. Am. 118, 1886 (2005).
[Crossref]

Soldovieri, F.

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Figures (6)

Fig. 1
Fig. 1 Pictorial view of the scattering scene. Invariance is assumed along the z-axis.
Fig. 2
Fig. 2 (a) Comparing normalized point spread functions returned by Eqs. (6) (denoted as BFNW) and (8) (denoted as psfBFNW). Cut view along the x axis is displayed for two scatterer’s locations, (0,0) (top panel) and (0,4λm) (middle panel) where λm is the wavelegth in a medium with v = c/3 and relative to the bottom frequency of the band [1,3] GHz. The bottom panel shows the difference between the two point spread functions. (b) Comparing point spread functions returned by Eqs. (13) (denoted as BFW) and (15) (denoted as pdfBFW). Cut view along the x axis is displayed for two scatterer’s locations, (0,0) (top panel) and (0,4λm) (bottom panel) where λm is the wavelegth in a medium with v = c/3 and relative to the bottom frequency of the band [1,3] GHz.
Fig. 3
Fig. 3 (a) Cut view along the x-axis of the reconstruction of a point objects obtained by the windowed beamforming procedure for the same parameters as in Fig. 2(b) for different frequency band width: [1,3] GHz (top panel); [1,5] GHz (middle panel); [1,10] GHz (bottom panel). (b) Cut view along the x-axis of the reconstruction of a point objects obtained by the windowed beamforming procedure for the same parameters as in Fig. 2(b) for different frequency allocation: [1,3] GHz, [3,5] GHz, [5,7] GHz and [7,9] GHz (from top to bottom).
Fig. 4
Fig. 4 (a) Level of the first side lobe normalized to the maximum of the PSF as the frequency band (top panel) and the central frequency (bottom panel) change. (b) Comparing the cut views along the x-axis of the reconstructions returned by Eqs. (13), (15) and (17) for [1,10] GHz (top panel) and Eqs. (13), (15) and (18) for [8,10] GHz (bottom panel).
Fig. 5
Fig. 5 (a) Cut view along the x-axis of the reconstruction of two scattering point objects obtained by the no-windowed beamforming procedure for the same parameters as in Fig. 2(a). The green and the dashed blue lines refer to the outcomes of Eqs. (19) and (20), respectively. Red lines instead reports Eq. (20) implemented by using the estimated point spread function retuned by Eq. (8). (b) Cut view along the x-axis of the reconstruction of two scattering point objects obtained by the windowed beamforming procedure for the same parameters as in Fig. 2(b). The green and the dashed blue lines refer to the outcomes of Eqs. (19) and (20), respectively. Red lines instead reports Eq. (20) implemented by using the estimated point spread function retuned by Eq. (15). (c) Cut view along the x-axis of the reconstruction of two scattering point objects for the same parameters as in Fig. 2(b). The green lines refer to holography given by eq. (28) and the blue lines to the windowed beamforming retuned by Eq. (13). For all three procedures the objects have been located at r ¯ 1 = ( 0 , 0 , 0 ) and r ¯ 2 = ( 0.3 , 0 , 0 ) λ m ( top panel ); r ¯ 1 = ( 0 , 0 , 0 ) λ m and r ¯ 2 = ( 0.18 , 0 , 0 ) λ m ( middle panel ); r ¯ 1 = ( 0 , 0 , 0 ) λ m and r ¯ 2 = ( 0.1 , 0 , 0 ) λ m ( bottom panel ).
Fig. 6
Fig. 6 In the left panel angular summation in Eq. (30) for different number of measurement points. (a) N = 45. (b) N = 22. (c) N = 17. (d) N = 8. According to theory, spurious artefacts start at | r ¯ r ¯ p | = N / 2 k. In the middle panel no-windowed beamforming for [1,3] GHz. (e) N = 22. (f) N = 17. (g) N = 8. In the right panel windowed beamforming for [1,3] GHz. (h) N = 22. (i) N = 17. (l) N = 8.

Equations (46)

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U S ( ω , θ ) = ( ω / 2 v ) 2 S ( ω ) D { H 0 ( 2 ) [ ω v R 2 + r 2 2 R r cos ( θ ϕ ) ] } 2 χ ( r ¯ ) d r ¯
U S ( ω , θ ) = ( j ω / 2 π v ) S ( ω ) D exp ( 2 j ω v R 2 + r 2 2 R r cos ( θ ϕ ) ) R 2 + r 2 2 R r cos ( θ ϕ ) χ ( r ¯ ) d r ¯
u S ( t , θ ) = D s [ t τ ( θ , r ¯ ) ] χ ( r ¯ ) d r ¯
I B F ( r ¯ ) = W ( t ) [ 0 2 π u s [ t T ( θ , r ¯ ) , θ ] d θ ] 2 d t
I B F N W ( r ¯ ) = Ω | 0 2 π U S ( ω , θ ) exp [ j ω τ ( θ , r ¯ ) ] d θ | 2 d ω
I B F N W ( r ¯ ) = Ω | ω / ( 2 π v ) S ( ω ) | 2 | D χ ( r ¯ ) 0 2 π exp { j ω [ τ ( θ , r ¯ ) τ ( θ , r ¯ ) ] } R 2 + r 2 2 R r cos ( θ ϕ ) d θ d r | 2 d ω
F N W : χ χ ˜ = I B F N W
P S F N W ( r ¯ , r ¯ p ) | v [ Φ ( 2 k max | r ¯ r ¯ p | ) Φ ( 2 k min | r ¯ r ¯ p | ) ] 2 π R 2 ( 2 | r ¯ r ¯ p | ) 3 |
Φ ( α ) = 0 α y 2 J 0 2 ( y ) d y = 1 / 8 × [ ( 2 y 3 + y ) J 0 2 ( y ) + 2 y 2 J 0 ( y ) J 1 ( y ) + 2 y 3 J 1 2 ( y ) | 0 α 0 α J 0 2 ( y ) d y ]
0 α J 0 2 ( y ) d y = y [ J 0 2 ( y ) + J 1 2 ( y ) ] | 0 α + 0 α J 1 2 ( y ) d y
I B F W ( r ¯ ) = | 0 2 π u S [ T W T ( θ , r ¯ ) , θ ] d θ | 2
I B F W ( r ¯ ) = | 0 2 π u S ( t , θ ) δ [ t τ ( θ , r ¯ ) ] d t d θ | 2
I B F W ( r ¯ ) = | Ω 0 2 π U S ( ω , θ ) exp [ j ω τ ( θ , r ¯ ) ] d θ d ω | 2
I B F W ( r ¯ ) = | D χ ( r ¯ ) Ω j ω / ( 2 π v ) × 0 2 π exp { 2 j ω / v [ | R ¯ r ¯ | | R ¯ r ¯ | ] } | R ¯ r ¯ | d θ d ω d r ¯ | 2
F W : χ χ ˜ = I B F W
P S F W ( r ¯ , r ¯ p ) { v [ Ψ ( 2 k max | r ¯ r ¯ p | ) Ψ ( 2 k min | r ¯ r ¯ p | ) ] R ( 2 | r ¯ r ¯ p | ) 2 } 2
Ψ ( α ) = 0 α y J 0 ( y ) d y = y J 1 ( y ) | 0 α
P S F W ( r ¯ , r ¯ p ) a p p = { v [ Ψ ( 2 k max | r ¯ r ¯ p | ) ] R ( 2 | r ¯ r ¯ p | ) 2 } 2
P S F W ( r ¯ , r ¯ p ) a p p = [ ( v / R ) k a v J 0 ( 2 k a v | r ¯ r ¯ p | ) ] 2
I B F i ( r ¯ ) = I B F i ( r ¯ , r ¯ 1 , χ 1 ) + I B F i ( r ¯ , r ¯ 2 , χ 2 ) + M B F i ( r ¯ , r ¯ 1 , r ¯ 2 , χ 1 , χ 2 )
I B F i ( r ¯ ) I B F i ( r ¯ , r ¯ 1 , χ 1 ) + I B F i ( r ¯ , r ¯ 2 , χ 2 )
U S ( ω , θ ) = D K ( ω , R , θ , r , ϕ ) χ ( r ¯ ) d r ¯
K ( ω , R , θ , r , ϕ ) = ( j ω / 2 π ν ) × exp ( 2 j ω ν R 2 + r 2 2 R r cos ( θ ϕ ) ) R 2 + r 2 2 R r cos ( θ ϕ )
U S ( ω , ε ) = D F T θ [ K ( ω , R , θ , r , ϕ ) ] χ ( r ¯ ) d r ¯
U ˜ S ( ω , ε ) = D exp [ j ψ ( ω , ε , R ) ] F T θ [ K ( ω , R , θ , r , ϕ ) ] χ ( r ¯ ) d r ¯
U ˜ S ( ω , θ ) = D exp [ 2 j k r cos ( θ ϕ ) ] χ ( r ¯ ) d r ¯
U ˜ S ( k x , k z ) = D exp [ j ( k x x + k z z ) ] χ ( x , z ) d x d z
o l : χ χ ˜
χ ˜ ( x , z ) = D Ω k x , k z exp { j [ k x ( x x ) + k z ( z z ) ] } d k x d k z χ ( x , z ) d x d z
P S F H o l ( r ¯ , r ¯ p ) = 2 π [ Ψ ( 2 k m a x | r ¯ r ¯ p | ) Ψ ( 2 k m i n | r ¯ r ¯ p | ) ] | r ¯ r ¯ p | 2
0 2 π H 0 ( 2 ) ( 2 k | R ¯ r ¯ | ) H 0 ( 1 ) ( 2 k | R ¯ r ¯ p | ) d θ
n = 0 N 1 H 0 ( 2 ) ( 2 k | R ¯ n r ¯ | ) H 0 ( 1 ) ( 2 k | R ¯ n r ¯ p | )
h , l H h ( 2 ) ( 2 k R ) H h l N ( 1 ) ( 2 k R ) J h ( 2 k r ) J h l N ( 2 k r p ) exp [ j h ( θ p θ ) ] exp [ j l N θ p ]
| H 0 ( 2 ) ( k R ) | 2 l j l N J l N ( 2 k | r ¯ r ¯ p | ) exp [ j l N a r g ( r ¯ r ¯ p ) ]
| H 0 2 ( 2 k R ) | 2 { J 0 ( 2 k | r ¯ r ¯ p | ) + l 0 j l N J l N ( 2 k | r ¯ r ¯ p | ) exp [ j l N a r g ( r ¯ r ¯ p ) ] }
N 4 k R b
N 4 k m a x R b
N 2 π 4 k m a x R R b / R 2 + R b 2
I B F N W ( r ¯ ) = ν | χ p | 2 4 K Ω k 4 | 0 2 π | R ¯ r ¯ | | R ¯ r ¯ p | × [ 2 j 2 k π × exp ( 2 j k | R ¯ r ¯ | ) | R ¯ r ¯ | ] × [ 2 j 2 k π × exp ( 2 j k | R ¯ r ¯ p | ) | R ¯ r ¯ p | ] d θ | 2 d k
I B F N W ( r ¯ ) = ν π | χ p | 2 2 K Ω k 4 × | n | H n ( 2 ) ( 2 k R ) | 2 J n ( 2 k r ) J n ( 2 k r p ) exp [ j n ( ϕ ϕ p ) ] | 2 d k
I B F N W ( r ¯ ) = ν π | χ p | 2 2 K Ω k 4 × | H 0 ( 2 ) ( 2 k R ) | 2 | L L J n ( 2 k r ) J n ( 2 k r p ) exp [ j n ( ϕ ϕ p ) ] | 2 d k
I B F N W ( r ¯ ) = ν | χ p | 2 2 π R 2 K Ω k 2 J 0 2 ( 2 k | r ¯ r ¯ p | ) d k
I B F W ( r ¯ ) = | ( j ν / 2 ) χ p K Ω k 2 0 2 π [ 2 j 2 k π exp ( 2 j k | R ¯ r ¯ | ) | R ¯ r ¯ | ] × [ 2 j 2 k π exp ( 2 j k | R ¯ r ¯ p | ) | R ¯ r ¯ p | ] d θ d k | 2
I B F W ( r ¯ ) = | ( ν / R ) χ p K Ω k J 0 ( 2 k | r ¯ r ¯ p | ) d k | 2
P S F ( r ¯ ) = Ω k x , k z exp ( j β ¯ Δ ¯ r ) d k x d k z
P S F ( r ¯ ) = 2 k m i n 2 k m a x 0 2 π exp [ j β Δ r cos ( ϕ β ϕ Δ ) ] β d β d ϕ β

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